# The universal rotation curve of low surface brightness galaxies - IV. The
interrelation between dark and luminous matter

Chiara Di Paolo,1,2,3 Paolo Salucci,1,2,3 Adnan Erkurt,4
1SISSA/ISAS, Via Bonomea 265, 34136 Trieste, Italy
2INFN Sez. Trieste, Via A. Valerio 2, 34127 Trieste, Italy
3IFPU, Via Beirut 2-4, 34151 Trieste, Italy
4Istanbul University, Beyazit, 34452 Fatih/Istanbul, Turkey
chiara.dipaolo<EMAIL_ADDRESS>
###### Abstract

We investigate the properties of the baryonic and the dark matter components
in low surface brightness (LSB) disc galaxies, with central surface brightness
in the B band $\mu_{0}\geq 23\,mag\,arcsec^{-2}$. The sample is composed by 72
objects, whose rotation curves show an orderly trend reflecting the idea of a
universal rotation curve (URC) similar to that found in the local high surface
brightness (HSB) spirals in previous works. This curve relies on the mass
modelling of the coadded rotation curves, involving the contribution from an
exponential stellar disc and a Burkert cored dark matter halo. We find that
the dark matter is dominant especially within the smallest and less luminous
LSB galaxies. Dark matter halos have a central surface density $\Sigma_{0}\sim
100\,M_{\odot}pc^{-2}$, similar to galaxies of different Hubble types and
luminosities. We find various scaling relations among the LSBs structural
properties which turn out to be similar but not identical to what has been
found in HSB spirals. In addition, the investigation of these objects calls
for the introduction of a new luminous parameter, the stellar compactness
$C_{*}$ (analogously to a recent work by Karukes & Salucci), alongside with
the optical radius and the optical velocity in order to reproduce the URC.
Furthermore, a mysterious entanglement between the properties of the luminous
and the dark matter emerges.

Key words : dark matter - galaxies: kinematics and dynamics, fundamental
parameters

††pagerange: The universal rotation curve of low surface brightness galaxies -
IV. The interrelation between dark and luminous matter–I

##  1 Introduction

Dark matter (DM) is a main actor in cosmology. It is believed to constitute
the great majority of the mass and to rule the processes of structure
formation in the Universe. 111In this paper we adopt the scenario of DM in
Newtonian gravity, leaving to other works the investigation in different
frameworks. The so-called $\Lambda$CDM ($\Lambda$ Cold Dark Matter) scenario,
in which one assumes a WIMP (Weakly Interactive Massive Particle) that
decouples from the primordial plasma when non-relativistic, successfully
reproduces the structure of the cosmos on large scales (Kolb & Turner, 1990).
However, some challenges to this scenario emerge at small galactic scales,
such as the ’missing satellite problem’ (e.g Klypin et al., 1999; Moore et
al., 1999; Zavala et al., 2009; Papastergis et al., 2011; Klypin et al., 2015)
and the ’too big to fail problem’ (e.g. Ferrero et al., 2012; Boylan-Kolchin
et al., 2012; Garrison-Kimmel et al., 2014; Papastergis et al., 2015).
Moreover, the galactic inner DM density profiles generally appear to be cored,
rather than cuspy as predicted in the $\Lambda$CDM scenario (e.g. Salucci,
2001; de Blok & Bosma, 2002; Gentile et al., 2004, 2005; Simon et al., 2005;
Del Popolo & Kroupa, 2009; Oh et al., 2011; Weinberg et al., 2015), in spirals
of any luminosity (see Salucci, 2019). In ellipticals and dSphs the question
is still uncertain ( Salucci, 2019).

Figure 1: LSBs rotation curves (each one in different color) ordered according
to increasing optical velocities $V_{opt}$. Note that the radial coordinate is
normalised with respect to the disc scale length $R_{d}$. A universal trend is
recognizable analogous to that emerged in normal spirals (see Fig. 20 in
Appendix A).

These issues suggest to study different scenarios from the ’simple’
$\Lambda$CDM, such as warm DM (e.g. de Vega et al., 2013; Lovell et al.,
2014), self-interacting DM (e.g Vogelsberger et al., 2014; Elbert et al.,
2015), or to introduce the effect of the baryonic matter feedbacks on the DM
distribution (e.g. Navarro et al., 1996; Read & Gilmore, 2005; Mashchenko et
al., 2006; Pontzen & Governato, 2014; Di Cintio et al., 2014).

One important way to investigate the properties of DM in galaxies is to study
rotational supported systems, such as spiral galaxies, since they have a
rather simple kinematics. The stars are mainly distributed in an exponential
thin disc with scale length $R_{d}$ (Freeman, 1970). Notice that related to
this scale length, in this paper, we will use the optical radius $R_{opt}$,
defined as the radius encompassing $83\%$ of the total luminosity and
proportional to the stellar disc scale length: $R_{opt}=3.2R_{d}$ (the details
of this choice are expressed at length in Persic et al., 1996). In order to
explain the observed rotation curves (RCs) of disc systems, it is necessary to
assume the presence of a spherical DM halo surrounding the galaxies ( Faber &
Gallagher, 1979; Rubin et al., 1985 and e.g. Salucci, 2019).

A very interesting feature of spiral galaxies is that the bigger they are, the
more luminous they are and the higher rotational velocities they show.
Moreover, when their RCs, with the radial coordinate normalised with respect
to their optical radius $R_{opt}$, are put together, they appear to follow a
universal trend (first shown in Fig. 4 in Rubin et al., 1985, then in Persic &
Salucci, 1991; Persic et al., 1996; Rhee, 1996; Roscoe, 1999; Catinella et
al., 2006; Noordermeer et al., 2007; Salucci et al., 2007; L pez Fune, 2018
and e.g. Salucci, 2019). From small to large galaxies, the RCs have higher and
higher velocities and profiles that gradually change. See also the top panel
in Fig. 20 in Appendix A.

By means of the "universal rotation curve (URC) method", which involves
groupings of similar RCs and their mass modelling, it is possible to construct
an analytic function that gives a good description of all the rotation curves
of the local spiral galaxies within a spherical volume $\simeq(100\,Mpc)^{3}$.
The URC method was applied for the first time in Persic & Salucci (1991). This
was followed by a series of three works: Persic et al. (1996) (Paper I),
Salucci et al. (2007) (Paper II) and Karukes & Salucci (2017) (Paper III) ,
where the URC method gave deeper results related to normal spirals, also
called high surface brightness (HSB) spirals, and dwarf disc (dd) galaxies. A
subsequent work confirmed the above results with up to 3100 disc galaxies and
highlighted the existence of tight scaling relations among the properties of
spirals with different size (Lapi et al., 2018).

Let us underline that the concept of universality in the RCs means that all of
them can be described by the same analytical function as long as expressed in
terms of the normalised radius and of one global parameter of the galaxies,
such as magnitude, luminosity, mass or velocity at the optical radius
($V_{opt}\equiv V(R_{opt})$). Therefore, the universal rotation curve (URC) is
the circular velocity at a certain radius $r$ given by (e.g.)
$V(r/R_{opt},L)$, where $L$ is the galaxy’s luminosity. See the bottom panel
in Fig. 20 in Appendix A. Obviously, the URC does not change even using,
instead of $R_{opt}$, any other radial coordinate proportional to the stellar
disc scale length $R_{d}$222The results of the paper remain unchanged for any
chosen radial coordinate if expressed in units of $\lambda R_{d}$, with any
$\lambda$ value ranging from 1 to 4..

The URC is a very powerful tool since, given the observation of few properties
(such as $R_{d}$ and $L$) of a certain galaxy, it is possible to deduce its
rotation curve and all its properties.

In this paper (IV), we investigate the concept of the URC, the resulting mass
models and the scaling relations in Low Surface Brightness (LSB) disc
galaxies, comparing them to the results of other disc galaxies of a different
Hubble type.

LSB galaxies are rotating disc systems which emit an amount of light per area
smaller than normal spirals. They are locally more isolated than other kinds
of galaxies (e.g. Bothun et al., 1993; Rosenbaum & Bomans, 2004) and likely
evolving very slowly with very low star formation rates. This is suggested by
colors, metallicities, gas fractions and extensive population synthesis
modelling (e.g. van der Hulst et al., 1993; McGaugh, 1994; de Block et al.,
1995; Bell et al., 2000). As we see in radio synthesis observations, LSB
galaxies have extended gas discs with low gas surface densities and high
$M_{HI}/L$ ratios (e.g. van der Hulst et al., 1993), where $M_{HI}$ is the
mass of the HI gaseous disc. The low metallicities make the gas cooling
difficult and in turn the stars difficult to form (e.g. McGaugh, 1994). LSBs
are required to be dominated by DM, as shown by the analysis of their Tully-
Fisher relation (e.g. Zwaan et al., 1995) and of their individual rotation
curves (e.g. de Blok et al., 2001; de Blok & Bosma, 2002).

The LSB sample used in this work involves 72 galaxies selected from
literature, whose optical velocities span from $\sim 24$ km/s to $\sim 300$
km/s, covering the values of the full population. Our analysis of LSBs by
means of the URC method is triggered by the result shown in Fig. 1, from which
we can see that the LSBs rotation curves gradually change very orderly from
small to large galaxies (or equally from objects with small to large optical
velocities $V_{opt}$). Following the URC method, the sample of galaxies is
divided in different velocity bins, according to their increasing values of
$V_{opt}$. A double normalisation of all the RCs is performed with respect to:
i) their own $R_{opt}$, along the radial axis, and ii) their own $V_{opt}$,
along the velocity axis. In these specific coordinates, in each velocity bin,
the RCs are all alike. Then, the double normalised coadded rotation curves, a
kind of average rotation curve for each velocity bin, are constructed. The
analysis continues with their mass modelling, yielding the distribution of
luminous and dark matter in structures with different $V_{opt}$. This is
followed by the denormalisation process, which gives the structural parameters
of each object of the sample, and allows us to obtain the related scaling
relations for the LSBs. The internal scatter of the found scaling
relationships is larger (three times or more) than the analogous ones in
normal spirals. A similar finding also emerged in the case of dwarf disc (dd)
galaxies (Karukes & Salucci, 2017). Remarkably, the scatter in the dd
relationships was reduced after the introduction of a new quantity, the
compactness of the luminous matter distribution $C_{*}$, that indicates how
the values of $R_{d}$ vary in galaxies with the same stellar disc mass.
Therefore, such results statistically suggest the introduction of the
compactness also in the analogous LSBs scaling relationships. The previous
steps lead to the construction of the URC for the LSBs, which is one of the
main goals of this work. Finally, in analogy to Karukes & Salucci (2017), we
also investigate the compactness of the dark matter distribution $C_{DM}$ and
its relation to $C_{*}$.

Figure 2: Optical velocity versus disc scale lengths in LSB galaxies (red) and
in normal spirals (blue) (Persic et al., 1996). The typical fractional
uncertainties are 5% in $V_{opt}$ and 15% in $R_{d}$, as shown in the right-
down corner.  Figure 3: LSBs rotation curves (each one in different color) in
physical units (first panel), in normalised radial units (second panel) and in
double normalised radial and velocity units (third panel). See also Appendix
C. Figure 4: LSBs rotation curves (in normalised radial units) grouped in five
optical velocity bins. In this and in the following figures, purple, blue,
green, orange and red colors are referred to the rotation curves of the I, II,
III, IV and V optical velocity bins, respectively.

The structure of this paper is the following: in Section 2, we describe our
sample of LSB galaxies; in Sections 3-4-5, the URC method and the analysis of
the LSBs structural properties are described in detail; in Section 6 we obtain
the LSBs scaling relations and we compare them to those of other disc systems;
in Sections 7-8, the concept of compactness is introduced and the URC-LSB is
built; finally, in Section 9, we comment on our main results.

The distances are evaluated from the recessional velocity assuming $H_{0}=72$
km s-1Mpc-1.

##  2 The LSB sample and the rotation curves universal trend

We consider 72 rotating disc galaxies classified as "low surface brightness"
in literature (see Tab. 3 in Appendix B). In the very majority of cases the
authors classify a galaxy as LSB when the face-on central surface brightness
$\mu_{0}\gtrsim 23\,mag\,arcsec^{-2}$ in the B band. We select our sample
according to the following criteria:
i) the rotation curves extend to at least $\simeq 0.8\,R_{opt}$ (when
$V_{opt}$ is not available from observation, it can be extrapolated since from
$\simeq 1/2\,R_{opt}$ to $2\,R_{opt}$, the RCs are linear in radius with a
small value of the slope);
ii) the RCs are symmetric, smooth (e.g. without strong signs of non circular
motions) and with an average fractional internal uncertainty lesser than
$20\%$. In short we eliminated RCs that in no way can be mass-modelled without
huge uncertainties;
iii) the galaxy disc scale length $R_{d}$ and the inclination function
$1/sin\,i$ are known within $30\%$ uncertainty.

The selected 72 LSBs have optical velocities $V_{opt}$ spanning from $\sim 24$
km/s to $\sim 300$ km/s; the sample of rotation curves consists of 1614
independent $(r,V)$ measurements. When the RCs, expressed in normalised radial
units, are put together, see Fig. 1, they show an universal trend analogous to
that of the the normal spirals (Fig. 20 in Appendix A). Then, given the
observed trend in LSBs and the relevance of the URC method, we search our
sample of LSBs for a universal rotation curve and for the related scaling
relations among the galaxy’s structural parameters.

In Fig. 2, the values of the stellar disc scale lengths $R_{d}$ and the
optical velocities $V_{opt}$ measured in LSBs are shown and compared to those
measured in normal spirals. A larger spread in the former case is clearly
recognizable. This feature will be used later to explain the need of
introducing a new structural variable, the compactness.

Finally, it is useful to stress that previous studies on individual LSB
galaxies reveal in the mass profiles of these objects the presence of an
exponential stellar disc, an extended gaseous disc at very low density (e.g.
de Blok et al., 1996) and the presence of a spherical DM halo, likely with a
core profile (e.g. de Blok et al., 2001; de Blok & Bosma, 2002; Kuzio de Naray
et al., 2008).

##  3 The coadded rotation curves of LSB galaxies

The individual rotation curves (in normalised radial units) shown in Fig. 1
motivate us to proceed, also in LSB, with the URC method, analogously to what
has been done on the high surface brightness spiral galaxies (Persic et al.,
1996; Lapi et al., 2018) and dwarf discs (Karukes & Salucci, 2017). It is
useful to anticipate here that the average scatter of the RCs data from a
fitting surface (as the URC in Fig. 15) is $\Delta V/\,V\simeq 8\%$ (taking
into account the observational errors, the systematics and the small non
circularities in the motion). This small value gives an idea of the
universality of the LSBs rotation curves expressed in normalised radial units.

Among the first steps, the URC method (Persic et al., 1996) requires to make
the galaxies RCs as similar as possible (in radial extension, amplitude and
profile) by introducing the normalisation of their coordinates and an eventual
galaxies binning. Let us notice that the justification for these starting
steps comes from the analogous process performed in spirals and from a
qualitative inspection of LSB RCs. Finally, the goodness of the results will
show the goodness of the method.

The characteristics of the RCs in physical and normalised units are visible in
Fig. 3:
i) in the first panel, the RCs are expressed in physical units, they appear to
be different in radial extension, amplitude and profile;
ii) in the second panel, the RCs are expressed in normalised radial units with
respect to their disc scale length $R_{d}$. Their radial extensions are made
more similar. Indeed, most of the data are extended up to $\simeq 5.5R_{d}$;
iii) in the third panel, the RCs are expressed in double normalised units with
respect to their disc scale length $R_{d}$ and optical velocity $V_{opt}$,
along the radial and the velocity axis respectively. The rotation curves in
such specific units are comparable also in their amplitude.

Overall, the double normalisation makes the 72 RCs more similar, apart from
their profiles. However, when these RCs are arranged in 5 optical velocity
bins according to their increasing $V_{opt}$ as in Fig. 4, we realise that the
double normalised RCs profiles belonging to one of these bins are very similar
among themselves but clearly different from those of the RCs in other optical
velocity bins (see Fig. 5).

We have chosen to build five $V_{opt}$ bins as a compromise between having a
large number of data for each coadded RC and a large number of coadded RCs.
Particularly, the binning in five groups is suggested by the fact that, since
the sample includes 72 objects, 10-15 galaxies are the minimum number in each
optical velocity bin in order to create suitable coadded RCs (that will be
described in the next paragraphs) and to eliminate statistically observational
errors and small non circularities from the individual RCs.

In detail, the number of galaxies in each bin, the span in $V_{opt}$, the
average optical velocity $\langle V_{opt}\rangle$, the average stellar disc
scale length $\langle R_{d}\rangle$, the number of galaxies and of the ($r$,
$V$) data are all reported in Tab. 1.

We also point out Fig. 22 in Appendix D, where the rotation curves, grouped in
their velocity bins, are compared in physical and double normalised units.

Figure 5: In each of the five panels: LSBs double normalised rotation curves
for each of the five optical velocity bins (grey points). Also shown the
corresponding coadded rotation curves (larger coloured points) for each of
these five bins. Notice that part of the scatter in the five profiles will be
eliminated by introducing the compactness in the URC. See Section 7.

After that all the RCs are double normalised, we perform the radial binning in
each of the five optical velocity bins. Similarly to the velocity binning
process, we have chosen $\simeq 11$ normalised radial bins as a compromise
between having a large number of data for each radial bin and a large number
of radial bins for each coadded RC. Moreover, we required that the inner
radial bins (for $r\leq 2R_{d}$) and the outer radial bins (for $r>2R_{d}$)
included a minimum of 13 and 5 measurements, respectively. In detail, for the
I, the II and the III optical velocity bins, the radial normalised coordinate
is divided in 12 bins; the first 5 have a width of 0.4 and the remaining a
width of 0.5. For the IV and the V velocity bins, for statistical reasons, we
adopt a different division of the radial coordinate. In the IV velocity bin we
adopt 3 radial bins of width 0.4, 5 of width 0.6 and the last one of width 1.
In the V velocity bin, we adopt 5, 4 and 2 radial bins of widths 0.4, 0.5 and
0.8, respectively. The number of data per radial bin is reported in Tab. 4 \-
5 in Appendix E. Reasonable variations of the positions and amplitudes of the
radial bins do not affect the resulting coadded RCs.

Therefore, for each of the five $V_{opt}$ bins , in every $k$-radial bin we
built there are $N_{k}$ double normalised velocities $v_{ik}$, with $i$
running from 1 to $N_{k}$. Their average value is given by:
$V_{k}=\frac{\sum_{i=1}^{N_{k}}v_{ik}}{N_{k}}$, as in Persic et al. (1996).
Then, by repeating this for all the radial bins of each of the five $V_{opt}$
bins, we obtain the five double normalised coadded RCs shown in Fig.5. The
standard error of the mean we consider in this work, is

$\displaystyle\delta
V_{k}=\sqrt{\frac{\sum_{i=1}^{N_{k}}(v_{ik}-V_{k})^{2}}{N_{k}(N_{k}-1)}}\quad.$
(1)

In short, the above coadded RCs can be considered as the average rotation
curves of galaxies of similar properties as, e.g., $V_{opt}$. It is worth
emphasizing the advantages of these RCs: their building erases the
peculiarities and much reduces the observational errors of the individual RCs.
This yields to a universal description of the kinematics of LSBs by means of 5
extended and smooth RCs whose values have an uncertainty at the level of
$5\%-15\%$. In Fig. 6 the five coadded RCs are shown together:

Figure 6: Coadded rotation curves for the five velocity bins in double normalised units (first panel), in physical velocity units (second panel) and in physical units along both the velocity and radial axes (third panel). The black empty triangles are the coadded RC for the dwarf disc galaxies (Karukes & Salucci, 2017). $V_{opt}$ bin |  $V_{opt}$ range |  N.galaxies |  $\langle V_{opt}\rangle$ |  $\langle R_{d}\rangle$ |  N.data
---|---|---|---|---|---
|  km/s |  |  km/s |  kpc |
(1) |  (2) |  (3) |  (4) |  (5) |  (6)
1 |  24-60 |  13 |  43.5 |  1.7 |  151
2 |  60-85 |  17 |  73.3 |  2.2 |  393
3 |  85-120 |  17 |  100.6 |  3.7 |  419
4 |  120-154 |  15 |  140.6 |  4.5 |  441
5 |  154-300 |  10 |  205.6 |  7.9 |  210
Table 1: LSB velocity bins. Columns: (1) i - velocity bin; (2) range values
for $V_{opt}$; (3) number of LSB galaxies in each velocity bin; (4) average
value of $V_{opt}$ evaluated from the individual galaxies; (5) average value
of $R_{d}$ evaluated from the individual galaxies; (6) number of ($r$, $V$)
data from the individual galaxies.

i) in the first panel, they are expressed in double normalised units covering
a very small region in the ($V/V_{opt},R/R_{opt}$) plane ;
ii) in the second panel, they are expressed in physical velocity units. These
coadded RCs are obtained by multiplying the previous coadded RCs by the
corresponding $\langle V_{opt}\rangle$ (reported in Tab. 1).
iii) in the third panel, the coadded RCs are expressed in physical units both
along the velocity and the radial axes. They are obtained by multiplying the
previous coadded RCs by the corresponding $\langle R_{d}\rangle$ reported in
Tab. 1.

In Fig. 6 the difference in the profiles corresponding to galaxies with
different optical velocities is evident. 333This is explained by the very
different luminous and dark mass distributions in LSBs of different sizes and
optical velocities, as shown in the next section.

All the data shown in Fig. 6 can be recast by means of Tab. 4 \- 5 (in
Appendix E) and Tab. 1.

##  4 The mass modelling of the coadded rotation curves

In this section we investigate the coadded rotation curves, normalised along
the radial axis (see second panel in Fig. 6), whose data are listed in Tab. 4
\- 5 in Appendix E. We model the coadded RCs data, as in normal spirals
(Salucci et al., 2007), with an analytic function $V(r)$ which includes the
contributions from the stellar disc $V_{d}$ and from the DM halo $V_{h}$:

Vel. Bin |  $\langle V_{opt}\rangle$ |  $\rho_{0}$ |  $R_{c}$ |  $M_{d}$ |  $M_{vir}$ |  $\alpha(R_{opt})$ |  $k$
---|---|---|---|---|---|---|---
|  $km/s$ |  $10^{-3}\,M_{\odot}/pc^{3}$ |  kpc |  $10^{11}\,M_{\odot}$ |  $10^{11}\,M_{\odot}$ |  |
(1) |  (2) |  (3) |  (4) |  (5) |  (6) |  (7) |  (8)
1 |  43.5 |  $3.7\pm 1.4$ |  $10.7\pm 4.3$ |  $(8.8\pm 1.8)\times 10^{-3}$ |  $1.0\pm 0.4$ |  0.37 |  0.36
2 |  73.3 |  $5.1\pm 1.1$ |  $12.8\pm 3.0$ |  $(3.8\pm 0.3)\times 10^{-2}$ |  $2.4\pm 0.9$ |  0.49 |  0.44
3 |  100.6 |  $3.7\pm 0.5$ |  $17.1\pm 1.9$ |  $(13.0\pm 0.5)\times 10^{-2}$ |  $4.0\pm 1.3$ |  0.52 |  0.47
4 |  140.6 |  $1.7^{+1.8}_{-1.1}$ |  $30^{+40}_{-22}$ |  $(4.2\pm 0.4)\times 10^{-1}$ |  $8.4\pm 3.5$ |  0.76 |  0.63
5 |  205.6 |  $0.8^{+0.7}_{-0.4}$ |  $99^{+213}_{-87}$ |  $1.7\pm 0.1$ |  $112\pm 55$ |  0.82 |  0.70
Table 2: Relevant parameters of the five coadded RCs. Columns: (1) i -
velocity bin; (2) average value of $V_{opt}$; best fit value of (3)
$\rho_{0}$; (4) $R_{c}$; (5)$M_{d}$; (6) estimated halo virial mass according
to Eq. 13; (7) fraction of baryonic component at $R_{opt}$ (Eq. 11); (8) $k$
values defined according to Eq. 10. Figure 7: In each of the five panels the
velocity best-fit models to the corresponding coadded RCs are shown. The
dashed, dot-dashed, dotted and solid lines indicate the stellar disc, the DM
halo, the stellar bulge and the model contribution to the circular velocities.

$\displaystyle V^{2}(r)=V_{d}^{2}(r)+V_{h}^{2}(r)\quad.$ (2)

Let us stress that in first approximation the inclusion in the model of a HI
gaseous disc component can be neglected. In fact, the gas contribution is
usually a minor component to the circular velocities, since the inner regions
of galaxies are dominated by the stellar component and in the external
regions, where the gas component overcomes the stellar one, the DM
contribution is largely the most important (Evoli et al., 2011). A direct test
in Appendix F shows that our assumption does not affect the mass modelling
obtained in this paper.

We describe the stellar and the dark matter component. The first one is given
by the well-known Freeman disc (Freeman, 1970), whose surface density profile
is

$\displaystyle\Sigma_{d}(r)=\frac{M_{d}}{2\pi R_{d}^{2}}\,exp(-r/R_{d})\quad,$
(3)

where $M_{d}$ is the disc mass. Eq. 3 leads to (Freeman, 1970):

$\displaystyle
V_{d}^{2}(r)=\frac{1}{2}\frac{G\,M_{d}}{R_{d}}\left(\frac{r}{R_{d}}\right)^{2}(I_{0}K_{0}-I_{1}K_{1})\quad,$
(4)

where $I_{n}$ and $K_{n}$ are the modified Bessel functions computed at
$1.6\,x$, with $x=r/R_{opt}$.

Finally, for the fifth optical velocity bin we will introduce a bulge
component. (Das, 2013).

Concerning the dark matter component, the presence of cored profiles in LSBs
is well known from individual rotation curves (see e.g. de Blok et al., 2001;
de Blok & Bosma, 2002; Kuzio de Naray et al., 2008, Bullock & Boylan-Kolchin,
2017 ). In this paper, we model the DM halo profile by means of the cored
Burkert profile (Burkert, 1995; Salucci & Burkert, 2000). This halo profile
has an excellent record in fitting the actual DM halos around disc systems of
any luminosity and Hubble Types (see Salucci, 2019; Lapi et al., 2018, Memola
et al., 2011, Salucci et al., 2012). In addition, the Burkert profile is in
agreement with weak lensing data at virial distances (Donato et al., 2009).

It is however worth to noticing that there is no sensible difference, in the
mass modelling inside $R_{opt}$, in adopting different cored DM density
profiles (Gentile et al., 2004). Then, we adopt the following density profile
(Burkert, 1995):

$\displaystyle\rho_{DM}(r)=\frac{\rho_{0}R_{c}^{3}}{(r+R_{c})(r^{2}+R_{c}^{2})}\quad,$
(5)

where $\rho_{0}$ is the central mass density and $R_{c}$ is the core radius.
Its mass distribution is:

$\displaystyle M_{DM}(r)$ $\displaystyle=$
$\displaystyle\int_{0}^{r}4\pi{\tilde{r}}^{2}\rho_{DM}({\tilde{r}})\;d{\tilde{r}}=$
$\displaystyle=$ $\displaystyle 2\pi\rho_{0}R_{c}^{3}\;[ln(1+r/R_{c})$
$\displaystyle-tg^{-1}(r/R_{c})+0.5\;ln(1+(r/R_{c})^{2})]\quad.$

The contribution to the total circular velocity is given by:

$\displaystyle V_{h}^{2}(r)=G\frac{M_{DM}(r)}{r}\quad.$ (7)

We fit the five coadded RCs by means of the URC model described above, which,
for each coadded RC, is characterized by three free parameters, $M_{d}$,
$\rho_{0}$ and $R_{c}$, all set to be larger than zero. Other limits for the
priors of the fitting arise from the amplitude and the profile of the coadded
RCs themselves. We require that: $10^{6}M_{\odot}\lesssim M_{d}\lesssim
10^{12}\,M_{\odot}$ from the galaxies luminosities, $R_{c}\lesssim 200\
\frac{R_{opt}}{30kpc}\,kpc$ to avoid solid body rotation curves in all objects
and $10^{-26}\lesssim\rho_{0}\lesssim 10^{-22}\,g/cm^{3}$ (the lower limit
guarantees that the dark component is able to fit the RC allied with the
luminous component, the upper limit is to make the dark matter contribution
important but not larger than the RCs amplitudes). Notice that these limits
well agree with the outcomes of the modelling of individual RCs as found in
literature.

The resulting best fit values for the three free parameters ($M_{d}$,
$\rho_{0}$, $R_{c}$) are reported in Tab. 2 and the best fit velocity models
are plotted alongside the coadded RCs in Fig. 7.

In the case of the V velocity bin, we introduce a central bulge (whose
presence is typical in the largest galaxies) (Das, 2013). We adopt for the
bulge velocity component the simple functional form:

$\displaystyle
V^{2}_{b}(r)=\alpha_{b}V^{2}_{in}\left(\frac{r}{r_{in}}\right)^{-1}\quad,$ (8)

where $\;V_{in}=127\,km/s\,$ and $\;r_{in}=0.2\,\langle R_{d}\rangle\simeq
1.6\,kpc$ are the values of the first velocity point of the V coadded RC.
Since $r_{in}$ is larger than the edge of the bulge, we consider the latter as
a point mass. $\alpha_{b}$ is a parameter which can vary from $0.2$ to $1$
(e.g. see Yegorova & Salucci, 2007). By fitting the V coadded RC we found:
$\alpha_{b}=0.8$; the other best fit parameters $M_{d}$, $\rho_{0}$, $R_{c}$
are reported in Tab. 2.

In Fig. 7 we realise that, in the inner regions of the LSB galaxies, the
stellar component (dashed line) is dominant; while, on the contrary, in the
external regions, the DM component (dot-dashed) is the dominant one. Moreover,
the transition radius444The transition radius is the radius where the DM
component, dot-dashed line, overcomes the luminous component, dashed line.
between the region dominated by the baryonic matter and the region dominated
by the dark matter increases with normalised radius when we move from galaxies
with the lowest $V_{opt}$ to galaxies with the highest $V_{opt}$. A similar
behaviour was also observed in normal spiral galaxies (Persic et al., 1996;
Lapi et al., 2018).

##  5 Denormalisation of the coadded rotation curves

The mass models found in the previous section provided us with the structural
parameters of the five coadded RCs. Now, we retrieve the properties from the
individual RCs by means of the denormalisation method. It relies on the facts
that, in each velocity bin, i) all the double normalised RCs are similar to
their coadded double normalised RC (see Fig. 5) and that ii) we have performed
extremely good fits of the coadded RCs (see Fig. 7). Thus, the relations
existing for the coadded RCs are assumed to approximately hold also for the
individual RCs that form each of the 5 coadded ones.

The first relation that we apply in the denormalisation process is shown in
Fig. 8; the stellar disc scale length and the DM core radius of the five
velocity models are strongly correlated. The best linear fit in logarithmic
scale is:

$\displaystyle Log\,R_{c}=0.60+1.42\,Log\,R_{d}\quad,$ (9)

The errors in the fitting parameter are shown in Tab. 8 in Appendix H. The
relation expressed by Eq. 9 means that, in each galaxy we can evaluate $R_{c}$
from its measured $R_{d}$. It is worth noting that a similar relation exists
also in normal spirals (Fig. 8).

Figure 8: Relationship between the DM halo core radius and the stellar disc
scale length (points) and its best fit (solid line) compared to that of the
normal spirals(dashed line) (e.g. Lapi et al., 2018). The black empty triangle
represents the relationship in dwarf disc galaxies (Karukes & Salucci, 2017).
Figure 9: Relationship between the stellar disc mass and the optical velocity.
The large points refer to the values of the five velocity bins, while the
small points refer to the values of each LSB galaxy. The solid and the dashed
lines are the best fit for LSBs and normal spirals (e.g. Lapi et al., 2018).
The black triangle represents the dwarf discs (Karukes & Salucci, 2017).

Figure 10: Left panel: the relationship between the central DM halo mass
density and its core radius. Right panel: surface density
$\Sigma_{0}=\rho_{0}R_{c}$ versus their optical velocities $V_{opt}$ (LSBs in
red points). Also shown the scaling relation obtained by Donato et al., 2009
(yellow shadowed area) and Burkert, 2015 (light blue shadowed area). The black
empty triangle represents the dwarf discs (Karukes & Salucci, 2017).

The second relation we use for the denormalisation assumes that for galaxies
belonging to each $V_{opt}$ bin:

$\displaystyle\frac{G\,M_{d}}{V_{opt}^{2}R_{opt}}=k\quad,$ (10)

where the $k$ values are reported in Tab. 2. $R_{opt}$ and $V_{opt}$ are
measured for all the galaxies, thus Eq. 10 allows us to evaluate the stellar
disc mass $M_{d}$ for each of them.

As the third step in the denormalisation process we evaluate at $R_{opt}$, for
each of the five coadded RCs, the fraction of the baryonic matter:

$\displaystyle\alpha(R_{opt})=\frac{V_{d}^{2}(R_{opt})}{V^{2}(R_{opt})}\quad.$
(11)

The $\alpha(R_{opt})$ values are reported in Tab. 2; we assume that all the
galaxies included in each optical velocity bin take the same value for
$\alpha(R_{opt})$. Then, for each galaxy, we write the DM mass inside the
optical radius as:

$\displaystyle
M_{DM}(R_{opt})=[1-\alpha(R_{opt})]V_{opt}^{2}R_{opt}G^{-1}\quad.$ (12)

Finally, by considering Eq. 4-11-12 together with the result from the first
denormalisation step, we evaluate the central density of the DM halo
$\rho_{0}$ for each galaxy.

The structural parameters of the dark and luminous matter of the galaxies of
our sample, inferred by the denormalisation procedure, are reported in Tab.
6-7 in Appendix G. Moreover, we have the basis to infer other relevant
quantities of the galaxies structure that will be involved, in the next
section, in building the scaling relations. The virial mass $M_{vir}$, that
practically encloses the whole mass of a galaxy, is evaluated according to:

$\displaystyle
M_{vir}=\frac{4}{3}\,\pi\,100\,\,\rho_{crit}\,R_{vir}^{3}\quad,$ (13)

where $R_{vir}$ is the virial radius and $\rho_{crit}=9.3\times
10^{-30}g/cm^{3}$ is the critical density of the Universe. The DM central
surface density $\Sigma_{0}$ is evaluated by the product $\rho_{0}$ and
$R_{c}$. The $M_{vir}$ and $\Sigma_{0}$ values for the objects in our sample
are shown in Tab. 6-7 in Appendix G.

##  6 The scaling relations

In this section, we work out the scaling relations among the structural
properties of dark and luminous matter in each LSB galaxy. Let us stress that
for many of the scaling relations we have no a priori insight of how they
should be; in this case, the goal is to find a statistically relevant
relationship. Then we fit the observational data with the simple power-law
model. The errors on the fitting parameters of the various scaling relations
and their standard scatters are reported in Tab. 8 in Appendix H. Hereafter,
the masses are expressed in $M_{\odot}$, the radial scale length in $kpc$, the
velocities in $km/s$ and the mass densities in $g/cm^{3}$.

We start with the relation between the stellar disc mass and the optical
velocity. Fig. 9 shows that the LSB data are well fitted by:

$\displaystyle Log\,M_{d}=3.12+3.47\,Log\,V_{opt}\quad.$ (14)

This relation holding for the LSBs is similar but not identical to the normal
spirals’ one. See the comparison with Lapi et al. (2018) in Fig. 9.

Next, in Fig. 10 (left panel) we show the relation between the DM halo central
density and the core radius, which indicates that the highest mass densities
are in the smallest galaxies, as also found in normal spirals (Salucci et al.,
2007). We find:

$\displaystyle Log\,\rho_{0}=-23.15-1.16\,Log\,R_{c}\quad.$ (15)

Moreover, we find that the central surface density follows the relationship
(see Fig. 10 (right panel)):

$\displaystyle Log\,\Sigma_{0}=Log\,(\rho_{0}R_{c})\simeq 1.9\quad,$ (16)

$\Sigma_{0}$ is expressed in units of $M_{\odot}/pc^{2}$.

Remarkably, this relationship extends itself over 18 blue magnitudes and in
objects spanning from dwarf to giant galaxies (Spano et al., 2008; Gentile et
al., 2009; Donato et al., 2009; Plana et al., 2010; Salucci et al., 2012; Li
et al., 2019; Chan, 2019).

Figure 11: Fraction of baryonic matter in LSBs versus their mass in stars
(points) compared with that of normal spirals (dashed line) (Lapi et al.,
2018), of other Hubble Types (black solid line) (Moster et al., 2010) and of
dwarf discs (black dot-dashed line) (Karukes & Salucci, 2017). Figure 12: LSBs
relationships between a) the stellar disc scale length, b) the DM core radius
and c) the central DM core density versus the optical velocity (first, second
and third panel).

Then, we consider the baryonic fraction (complementary to the DM fraction)
relative to the entire galaxies, namely, the ratio between the stellar mass
$M_{*}\equiv M_{d}$ in LSBs and the virial mass $M_{vir}$, that practically
represents the whole dark mass of a galaxy. Fig. 11 shows that the lowest
fraction of baryonic content is in the smallest galaxies (with the smallest
stellar disc mass $M_{d}$). We note that this ratio increases going towards
larger galaxies and then reaches a plateau from which it decreases for the
largest galaxies. This finding is in agreement with the inverse "U-shape" of
previous works relative to normal spirals (Lapi et al., 2018). Furthermore,
our result seems to follow a trend similar to that found in Moster et al.
(2010), concerning all Hubble Types 555In Moster et al. (2010), the stellar
mass $M_{*}$ can indicate the mass enclosed in a disc and/or in a bulge).. The
result points to a less efficient star formation in the smallest LSBs.

Finally, we work out the relationships needed to establish
$V_{URC}(R;R_{opt},V_{opt})$, the URC-LSB in physical units (as in Persic et
al., 1996). Straightforwardly, we are looking for the universal function
$V_{URC}(r/R_{opt},V_{opt})$666Hereafter, we express the normalised radial
coordinate in terms of the optical radius $R_{opt}$, instead of $R_{d}$, in
order to facilitate the comparison with previous works on the URC., able to
reproduce analytically the LSBs RCs in Fig. 1.

This implies that $M_{d}$, $R_{d}$, $R_{c}$ and $\rho_{0}$ have to be
expressed as a function of $V_{opt}$. Thus, we use Eq. 14 and the following
relations, obtained after the denormalisation process:

$\displaystyle Log\,R_{d}=-1.65+1.07\,Log\,V_{opt}$ (17) $\displaystyle
Log\,R_{c}=-1.75+1.51\,Log\,V_{opt}$ $\displaystyle
Log\,\rho_{0}=-22.30-1.16\,Log\,V_{opt}$

See Fig. 12. We note that the above relations (Eq. 14-17) show a large
scatter, on average $\sigma\simeq 0.34$ dex, more than three times the value
($\sigma\simeq 0.1$ dex in Lapi et al., 2018, Yegorova & Salucci, 2007) found
in normal spiral galaxies for the respective relations. This poses an issue to
the standard procedure (Persic et al., 1996) to build the URC in physical
units.

In the previous sections we have found a universal function to reproduce the
double normalised rotation curve of LSBs $V(r/R_{opt})/V(R_{opt})$. Now we are
looking for a universal function to reproduce the RC in physical units $V(r)$.
In spiral galaxies this is simple since $M_{d}$, $R_{d}$, $R_{c}$ and
$\rho_{0}$ are closely connected.

##  7 The compactness as the third parameter in the URC

We can reduce the scatter in the LSBs scaling relations and proceed with the
URC building by introducing a new parameter, the compactness of the stellar
mass distribution $C_{*}$. This parameter was first put forward in Karukes &
Salucci (2017) to cope with a similar large scatter in the above scaling
relations of the dd galaxies. In short the large scatter in the previous
relationships is due to the fact that galaxies with the same stellar disc mass
$M_{d}$ (or $V_{opt}$) can have a very different size for $R_{d}$ (i.e.
$Log\,R_{d}$ can vary almost 1 dex). We define this effect with the fact that
LSBs have a different "stellar compactness" $C_{*}$; see Fig. 2 and Fig. 13.

We define $C_{*}$, starting from the best fit linear relation (see Fig. 13):

$\displaystyle Log\,R_{d}=-3.19+0.36\,Log\,M_{d}\quad$ (18)

and, according to Karukes & Salucci (2017), we set the stellar compactness
through the following relation:

$\displaystyle C_{*}=\frac{10^{(-3.19+0.36\,Log\,M_{d})}}{R_{d}}\quad,$ (19)

where, let us remind, $R_{d}$ is measured from photometry. By means of Eq. 19,
$C_{*}$ measures, for a galaxy with a fixed $M_{d}$, the deviation between the
observed $R_{d}$ and the "expected" $R_{d}$ value from Eq. 18 (obtained by
using the best fit line in Fig. 13). In short, at fixed $M_{d}$, galaxies with
the smallest $R_{d}$ have a high compactness ($Log\,C_{*}>0$), while galaxies
with the largest $R_{d}$ have low compactness ($Log\,C_{*}<0$).

The $Log\,C_{*}$ values for the galaxies of our sample are shown in Tab. 6-7
in Appendix G and span from -0.45 to 0.35.

Figure 13: Relationship between the stellar disc scale length and the stellar
disc mass. Figure 14: In the panels we show the relationships between a) the
stellar disc mass, b) the stellar disc scale length, c) the DM core radius and
d) the central DM core density versus the optical velocity and the compactness
of the stellar distribution. The errors in the distance of galaxies, which
propagates on $R_{c}$, $M_{d}$, $R_{d}$, $\rho_{0}$, are negligible in the
above for 3D relationships.

By introducing the compactness we reduce the scatter in the relations needed
to establish the analytical function of the URC-LSB in physical units. This is
highlighted in Fig. 14, where the data are shown alongside with their best fit
plane.

$\displaystyle Log\,M_{d}$ $\displaystyle=$ $\displaystyle
2.52+3.77\,Log\,V_{opt}-1.49\,Log\,C_{*}$ (20) $\displaystyle Log\,R_{d}$
$\displaystyle=$ $\displaystyle-2.27+1.38\,Log\,V_{opt}-1.55\,Log\,C_{*}$
$\displaystyle Log\,R_{c}$ $\displaystyle=$
$\displaystyle-2.62+1.96\,Log\,V_{opt}-2.20\,Log\,C_{*}$ $\displaystyle
Log\,\rho_{0}$ $\displaystyle=$
$\displaystyle-20.95-1.84\,Log\,V_{opt}+3.38\,Log\,C_{*}\quad.$

We find that, by using Eq. 20, the internal scatter of data with respect to
the planes is always reduced compared to the case in which $M_{d}$, $R_{d}$,
$R_{c}$ and $\rho_{0}$ were expressed only in terms of $V_{opt}$. The previous
average scatter $\sigma\simeq 0.34$ dex of the 2D relations (Eq. 14-17), in
the 3D relations (Eq. 20), is reduced to $\sigma\simeq 0.06$ dex smaller than
the typical values obtained for normal spirals.

We now evaluate the analytic expression for the universal rotation curve
(expressed in physical units). By using Eq. 2 alongside with with Eqs. 4, 4, 7
and expressing $M_{d}$, $R_{d}$, $R_{c}$ and $\rho_{0}$ as in Eq. 20, we
obtain:

$\displaystyle V^{2}(x,\,V_{opt},C_{*})=2.2\,x^{2}\times
10^{f_{1}(V_{opt},C_{*})}$ (21)
$\displaystyle\times\,[I_{0}K_{0}-I_{1}K_{1}]+1.25/x\times
10^{f_{2}(V_{opt},C_{*})}$ $\displaystyle\times\\{-tg^{-1}[3.2\,x\times
10^{f_{3}(V_{opt},C_{*}))}]$ $\displaystyle+\,ln[1+3.2\,x\times
10^{f_{3}(V_{opt},C_{*})}]$ $\displaystyle+0.5\,ln[1+10.24\,x^{2}\times
10^{2\,f_{3}(V_{opt},C_{*})}]\\}\quad,$

where $I_{n},K_{n}$ are the modified Bessel functions evaluated at $1.6\,x$,
with $x=r/R_{opt}$ and

$\displaystyle f_{1}(V_{opt},C_{*})$ $\displaystyle=$ $\displaystyle
9.79+2.39Log\,V_{opt}+0.05Log\,C_{*}$ (22) $\displaystyle
f_{2}(V_{opt},C_{*})$ $\displaystyle=$
$\displaystyle-0.55+2.65Log\,V_{opt}-1.67Log\,C_{*}$ $\displaystyle
f_{3}(V_{opt},C_{*})$ $\displaystyle=$ $\displaystyle
0.35-0.58Log\,V_{opt}+0.65Log\,C_{*}\quad.$

Finally, we plot in Fig. 15 the URC (Eq. 21-22) considering $Log\,C_{*}=0$,
corresponding to the case in which all the LSBs data in Fig. 13 were lying on
the regression line (or, analogously, the case in which the spread of LSBs
data in Fig. 2 was small). The curve shown in Fig. 15 is in good agreement
with the LSBs rotation curves data. On average, the uncertainty between the
velocity data and the URC velocity predicted values is $\Delta V/\,V\simeq
19\%$, which can be reduced to $\Delta V/\,V\simeq 8\%$, when the
observational errors, the systematics, the small non circularities and the
prominent bulge component777The bulge component is taken into account in the
coadded RCs modelling, but not in the final URC, going beyond the scope of the
paper. (as in ESO534-G020) are taken into account in the individual RCs. This
result, approximately equal to that found in normal spirals (Persic et al.,
1996), highlights the success of the URC method also in LSBs galaxies. The
smallness of the uncertainty achieved in the URC-LSB (physical) is evident is
Appendix I, where the individual RCs are tested. As a gauge we point out that
F583-4, NGC4395, UGC5005, F568V1, ESO444-G074 have a value of $\Delta V/\,V$
$\simeq 8\%$. Moreover, in Appendix I, the individual RCs are tested by
assuming i) $Log\,C_{*}=0$ and ii) their values of $C_{*}$ (reported in
Tab.6-7 in Appendix G).

Figure 15: LSBs universal rotation curve (URC), with compactness
$Log\,C_{*}=0$, and the individual 72 LSBs rotation curves.

Finally, in Fig. 16 we show the URC obtained with three significant different
values of stellar compactness. The central yellow surface has
$Log\,C_{*}=0.00$ (standard case) and the other two surfaces have
$Log\,C_{*}=-0.45$ (the minimum value achieved in the LSB sample) and
$Log\,C_{*}=+0.35$ (the maximum one). The three surfaces appear similar,
however when we normalise them with respect to $V_{opt}$ along the velocity
axis, their profiles appear different. See Fig. 17. Nevertheless, the
differences between the URC with $Log\,C_{*}=0.00$ and the URC with the
appropriate values of $C_{*}$ for each individual object lie within the URC
errorbars for most of the objects (see Appendix I).

Figure 16: Universal rotation curves (URC) in physical velocity units for
three different values of stellar compactness; low ($Log\,C_{*}=-0.45$),
standard ($Log\,C_{*}=0.00$) and high ($Log\,C_{*}=+0.35$) stellar
compactness, respectively in blue, yellow and red colors. The figure in the
second panel corresponds to that of the first panel when rotated by
$180^{\circ}$ around the velocity axis. Figure 17: Universal rotation curves
(URC) in double normalised units for three values of the stellar compactness:
$Log\,C_{*}=-0.45\,,\,0.00\,,\,+0.35$, respectively in blue, yellow and red
colors. Notice that the figure in the second panel corresponds to that of the
first panel rotated of $180^{\circ}$ around the velocity axis.

###  7.1 The relevance of $C_{*}$ in LSB galaxies

Completing our analysis, we have discovered the relevance of $C_{*}$ in the
LSB galaxies. By resuming, this work shows that:
i) the compacteness is linked to the spread in the $V_{opt}$ \- $R_{d}$ plot
(Fig 2). Galaxies at fixed $V_{opt}$ can have smaller $R_{d}$ (higher $C_{*}$)
or larger $R_{d}$ (lower $C_{*}$) than the average. The range of $Log\,R_{d}$
at fixed $V_{opt}$ can reach almost 1 dex;
ii) the profiles of the various RCs can be affected by the compactness (see
e.g. Fig. 17). Thus, the spread in the profiles of the RCs in each velocity
bin, (see Fig. 4-5) is not only due to the large width of the optical velocity
bins888Given the limited number of available RCs, each optical velocity bin
includes galaxies with a certain range in $V_{opt}$, causing the corresponding
RCs to have (moderately) different profiles, analogously to normal spirals.,
but it is also due to the different values of the galaxies compactness.
iii) the compactness is a main source for the large scatter ($\sigma\simeq
0.34$) in the 2D scaling relations (see Fig. 9\- 12-14).

Taking all this into account, we point out that in the URC-LSB building
procedure, having an improved statistic, the optimal approach would be
considering from the start to bin the available RCs in $C_{\star}$ (obtained
by the spread of data in the $V_{opt}$ \- $R_{d}$ plot in Fig. 2)
contemporaneously to $V_{opt}$. Moreover, with a sufficiently higher
statistics, we can also increase the number of the velocity bins and
characterise each of them with a smaller $V_{opt}$ range to reach the
performance of Persic et al. (1996).

Finally, we stress that in the LSBs there is no one-to-one correspondence
among the optical velocity, the optical radius, the luminosity, the virial
mass and other galaxies quantities. Then, if we order the RCs normalised in
radial units, according to quantities different from the optical velocity (as
in Fig.1), they would not lie on a unique surface but, according to the spread
of the stellar compactness among the objects, will give rise to a spread of RC
data lying on different surfaces.

##  8 The correlation between the compactnesses of the stellar and the DM mass
distributions

Following Karukes & Salucci (2017), we evaluate also the compactness of the DM
halo $C_{DM}$, i.e. we investigate the case in which the galaxies with the
same virial (dark) mass $M_{vir}$ exhibit different core radius $R_{c}$.

Figure 18: Relationship between the DM halo core radius and the virial mass.

The $M_{vir}$ vs $R_{c}$ relationship is shown in Fig.18 alongside with the
best fit linear relation, described by

$\displaystyle Log\,R_{c}=-5.32+0.56\,Log\,M_{vir}\quad.$ (23)

Then, according to Karukes & Salucci (2017), we define the compactness
$C_{DM}$ of the DM halo as:

$\displaystyle C_{DM}=\frac{10^{(-5.32+0.56\,Log\,M_{vir})}}{R_{c}}\quad.$
(24)

Thus, at fixed $M_{vir}$, galaxies with smaller $R_{c}$ have higher
compactness ($Log\,C_{DM}>0$), while galaxies with larger $R_{c}$ have lower
compactness ($Log\,C_{DM}<0$).

The values obtained for $Log\,C_{DM}$ are reported in Tab. 6-7 in Appendix G
and span from -0.57 to 0.30.

Then, we plot the compactness of the stellar disc versus the compactness of
the DM halo in Fig. 19. We note that $C_{*}$ and $C_{DM}$ are strictly
related: galaxies with high $C_{*}$, also have high $C_{DM}$. The logarithmic
data are well fitted by the linear relation:

$\displaystyle Log\;C_{*}=0.00+0.90\,Log\,C_{DM}\quad.$ (25)

The results are in very good agreement with those obtained for dd galaxies
(Karukes & Salucci, 2017), whose best-fitting relation is given by:
$Log\;C_{*}=0.77\,Log\,C_{DM}+0.03$. In the figure we realize that the average
difference between the two relationship is just about 0.1 dex.

This result is remarkable because the same relation is found for two very
different types of galaxies (LSBs and dds). The strong relationship between
the two compactnesses certainly indicates that the stellar and the DM
distributions follow each other very closely. In a speculative way, given the
very different distribution of luminous matter in an exponential thin disc and
the distribution of DM in a spherical cored halo, such strong correlation in
Eq. 25 might point to a non-standard interaction between the baryonic and the
dark matter; a velocity dependent self-interaction in the dark sector; or a
fine tuned baryonic feedback (e.g. Di Cintio et al., 2014; Chan et al., 2015).

Figure 19: Relationship between the compactness of the stellar disc and the
compactness of the DM halo (red points). The black triangles refer to the
dwarf discs of Karukes & Salucci, 2017. The solid and the dot-dashed lines are
the best fit relations for LSBs and dwarf discs.

##  9 Conclusions

We analyzed a sample of 72 low surface brightness (LSB) galaxies selected from
literature, whose optical velocities $V_{opt}$ span from $\sim 24$ km/s to
$\sim 300$ km/s. Their rotation curves, normalised in the radial coordinate
with respect to the stellar disc scale length $R_{d}$ (or the optical radius
$R_{opt}$) and ordered according to the increasing optical velocity $V_{opt}$,
follow a universal trend (Fig. 1), analogously to the normal (high surface
brightness) spirals. This led us to build the universal rotation curve (URC)
of LSBs as in Persic et al. (1996), i.e. to find an analytic expression to
reproduce any circular velocity by means of only few observable parameters
(e.g. $R_{opt}$ and $V_{opt}$).

The building of the URC allows us to obtain the properties of the stellar and
DM distribution and to evaluate the scaling relations valid for the whole
population of objects. The analysis on the LSBs RCs leads us to a scenario
which is very similar qualitatively, but not quantitatively, to that of the
normal spirals. In detail, in both cases, we observe that the main
contribution to the circular velocity, in the innermost galactic region, is
given by the stellar disc component, while, in the external region is given by
a cored DM spherical halo. Moreover, the fraction of DM that contributes to
the RCs is more relevant as lower $V_{opt}$ is, i.e. in smaller galaxies (Fig.
7).

The scaling relations among the galactic properties seem to follow a similar
trend in LSB galaxies and high surface brightness (HSB) spirals (Fig. 8-9-10).

On the other side, there is a clear difference: we realise the presence of a
large scatter in the LSBs relationships with respect to that found in normal
spirals (see Lapi et al., 2018). Such difference can be traced back to the
large spread of the $V_{opt}$-$R_{d}$ data (see Fig. 2) or, analogously, to
the large spread of the $R_{d}$-$M_{d}$ relationship in Fig. 13. This finding
leads us to introduce the concept of compactness of the luminous matter
distribution $C_{*}$, involved for the first time in Karukes & Salucci (2017)
to cope with a similar issue in the case of dwarf disc (dd) galaxies.

We have that in galaxies with a fixed value for $M_{d}$, the smaller is
$R_{d}$, the higher is $C_{*}$. By considering $C_{*}$ in the scaling
relations, the scatter is much reduced (it becomes smaller than that of the
normal spirals). By involving this new parameter, we proceed with the building
of the analytic universal expression to describe all the LSBs rotation curves
(in physical units, km/s vs kpc). The resulting URC,
$V(r;R_{opt},\,V_{opt},\,C_{*})$ in Eq. 21-22, well describes all the rotation
curves of our sample (Fig. 15-24-25-26-27-28). The average scatter of the RCs
data from the fitting surface in Fig. 15 achieves the small value of $\Delta
V/\,V\simeq 0.08$, taking into account the observational errors, the
systematics and the small non-circularities in the motion. This result remarks
the success of the method leading to the URC and of the relevance of $C_{*}$
in the RCs profiles (Fig. 17) and in the scaling relations, which has been
discovered in building the URC.

With larger statistics, one should subdivide the RCs according to the galaxies
$C_{*}$ and $V_{opt}$.

An important finding concerns the compactness of the DM distribution $C_{DM}$,
indicating galaxies with the same virial mass and different core radius (Fig.
18). We find a strong correlation between $C_{*}$ and $C_{DM}$ as also found
in Karukes & Salucci (2017) (Fig 19): the distributions of stellar disc and of
its enveloping dark matter halo are entangled. In a speculative way, this
finding appears to be of very important relevance for the nature of DM. In
fact, the strong correlation between $C_{*}$ and $C_{DM}$ may hint to the
existence of non standard interactions between the luminous matter and the
dark matter, or non trivial self-interaction in the DM sector or a (hugely)
fine-tuned baryonic feedback on the collisionless DM distribution.

Finally, the LSBs URC provides us with the best observational data to test
specific density profiles (e.g. NFW, WDM, Fuzzy DM) or alternatives to dark
matter (e.g. MOND). The normal spirals’ URC, in connection with the normal
spirals’ $R_{opt}$ vs $V_{opt}$ relationship, is a function of $V_{opt}$:
$V_{URC\,(ns)}(r/R_{opt},V_{opt})$ (ns stands for normal spirals). Therefore,
to represent all the normal spirals’ individual RCs it is sufficient to
evaluate $V_{URC\,(ns)}(r/R_{opt},V_{opt})$ for a reasonable number $j$ of
$V_{opt}$ values, homogeneously spread in the spirals $V_{opt}$ range. Any
mass model under test must reproduce the $V_{opt}$ dependent URC. Instead, the
LSBs URC, in connection with the LSBs $R_{opt}$ vs $V_{opt}$ and $C_{*}$
relationship, is a function of two galaxy structural properties: $V_{opt}$ and
$C_{*}$. In this case, to represent all the LSBs RCs we have to build
$V_{URC\,(LSB)}(r/R_{opt},V_{opt},C_{*})$. We need a large sample of RCs of
galaxies of different $V_{opt}$ and $C_{*}$ yielding a reasonable number of
RCs in each of the more numerous ($V_{opt};\,C_{*}$) bins we have to employ.
The galaxies model under test must reproduce a much complex (observational
driven) URC than that of normal spirals which depends on just the structural
parameter $V_{opt}$.

## acknowledgments

We thank E. Karukes and A. Lapi for useful discussions. We thank Brigitte
Greinoecker for improving the text. We thank the referee for her/his several
inputs that have improved the paper.

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##  Appendix A Universality in normal spirals

Fig. 20, from Persic et al. (1996) and Catinella et al. (2006), allows us to
appreciate the universality of the rotation curves in normal spirals after the
radial normalisation. Let us point out the trend of the RCs from small to
large galaxies.

Figure 20: Top panel: coadded rotation curves from 3100 normal spirals,
obtained by plotting together the results by Persic et al., 1996 and Catinella
et al., 2006 (originally in the slides by Salucci, 2010). Also indicated the
absolute I-magnitudes. Bottom panel: universal rotation curve (URC) (Persic et
al., 1996).

##  Appendix B LSB galaxies sample and references

In Tab. 3, we report the list of the LSB galaxies of this work with their
related references.

Galaxy |  Reference |  Galaxy |  Reference
---|---|---|---
NGC 100 |  de Blok & Bosma (2002) |  UGC 11557 |  Swaters et al. (2003)
NGC 247 |  Carignan & Puche (1990) |  UGC 11583 |  de Blok et al. (2001)
NGC 959 |  Kuzio de Naray et al. (2008) |  UGC 11616 |  de Blok et al. (2001)
NGC 2552 |  Kuzio de Naray et al. (2008) |  UGC 11648 |  de Blok et al. (2001)
NGC 2552 |  de Blok & Bosma (2002) |  UGC 11748 |  de Blok et al. (2001)
NGC 2552 |  Swaters et al. (2003) |  UGC 11819 |  de Blok et al. (2001)
NGC 2552 |  van den Bosch & Swaters (2001) |  ESO 186-G055 |  Pizzella et al. (2008)
NGC 3274 |  de Blok & Bosma (2002) |  ESO 206-G014 |  Pizzella et al. (2008)
NGC 3274 |  Swaters et al. (2003) |  ESO 215-G039 |  Palunas & Williams (2000)
NGC 3347B |  Palunas & Williams (2000) |  ESO 234-G013 |  Pizzella et al. (2008)
NGC 4395 |  de Blok & Bosma (2002) |  ESO 268-G044 |  Palunas & Williams (2000)
NGC 4395 |  van den Bosch & Swaters (2001) |  ESO 322-G019 |  Palunas & Williams (2000)
NGC 4455 |  de Blok & Bosma (2002) |  ESO 323-G042 |  Palunas & Williams (2000)
NGC 4455 |  Marchesini et al. (2002) |  ESO 323-G073 |  Palunas & Williams (2000)
NGC 4455 |  van den Bosch & Swaters (2001) |  ESO 374-G003 |  Palunas & Williams (2000)
NGC 5023 |  de Blok & Bosma (2002) |  ESO 382-G006 |  Palunas & Williams (2000)
NGC 5204 |  Swaters et al. (2003) |  ESO 400-G037 |  Pizzella et al. (2008)
NGC 5204 |  van den Bosch & Swaters (2001) |  ESO 444-G021 |  Palunas & Williams (2000)
NGC 7589 |  Pickering et al. (1997) |  ESO 444-G047 |  Palunas & Williams (2000)
UGC 628 |  de Blok & Bosma (2002) |  ESO 488-G049 |  Pizzella et al. (2008)
UGC 634 |  van Zee et al. (1997) |  ESO 509-G091 |  Palunas & Williams (2000)
UGC 731 |  de Blok & Bosma (2002) |  ESO 534-G020 |  Pizzella et al. (2008)
UGC 731 |  Swaters et al. (2003) |  F561-1 |  de Blok et al. (1996)
UGC 731 |  van den Bosch & Swaters (2001) |  F563-V1 |  de Blok et al. (1996)
UGC 1230 |  de Blok & Bosma (2002) |  F563-V2 |  Kuzio de Naray et al. (2006)
UGC 1230 |  van der Hulst et al. (1993) |  F563-V2 |  de Blok et al. (1996)
UGC 1281 |  Kuzio de Naray et al. (2006) |  F565-V2 |  de Blok et al. (1996)
UGC 1281 |  de Blok & Bosma (2002) |  F568-1 |  Swaters et al. (2000)
UGC 1551 |  Kuzio de Naray et al. (2008) |  F568-3 |  Kuzio de Naray et al. (2006)
UGC 2684 |  van Zee et al. (1997) |  F568-3 |  de Blok et al. (2001)
UGC 2936 |  Pickering et al. (1999) |  F568-3 |  Swaters et al. (2000)
UGC 3137 |  de Blok & Bosma (2002) |  F568-6 |  Pickering et al. (1997)
UGC 3174 |  van Zee et al. (1997) |  F568-V1 |  Swaters et al. (2000)
UGC 3371 |  de Blok & Bosma (2002) |  F571-8 |  Marchesini et al. (2002)
UGC 3371 |  van den Bosch & Swaters (2001) |  F571-8 |  de Blok et al. (2001)
UGC 4115 |  de Blok et al. (2001) |  F571-V1 |  de Blok et al. (1996)
UGC 4278 |  de Blok & Bosma (2002) |  F574-1 |  Swaters et al. (2000)
UGC 5005 |  de Blok & McGaugh (1997) |  F574-2 |  de Blok et al. (1996)
UGC 5272 |  Kuzio de Naray et al. (2008) |  F579-V1 |  de Blok et al. (2001)
UGC 5272 |  de Blok & Bosma (2002) |  F583-1 |  Kuzio de Naray et al. (2008)
UGC 5716 |  van Zee et al. (1997) |  F583-1 |  Marchesini et al. (2002)
UGC 5750 |  Kuzio de Naray et al. (2006) |  F583-1 |  de Blok et al. (2001)
UGC 5750 |  de Blok & Bosma (2002) |  F583-1 |  de Blok et al. (1996)
UGC 5999 |  van der Hulst et al. (1993) |  F583-4 |  Kuzio de Naray et al. (2006)
UGC 7178 |  van Zee et al. (1997) |  F583-4 |  de Blok et al. (2001)
UGC 8837 |  de Blok & Bosma (2002) |  F730-V1 |  de Blok et al. (2001)
UGC 9211 |  van den Bosch & Swaters (2001) |  PGC 37759 |  Morelli et al. (2012)
UGC 11454 |  de Blok et al. (2001) |  |
Table 3: LSB sample: galaxy names and references of their RCs and photometric
data. Note that some galaxies have multiple rotation curve data.

##  Appendix C Rotation curves in physical units

In Fig. 21, the 72 LSB RCs are shown in physical units. Here, all the data are
included, while, the first panel of Fig. 3 includes only data with $r\leq 30$
kpc.

Figure 21: The 72 LSB RCs in physical units (all data).

##  Appendix D Rotation curves in velocity bins

In Fig. 22 we show the LSBs rotation curves separately in the five velocity
bins, both in physical units and in double normalised units (i.e. along the
radial and the velocity axes).

Figure 22: LSBs rotation curves belonging to each of the five optical velocity
bins.

##  Appendix E Construction of the coadded rotation curves

In Tab. 4-5 we show the data related to the five coadded rotation curves
obtained in Section 3. In detail, the first column describes the center of the
radial bins represented by the coloured points in Fig. 5-6-7. The second
column indicates the number of RCs data (grey points in Fig. 5) belonging to
each radial bin. Finally, the third and the fourth columns show the velocity
data and their error-bars in physical units, related to the coadded RCs in the
second panel of Fig. 6 and in Fig. 7.

$r/R_{D}$ |  N. data |  V |  ErrorBar
---|---|---|---
|  |  km/s |  km/s
(1) |  (2) |  (3) |  (4)
I velocity bin |  |  |
0.2 |  15 |  7.3 |  1.2
0.6 |  19 |  16.7 |  1.7
1.0 |  18 |  25.6 |  2.4
1.4 |  19 |  34.2 |  3.1
1.8 |  13 |  37.1 |  1.6
2.25 |  13 |  42.4 |  1.4
2.75 |  10 |  41.9 |  2.1
3.25 |  13 |  45.5 |  0.9
3.75 |  5 |  48.4 |  0.7
4.25 |  5 |  51.1 |  1.4
4.75 |  5 |  49.9 |  1.1
5.25 |  5 |  56.4 |  4.2
II velocity bin |  |  |
0.2 |  62 |  25.0 |  1.5
0.6 |  70 |  40.0 |  1.3
1.0 |  39 |  52.0 |  1.9
1.4 |  26 |  56.5 |  1.9
1.8 |  26 |  62.3 |  1.2
2.25 |  23 |  64.8 |  1.3
2.75 |  23 |  70.7 |  0.8
3.25 |  16 |  74.3 |  0.5
3.75 |  15 |  76.3 |  1.2
4.25 |  12 |  78.6 |  1.4
4.75 |  12 |  81.0 |  1.6
5.25 |  9 |  81.7 |  2.1
III velocity bin |  |  |
0.2 |  86 |  25.3 |  1.8
0.6 |  56 |  53.7 |  1.9
1.0 |  46 |  71.8 |  2.7
1.4 |  45 |  81.1 |  2.8
1.8 |  35 |  89.9 |  3.2
2.25 |  39 |  93.6 |  1.3
2.75 |  29 |  97.2 |  1.7
3.25 |  20 |  101.3 |  0.5
3.75 |  10 |  104.0 |  0.8
4.25 |  8 |  106.9 |  1.0
4.75 |  10 |  107.8 |  1.4
5.25 |  6 |  107.9 |  2.0
Table 4: Rotation curves for each optical velocity bin of LSB galaxies. Columns: (1) center of the radial bin; (2) number of data in each bin; (3) coadded velocity for each bin; (4) velocity error. In order to express the radial coordinate in physical units, the data of the first column relative to each velocity bin must be multiplied by the respective average value of disc scale length $\langle R_{D}\rangle$, reported in Tab. 1. $r/R_{D}$ |  N. data |  V |  ErrorBar
---|---|---|---
|  |  km/s |  km/s
(1) |  (2) |  (3) |  (4)
IV velocity bin |  |  |
0.2 |  141 |  47.9 |  2.2
0.6 |  81 |  90.4 |  2.0
1.0 |  54 |  112.2 |  2.6
1.5 |  58 |  121.8 |  2.2
2.1 |  41 |  128.6 |  3.1
2.7 |  28 |  133.7 |  2.9
3.3 |  17 |  136.0 |  2.5
3.9 |  9 |  138.9 |  3.0
4.7 |  8 |  129.5 |  2.8
V velocity bin |  |  |
0.2 |  71 |  127.1 |  7.2
0.6 |  32 |  148.7 |  6.1
1. |  23 |  173.9 |  3.5
1.4 |  14 |  197.6 |  3.7
1.8 |  16 |  194.8 |  4.9
2.25 |  14 |  198.2 |  3.4
2.75 |  5 |  199.3 |  5.2
3.25 |  9 |  205.5 |  1.5
3.75 |  6 |  203.2 |  4.0
4.4 |  8 |  199.6 |  5.3
5.2 |  5 |  195.2 |  6.9
Table 5: It continues from Tab. 4.

##  Appendix F The gas component in the Rotation Curves

The gas disc component in galaxies is an additional component to the stellar
disc and the DM halo giving a contribution to the circular velocities. At any
rate, by performing a suitable test, it is possible to realize that the gas is
(moderately) important only in the first optical velocity bin, where, in any
case, in the inner regions the stellar component overcomes the gaseous one,
while in the external region the DM component overcomes the gaseous one; thus,
the gas component gives a modest contribution to the RC. In Fig. 23, for the
first velocity bin co-added rotation curve, we compare the mass-velocity model
fit that includes the contribution from a HI disc with the velocity-mass model
which does not. The estimated masses of the stellar disk and of the DM halo
show, in the two cases, only a moderate change .

By modelling the co-added rotation curve of the first $V_{opt}$ bin by means
of the stellar/HI disc + DM halo model we get:

$M_{d}=8.0\times 10^{8}M_{\odot}$ ;

$r_{0}=10.7\,kpc$ ;

$\rho_{0}=3.2\times 10^{-3}M_{\odot}/pc^{3}$ ;

$M_{vir}=8.2\times 10^{10}M_{\odot}$ ;

$M_{HI}=1.0\times 10^{9}M_{\odot}$.

By removing the gaseous disc, we get:

$M_{d}=8.8\times 10^{8}M_{\odot}$ ;

$r_{0}=10.7\,kpc$ ;

$\rho_{0}=3.7\times 10^{-3}M_{\odot}/pc^{3}$ ;

$M_{vir}=1.0\times 10^{11}M_{\odot}$ .

We remind that $M_{d},\,r_{0},\,\rho_{0},\,M_{HI}$ (all quantities inferred by
the fit) are the stellar disc mass, the DM halo core radius, the central core
mass density, the HI gaseous disc mass (including the correction for the
helium contribution), respectively. $M_{vir}$ is the virial mass. The
differences in the values of $M_{d},\,r_{0},\,\rho_{0},M_{vir}$, when we
include gas or we exclude the gaseous component, are inside the error bars
reported in Tab. 2 related to the fit without the HI disk.

Figure 23: I velocity bin rotation curve best-fitted with gas (left panel) and
without gas (right panel). The dashed, dot-dashed, dotted and solid lines
stand for the stellar disc, the DM halo, the gaseous disc and the total
contributions to the rotation curve.

##  Appendix G Structural properties of LSB galaxies

In Tab. 6-7 we report: the names of the LSB galaxies in our sample alongside
their distances $D$, the stellar disc scale lengths $R_{d}$ and the optical
velocities $V_{opt}$ (all taken from literature). Furthermore, the table shows
the values of the stellar disc mass $M_{d}$, the DM core radius $R_{c}$, the
central density of the DM halo $\rho_{0}$, the virial mass $M_{vir}$, the
central surface density $\Sigma_{0}=\rho_{0}\,R_{c}$, the compactness of the
stellar mass distribution $C_{*}$ and that of the DM mass distribution
$C_{DM}$, all evaluated in this work.

Name |  $D$ |  $R_{d}$ |  $V_{opt}$ |  $M_{d}$ |  $R_{c}$ |  $Log\,\rho_{0}$ |  $M_{vir}$ |  $Log\Sigma_{0}$ |  $LogC_{*}$ |  $LogC_{DM}$
---|---|---|---|---|---|---|---|---|---|---
|  Mpc |  kpc |  km/s |  $10^{7}\,M_{\odot}$ |  kpc |  $g/cm^{3}$ |  $10^{9}\,M_{\odot}$ |  $M_{\odot}/pc^{2}$ |  |
(1) |  (2) |  (3) |  (4) |  (5) |  (6) |  (7) |  (8) |  (9) |  (10) |  (11)
UGC4115 |  7.8 |  0.4 |  24.2 |  6.3 |  1.1 |  -23.57 |  1.6 |  1.63 |  0.06 |  -0.15
F563V1 |  51.0 |  2.4 |  27.3 |  48 |  14 |  -25.30 |  27 |  1.01 |  -0.40 |  -0.57
UGC11583 |  5.9 |  0.3 |  27.9 |  6.5 |  0.7 |  -23.17 |  1.6 |  1.88 |  0.17 |  -0.00
UGC2684 |  8.2 |  0.8 |  36.7 |  29 |  2.9 |  -23.95 |  12 |  1.69 |  -0.00 |  -0.10
F574-2 |  66.0 |  4.5 |  40.0 |  192 |  33 |  -25.57 |  171 |  1.13 |  -0.45 |  -0.50
F565V2 |  36.0 |  2.0 |  45.2 |  110 |  11 |  -24.69 |  76 |  1.51 |  -0.19 |  -0.21
UGC5272 |  6.1 |  1.2 |  48.8 |  77 |  5.2 |  -24.11 |  42 |  1.77 |  -0.02 |  -0.04
UGC8837 |  5.1 |  1.2 |  49.6 |  79 |  5.2 |  -24.10 |  44 |  1.78 |  -0.02 |  -0.03
F561-1 |  63.0 |  3.6 |  50.8 |  250 |  25 |  -25.15 |  244 |  1.41 |  -0.31 |  -0.28
UGC3174 |  11.8 |  1.0 |  51.7 |  72 |  4.0 |  -23.88 |  36 |  1.89 |  0.04 |  0.04
NGC4455 |  6.8 |  0.9 |  53.0 |  68 |  3.4 |  -23.75 |  33 |  1.96 |  0.08 |  0.08
UGC1281 |  5.5 |  1.7 |  55.0 |  138 |  8.5 |  -24.36 |  96 |  1.74 |  -0.08 |  -0.05
UGC1551 |  20.2 |  2.5 |  55.8 |  211 |  15 |  -24.73 |  182 |  1.61 |  -0.18 |  -0.14
UGC9211 |  12.6 |  1.3 |  61.9 |  165 |  5.9 |  -24.10 |  66 |  1.84 |  0.06 |  0.01
F583-1 |  1.6 |  1.6 |  62.0 |  201 |  7.8 |  -24.29 |  90 |  1.77 |  0.00 |  -0.03
UGC5716 |  24.1 |  2.0 |  66.4 |  288 |  11 |  -24.45 |  150 |  1.75 |  -0.03 |  -0.04
UGC7178 |  24.0 |  2.3 |  69.9 |  367 |  13 |  -24.54 |  210 |  1.74 |  -0.06 |  -0.04
ESO400-G037 |  37.5 |  4.1 |  69.9 |  651 |  29 |  -25.09 |  502 |  1.55 |  -0.21 |  -0.18
NGC3274 |  0.47 |  0.5 |  68.0 |  75 |  1.5 |  -23.01 |  18 |  2.33 |  0.35 |  0.30
F583-4 |  49.0 |  2.7 |  70.5 |  438 |  16 |  -24.69 |  275 |  1.69 |  -0.10 |  -0.08
F571V1 |  79.0 |  3.2 |  72.4 |  549 |  21 |  -24.83 |  382 |  1.66 |  -0.14 |  -0.10
NGC5204 |  4.9 |  0.7 |  73.1 |  115 |  2.2 |  -23.24 |  33 |  2.27 |  0.30 |  0.27
UGC731 |  8.0 |  1.7 |  73.3 |  298 |  8.5 |  -24.20 |  147 |  1.90 |  0.04 |  0.05
NGC959 |  7.8 |  0.9 |  75.3 |  172 |  3.6 |  -23.57 |  60 |  2.15 |  0.21 |  0.21
NGC100 |  11.2 |  1.2 |  77.2 |  233 |  5.2 |  -23.81 |  96 |  2.07 |  0.15 |  0.16
NGC5023 |  4.8 |  0.8 |  78.4 |  160 |  2.9 |  -23.38 |  52 |  2.25 |  0.27 |  0.27
UGC5750 |  56.0 |  5.6 |  80.0 |  1171 |  46 |  -25.27 |  1125 |  1.56 |  -0.26 |  -0.18
UGC3371 |  12.8 |  3.1 |  82.0 |  681 |  20 |  -24.69 |  494 |  1.78 |  -0.09 |  -0.02
NGC4395 |  3.5 |  2.3 |  82.3 |  509 |  13 |  -24.40 |  312 |  1.89 |  -0.00 |  0.05
UGC11557 |  23.8 |  3.1 |  83.7 |  710 |  20 |  -24.67 |  520 |  1.80 |  -0.08 |  -0.01
UGC1230 |  51.0 |  4.5 |  90.0 |  1278 |  34 |  -24.99 |  1027 |  1.71 |  -0.15 |  -0.07
ESO206-G014 |  60.3 |  5.2 |  91.3 |  1531 |  42 |  -25.12 |  1338 |  1.67 |  -0.19 |  -0.10
NGC2552 |  10.1 |  1.6 |  92.0 |  475 |  7.8 |  -23.97 |  213 |  2.09 |  0.14 |  0.18
UGC4278 |  10.5 |  2.3 |  92.6 |  691 |  13 |  -24.32 |  386 |  1.96 |  0.04 |  0.10
UGC634 |  35.0 |  3.1 |  95.1 |  984 |  20 |  -24.59 |  662 |  1.88 |  -0.03 |  0.05
ESO488-G049 |  23.0 |  4.4 |  95.3 |  1410 |  33 |  -24.92 |  1159 |  1.76 |  -0.13 |  -0.03
UGC5005 |  52.0 |  4.4 |  95.5 |  1406 |  33 |  -24.92 |  1153 |  1.77 |  -0.13 |  -0.03
UGC3137 |  18.4 |  2.0 |  97.7 |  669 |  11 |  -24.14 |  350 |  2.06 |  0.10 |  0.17
F574-1 |  96.0 |  4.5 |  99.0 |  1546 |  34 |  -24.91 |  1306 |  1.79 |  -0.12 |  -0.01
F568-3 |  77.0 |  4.0 |  100.5 |  1416 |  29 |  -24.78 |  1130 |  1.84 |  -0.08 |  0.02
ESO322-G019 |  45.2 |  2.5 |  100.7 |  878 |  14 |  -24.32 |  528 |  2.01 |  0.05 |  0.14
F563V2 |  61.0 |  2.1 |  101.3 |  755 |  11 |  -24.15 |  412 |  2.07 |  0.10 |  0.18
NGC 247 |  2.5 |  2.9 |  106.6 |  1156 |  18 |  -24.42 |  784 |  2.00 |  0.02 |  0.13
ESO444-G021 |  60.7 |  6.4 |  107.4 |  2603 |  56 |  -25.17 |  2760 |  1.75 |  -0.19 |  -0.05
F579V1 |  85.0 |  5.1 |  111.5 |  2223 |  40 |  -24.92 |  2134 |  1.85 |  -0.12 |  0.03
F568V1 |  80.0 |  3.2 |  115.8 |  1505 |  21 |  -24.44 |  1119 |  2.04 |  0.02 |  0.16
ESO374-G003 |  43.2 |  4.2 |  118.3 |  2084 |  31 |  -24.70 |  1856 |  1.97 |  -0.05 |  0.11
F568-1 |  85.0 |  5.3 |  130.0 |  4218 |  43 |  -25.13 |  1354 |  1.67 |  -0.03 |  -0.10
UGC628 |  65.0 |  4.7 |  130.0 |  3740 |  36 |  -25.02 |  1132 |  1.71 |  0.00 |  -0.07
UGC11616 |  72.8 |  4.9 |  133.2 |  4094 |  38 |  -25.04 |  1282 |  1.71 |  -0.00 |  -0.07
ESO186-G055 |  60.1 |  3.6 |  133.2 |  3041 |  25 |  -24.76 |  813 |  1.81 |  0.08 |  0.00
ESO323-G042 |  59.7 |  4.4 |  138.7 |  4020 |  33 |  -24.91 |  1221 |  1.78 |  0.04 |  -0.02
PGC37759 |  193.2 |  6.8 |  139.4 |  6195 |  60 |  -25.30 |  2318 |  1.65 |  -0.08 |  -0.12
ESO234-G013 |  60.9 |  3.7 |  139.4 |  3425 |  26 |  -24.74 |  949 |  1.84 |  0.08 |  0.02
F571-8 |  48.0 |  5.2 |  139.5 |  4765 |  42 |  -25.05 |  1577 |  1.73 |  -0.00 |  -0.05
F730V1 |  144.0 |  5.8 |  141.6 |  5523 |  49 |  -25.15 |  1953 |  1.71 |  -0.03 |  -0.07
UGC11648 |  46.7 |  3.8 |  142.2 |  3620 |  27 |  -24.74 |  1022 |  1.85 |  0.09 |  0.03
ESO215-G039 |  61.3 |  4.2 |  142.9 |  4037 |  31 |  -24.83 |  1208 |  1.82 |  0.06 |  0.01
ESO509-G091 |  72.8 |  3.7 |  146.8 |  3735 |  25 |  -24.68 |  1050 |  1.89 |  0.11 |  0.06
Table 6: Individual properties of LSBs. Columns: (1) galaxy name; (2) distance; (3) disc scale length; (4) optical velocity; (5) disc mass; (6) core radius; (7) central DM density; (8) virial mass; (9) central surface density; (10) compactness of stellar mass distribution; (11) compactness of the DM mass distribution. Name |  $D$ |  $R_{d}$ |  $V_{opt}$ |  $M_{d}$ |  $R_{c}$ |  $Log\,\rho_{0}$ |  $M_{vir}$ |  $Log\Sigma_{0}$ |  $LogC_{*}$ |  $LogC_{DM}$
---|---|---|---|---|---|---|---|---|---|---
|  Mpc |  kpc |  km/s |  $10^{7}\,M_{\odot}$ |  kpc |  $g/cm^{3}$ |  $10^{9}\,M_{\odot}$ |  $M_{\odot}/pc^{2}$ |  |
(1) |  (2) |  (3) |  (4) |  (5) |  (6) |  (7) |  (8) |  (9) |  (10) |  (11)
ESO444-G047 |  62.4 |  2.7 |  148.4 |  2809 |  16 |  -24.38 |  662 |  2.01 |  0.19 |  0.13
UGC11454 |  92.1 |  4.5 |  150.3 |  4787 |  34 |  -24.85 |  1525 |  1.85 |  0.06 |  0.03
UGC5999 |  45.0 |  4.4 |  153.0 |  4851 |  33 |  -24.82 |  1540 |  1.87 |  0.07 |  0.04
UGC11819 |  59.2 |  5.3 |  154.6 |  6578 |  43 |  -25.10 |  1490 |  1.70 |  0.04 |  -0.08
ESO382-G006 |  65.4 |  2.3 |  160.0 |  3097 |  13 |  -24.29 |  449 |  2.01 |  0.27 |  0.13
ESO323-G073 |  69.6 |  2.1 |  165.3 |  2923 |  11 |  -24.14 |  398 |  2.08 |  0.32 |  0.18
NGC3347B |  46.2 |  8.1 |  167.0 |  11760 |  78 |  -25.43 |  3369 |  1.63 |  -0.05 |  -0.14
ESO268-G044 |  49.9 |  1.9 |  175.6 |  3057 |  10 |  -24.01 |  406 |  2.16 |  0.36 |  0.23
ESO534-G020 |  226.7 |  16.7 |  216.6 |  40638 |  218 |  -25.86 |  17351 |  1.64 |  -0.17 |  -0.18
NGC7589 |  115.0 |  12.6 |  224.0 |  32831 |  146 |  -25.58 |  13657 |  1.75 |  -0.08 |  -0.07
UGC11748 |  73.1 |  3.1 |  240.7 |  9418 |  20 |  -24.22 |  1911 |  2.26 |  0.32 |  0.31
UGC2936 |  43.6 |  8.4 |  255.0 |  28363 |  82 |  -25.09 |  10784 |  1.99 |  0.07 |  0.12
F568-6 |  201.0 |  18.3 |  297.0 |  83839 |  249 |  -25.67 |  49173 |  1.89 |  -0.10 |  0.01
Table 7: It continues from Tab. 6.

##  Appendix H Errors and scatter in the scaling relations

In Tab. 8, the errors on the best fitting parameters of the scaling relations
evaluated in this work are shown. The standard scatter $\sigma$ of individual
galaxies data of the various scaling relations is also reported. In the 2D
scaling relations, it is evaluated according to:
$\sigma=\sqrt{\sum_{i=1}^{N}(y_{i}-f(x_{i}))^{2}/N}$, where $N=72$, $y_{i}$
and $x_{i}$ are the logarithmic data in the $y$ and $x$ axes, respectively,
and $f$ is the considered scaling function (a line). In the 3D scaling
relations, the standard scatter is evaluated according to:
$\sigma=\sqrt{\sum_{i=1}^{N}(z_{i}-{\tilde{f}}(x_{i},y_{i}))^{2}/N}$, where
$z_{i}$, $y_{i}$, $x_{i}$ are the logarithmic data in the $z$, $y$ and $x$
axes, respectively, and ${\tilde{f}}$ is the considered scaling function (a
plane).

Fitted relation |  $\Delta\,a$ |  $\Delta\,b$ |  $\Delta\,c$ |  $\sigma$
---|---|---|---|---
(1) |  (2) |  (3) |  (4) |  (5)
Eq.9: $Log\,R_{c}(Log\,R_{d})$ |  0.15 |  0.26 |  - |  -
Eq.14: $Log\,M_{d}(Log\,V_{opt})$ |  0.25 |  0.12 |  - |  0.24
Eq.15: $Log\,\rho_{0}(Log\,R_{c})$ |  0.07 |  0.05 |  - |  0.21
Eq.17: $Log\,R_{d}(Log\,V_{opt})$ |  0.25 |  0.13 |  - |  0.24
Eq.17: $Log\,R_{c}(Log\,V_{opt})$ |  0.36 |  0.18 |  - |  0.34
Eq.17: $Log\,\rho_{0}(Log\,V_{opt})$ |  0.56 |  0.28 |  - |  0.54
Eq.18: $Log\,R_{d}(Log\,M_{d})$ |  0.23 |  0.02 |  - |  0.16
Eq.20: $Log\,M_{d}(Log\,V_{opt},Log\,C_{*})$ |  0.06 |  0.03 |  0.04 |  0.06
Eq.20: $Log\,R_{d}(Log\,V_{opt},Log\,C_{*})$ |  0.02 |  0.01 |  0.02 |  0.02
Eq.20: $Log\,R_{c}(Log\,V_{opt},Log\,C_{*})$ |  0.03 |  0.02 |  0.02 |  0.03
Eq.20: $Log\,\rho_{0}(Log\,V_{opt},Log\,C_{*})$ |  0.15 |  0.07 |  0.10 |  0.13
Eq.23: $Log\,R_{c}(Log\,M_{vir})$ |  0.26 |  0.02 |  - |  0.15
Eq.24: $Log\,C_{*}(Log\,C_{DM})$ |  0.01 |  0.06 |  - |  0.15
Table 8: Errors and scatters on the various scaling relations. Columns: (1)
relation; (2)-(3)-(4) error bars on the first, second and third (when present)
fitting parameters; (5) standard scatter of the 72 individual galaxies data
from the scaling relations.

##  Appendix I LSB rotation curves with their URC

We show in Fig. 24-25-26-27-28 the LSBs rotation curves data together with
their URC, taking into account Eq. 21-22 and the values of $R_{opt}\equiv
3.2\,R_{d}$, $V_{opt}$ and $C_{*}$ reported in Tab. 6-7 in Appendix G. We also
show the URC for the case $Log\,C_{*}=0$ in Fig. 24-25-26-27-28. In comparing
the URC model with the 72 individual rotation curves, in 21 of them we have
assumed a random systematic error running from $\simeq 3\%$ to $\simeq 16\%$
in their amplitudes (velocity measurements). In Tab. 9, the changes applied
are specified. Removing such systematics improves fits which were already
successful. Let us stress that the URC can help determining how well an
individual RC correctly reflects the mass distribution of the galaxy.

Figure 24: LSBs rotation curves data with their URC given by Eq. 21-22. The solid line is obtained for the $Log\,C_{*}$ values reported in Tab. 6-7 in Appendix G and is compared with the dashed line obtained for $Log\,C_{*}=0$. For each galaxy, we show the URC fit up to the farthest measurements (left) and up to the virial radius (right). Figure 25: It continues from Tab. 24. Figure 26: It continues from Tab. 25. Figure 27: It continues from Tab. 26. Figure 28: It continues from Tab. 27. Name |  $f\%$
---|---
(1) |  (2)
UGC2684 |  \+ 10.9
F565V2 |  \+ 8.8
F561-1 |  \- 7.9
UGC3174 |  \- 9.7
UGC1551 |  \- 14.3
UGC9211 |  \+ 4.8
F583-1 |  \- 8.1
ESO400-G037 |  \- 7.1
NGC959 |  \- 15.9
F574-1 |  \- 8.1
ESO444-G021 |  \- 9.3
F579V1 |  \- 16.1
ESO374-G003 |  \- 5.9
F568-1 |  \- 9.2
UGC11616 |  \- 7.5
PGC37759 |  \- 10.8
F730V1 |  \- 9.2
ESO215-G039 |  \- 10.5
UGC11454 |  \- 3.3
NGC3347B |  \- 6.0
ESO268-G044 |  \- 5.7
Table 9: List of galaxies in which we have left the amplitude of the RC to
freely vary by $f\%$. Columns: (1) galaxy name; (2) correction to the velocity
values of the RCs data, expressed in $f\%$.