The universal rotation curve of low surface brightness galaxies IV: the interrelation between dark and luminous matter

The universal rotation curve of low surface brightness galaxies IV: the interrelation between dark and luminous matter

C. Di Paolo chiara.dipaolo@sissa.it SISSA/ISAS, Via Bonomea 265, 34136 Trieste, Italy    P. Salucci salucci@sissa.it SISSA/ISAS, Via Bonomea 265, 34136 Trieste, Italy INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy
Abstract

We apply the concept of spiral rotation curves universality  (Persic et al., 1996) in order to investigate the properties of the baryonic and dark matter components of low surface brightness galaxies (LSB). The sample is composed by 72 objects, whose rotation curves are selected from literature. After a galaxies’ division in five velocity bins according to their increasing optical velocity, we observe that in specifically normalized units the rotation curves are all alike in each selected velocity bin, i.e. it reflects the idea of the universal rotation curve (URC) found in Persic et al.  (Persic et al., 1996). From the mass modeling of our galaxies, we show that the dark matter component is dominant rather than the baryonic one, especially within the smallest and less luminous LSB galaxies. The Burkert profile results to be an optimal model fit for the dark matter halos and it is shown that the central surface density , similar to galaxies of different Hubble types and luminosities. Our analysis leads to a strong correlation between the structural properties of the dark and luminous matter. In particular, when we also evaluate the compactness for stars and dark matter, a strong correlation emerges between the stellar disc and dark matter halo. Finally, the introduction of the stellar compactness as a new parameter in the ballpark of the luminous matter besides the optical radius and the optical velocity improves the URC.

Key words : dark matter - galaxies: LSB - Universal Rotation Curve - compactness

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I 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. The so-called CDM paradigm, in which one assumes a cold dark matter (CDM) WIMP that decouples from the primordial plasma when non-relativistic, successfully reproduces the structure of the cosmos on large scales  (Kolb and Turner, 1990). But some challenges to this scenario emerge on galactic scales, such as the ’missing satellite problem’  (Klypin et al., 1999; Moore et al., 1999; Papastergis et al., 2011; Zavala et al., 2009; Klypin et al., 2015) and the ’too big to fail problem’  (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 CDM scenario  (Salucci, 2001; de Blok and Bosma, 2002; Gentile et al., 2005; Weinberg et al., 2015; Bosma, 2003; Simon et al., 2005; Gentile et al., 2004, 2007; Del Popolo and Kroupa, 2009; Oh et al., 2011).

These issues suggest to study different scenarios from the ’simple’ CDM, such as warm DM  (de Vega et al., 2013, 2014; Lovell et al., 2014), self-interacting DM  (Vogelsberger et al., 2014; Elbert et al., 2015), or to introduce the influence of baryonic matter on DM in the galaxy formation process  (Navarro et al., 1996; Read and Gilmore, 2005; Mashchenko et al., 2006; Pontzen and Governato, 2012, 2014; Di Cintio et al., 2014).

One important way to investigate the properties of DM in galaxies is to study rotational supported systems, since they have a rather simple kinematics. In normal spirals, it is useful to apply the concept of a universal rotation curve (URC) to represent and model the galaxies rotation curves (RCs)  (Persic et al., 1996). Let us underline that the concept of universality in RCs doesn’t mean that all of them have a unique profile, but that all the RCs of local spirals (within ) can be described by a same function of the normalized radius with respect the optical radius , which is a characteristic radius of the luminous matter, and of one global parameter of galaxies, such as magnitude, luminosity, mass or velocity at the optical radius. Therefore )  (Rubin et al., 1985; Persic and Salucci, 1991; Persic et al., 1996; Salucci et al., 2007). The outliers of this paradigm are expected to be taken care by few other parameters.

In this paper, we investigate the concept of URC in the low surface brightness galaxies, special galaxies which emit an amount of light per area smaller than normal ones. They are locally more isolated than other galaxies and likely evolving very slowly with very low star formation rates. As we see in radio synthesis observations, LSB galaxies have extended gas disks with low gas surface densities and high ratios. The low metallicities makes gas cooling difficult and the stars difficult to form. LSB galaxies can be said to be trapped in their current evolutionary state. Furthermore, LSBs seem to be dark matter dominated almost all the way into their centers and it is likely that the large dark matter dominance makes the stellar disks of these galaxies extremely stable, making it possible for the disk to exist at such low surface densities. It’s worth saying that investigations of individual LSB galaxies show that they form an alternative track of galaxy evolution, free from the instabilities and interactions that have shaped the Hubble sequence. They give us the opportunity to study almost unevolved galaxies in great detail  (de Block, 2000).

In this work, we use the results obtained through the URC concept applied to LSBs to investigate the DM distribution in these galaxies and in particular to study the relation between dark and luminous matter distributions. Our sample is made of 72 LSB galaxies selected from literature, whose optical velocities span from km/s to km/s, covering the full population. We divide the sample of galaxies in five different velocity bins, according to their increasing , so that we investigate the different peculiarities of the RCs’ profiles and different fractions of dark and luminous matter. We build five double normalized synthetic rotation curves. Our analysis goes on with their modeling, finally we denormalize as Karukes at al. 2016  (Karukes and Salucci, 2017). In this way, we are able to investigate each object of the sample and to find out its scaling relations. In particular, we evaluate the compactness of dark and luminous matter distribution and compare it in the light of the result of a similar study on dwarf disk (dd) galaxies  (Karukes and Salucci, 2017) . Let’s notice that is the velocity evaluated at the optical radius , where is defined as the radius encompassing 83% of the total luminosity and is the galaxy disc scale length.

The structure of the paper is the following: in Section II, we describe our sample of LSB galaxies; in Section III, we present the analysis needed to build the synthetic RCs; then, in Section IV we apply the mass modeling to the synthetic RCs and we establish the URC for its objects; in Section V, we treat the denormalization of the structure properties obtained by the mass modeling, so that, in Section VI, each single galaxy can be described individually and their properties are used to define the LSBs’ scaling relations; in Section VII, we introduce the concept of compactness and show the relative results from LSB galaxies; in Section VIII, we built the 3D URC taking into account the compactness; finally, in Section IX, we comment our main results.

Ii The sample

We consider the following sample: LSB galaxies selected from literature with optical velocities spanning from km/s to km/s and extending to at least one (or when it is possible to extrapolate this quantity). The sample is composed by 72 galaxies, which we divide in 5 velocity bins in order to describe their structural properties. We have 1614 independent measurements for our objects distributed in the different velocity bins as in Tab. 1. We have a quite good statistics for all the 5 velocity bins.

In this paper we report in Tab. 3 the galaxies names with their related disc scale lengths and optical velocities. The rotation curves and all the global properties of the involved galaxies such the luminosity and the inclination of the velocity field will be soon published on line.

Iii The co-added rotation curves of LSB galaxies

We start by plotting the rotation curves data in the space - - (see Fig. 1). Galaxies occupy a broadly well defined region and lay on a quite well defined surface. All this becomes much more apparent when we plot the double-normalized rotation curves in the space - - (see Fig. 1), the data seem to be alligned along a straight line. This is what we also find in high-surface brightness spiral galaxies  (Persic et al., 1996) and dwarf discs  (Karukes and Salucci, 2017).

Figure 1: of the LSB RCs data (black points) and of the double normalized RCs (red points). The radial axis is restricted to range where both normalized and non-normalized data were present.

For each of the five velocity bins defined in Tab.1, we build, from the 1614 kinematical measurements of our sample, five co-added rotation curves as in PSS 96  (Persic et al., 1996). It’s worth to underline that the utility of the co-added curves is to erase the peculiarities of the individual rotation curves in order to give a more generic (universal) description of galaxies. Moreover, this procedure leads to extendend RCs with good statistics.

Figure 2: Co-addeded double normalized RCs for the five velocity bins. Hereafter purple, blue, green, orange and red colors are referred to the five velocity bins respectively. The black empty triangles are the co-added RC for the dwarf disc galaxies  (Karukes and Salucci, 2017).
Vel.Bin Vel.Range N.galaxies 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: Grouping of LSB galaxies velocity bins according to their . Columns: (1) i - velocity bin; (2) range of velocity bin to which a certain should belong; (3) number of LSB galaxies in each velocity bin; (4) average value of for the galaxies of the i - velocity bin; (5) average value of for the galaxies of the i - velocity bin; (6) number of data points availaible for each velocity bin.

Then we start by assigning each rotation curve of the sample in the corresponding bin according to its own optical velocity, as indicated in Tab.1.

For each of the five velocity bins we build the double normalized co-added RCs. In order to achieve this goal, the RCs data of the 72 galaxies are double normalized, i.e. for each single galaxy the radial and the velocity data are normalized with respect its own disc scale length and its own optical velocity, respectively.

After that, we perform a radial binning for each of the five velocity bins. In detail, for the I, the II and the III velocity group, the radial (normalized) coordinate is divided in 12 bins : the first 5 have amplitude of 0.4 and the remaining an amplitude of 0.5 (units of ). While for the IV and the V velocity bins we adopt a different subdivision of the radial coordinate (as reported in 7) so that a quite good statistic is still present in the external radial bins, where data are not so numerous.

Finally, in each radial bin of each velocity group there is a set of double normalized data, from which we evaluate the average velocity. Thus, by repeating the procedure for all the radial bins of each velocity group we obtain the five double normalized co-added RCs shown in Fig. 2.
From this figure, the different profiles related to a different galaxies velocity (and luminosity) are very evident. Furthermore, the co-added RCs plotted in Fig. 3 and listed in Tab. 6 - 7, are obtained by multiplying the velocity values by , i.e. the average of the optical velocities of all galaxies of a certain velocity bin. The uncertainties on these co-added RCs are very small. Moreover, by multiplying the radial coordinate by the average disc scale length of all galaxies of each velocity bin, we find the co-added RCs of Fig. 4, which are useful to have an idea of the results in physical units.

Figure 3: Co-addeded LSB rotation curves as function of normalized radial coordinate. The black empty triangles describe the co-added rotation curves of dwarf spirals with  (Karukes and Salucci, 2017).
Figure 4: Co-addeded LSB rotation curves as function of radial coordinate. The black triangles describe the co-added rotation curves of dwarf spirals with and  (Karukes and Salucci, 2017).

Iv The universal rotation curves of LSB galaxies

In this section, we build the URCs of LSB galaxies by modeling the co-added RCs data with a specific function as in normal spirals (see PSS  (Persic et al., 1996)).

Vel. Bin
kpc
(1) (2) (3) (4) (5) (6) (7)
1 0.37 0.36
2 0.49 0.44
3 0.52 0.47
4 0.76 0.63
5 0.82 0.70
Table 2: Parameters of the best fit values and some useful quantities for the five velocity bins. Columns: (1) i - velocity bin; (2) fitting value of in each velocity bin; (3) fitting value of ; (4) fitting value of ; (5) estimated virial mass; (6) fraction of baryonic component to circular velocity at (eq. 12); (7) values defined according to eq. 9.
Figure 5: Velocity model fit for the co-added RCs of the first four velocity bins, respectively in panel (a), (b), (c) and (d). The dashed lines are the stellar contribution, the dot-dashed lines are the DM contribution and the solid lines are the total contribution to the circular velocities.

The circular velocity model consists into the sum in quadrature of two terms: and , that describe the contribution from the stellar disc and the dark halo, respectively. Then:

(1)

Let us stress that in first approximation and also because we investigate the inner region of these galaxies (), we can leave the introduction in the model of the HI disc gas component to further studies. Indeed, the gas contribution is usually a minor component to the circular velocities, since the inner regions of galaxies are dominated by the star component while in the external regions, where the gas component overcomes the stellar one, the DM contribution is largely the most important  (Evoli et al., 2011). Consequently our approximation doesn’t alter the mass modelling and the result of this paper.
Moreover we will introduce the bulge component only for the last velocity bin, related to galaxies with the largest optical velocities  (Salucci et al., 2000; Das, 2013).

We describe the stellar component through the wellknown Freeman disc  (Freeman, 1970), whose surface density profile is

(2)

Eq. 2 leads to:

(3)

where is the mass disc, and are the modified Bessel functions computed at , with .

We mass model the DM halo with the cored Burkert profile  (Burkert, 1995):

(4)

where is the central density and is the core radius. Its mass distribution is:

Let us stress that this density profile has an excellent record in fitting the DM halo in galaxies  (Salucci and Burkert, 2000) and in particular it was tested to be preferred rather than the cuspy structure also by LSBs’ halos  (de Blok and Bosma, 2002). The contribution to the total circular velocity given by the dark matter is simply

(6)


We fit the five co-added RCs with help of the model described above, which is characterized for each co-added RC by three free parameters: , , . The resulting best fitting parameters are reported in Tab. 2. These must be seen as the average values referred to the central values of each velocity bin. The URCs of LSBs resulting from the fit are plotted alongside the co-added RCs in Fig. 5.

Figure 6: Velocity model fit for the co-added RCs of the V velocity bin. The dashed, dot-dashed, dotted and solid lines are the stellar, dark matter, bulge and total contributions to the circular velocities, respectively.

Finally, we report a fit of the fifth velocity bin in Fig.6. However, here we introduce the presence of a central bulge (which is typical of the largest galaxies)  (Das, 2013). Thus, we took into account the bulge velocity component to the total circular velocity through

(7)

where , and are values referred to the first data points of the V co-added RC and is a parameter which in principle can vary from to (e.g.  (Yegorova and Salucci, 2007)). The best fitting parameters , , are reported in Tab. 2 and moreover, for this last velocity bin, we find , with and .
We note, as also observed in normal spiral galaxies  (Persic et al., 1996), that in the inner region of the LSB galaxies, the stellar component is dominant, while in the external region the DM is dominant. Moreover, we note that the galaxies with the lowest have the highest fraction of DM and this component decreases going to galaxies with higher . This observation can be made more quantitative by considering the fraction that the baryonic matter contributes to the total circular velocity through the ratio between the stellar disc mass (plus the bulge mass for the last velocity bin) and the virial mass , which practically encloses the whole galaxy mass. Fig. 7 shows that the lowest fraction of baryonic content is in the smallest galaxies (with the smallest stellar mass ), it increases going toward larger galaxies, but at a certain point the baryonic fraction comes back to decrease towards the largest ones. The result is in agreement with the "U-shape" of previous works (e.g.  (Moster et al., 2010)): the highest fraction of DM content is likely due to supernovae feedback in the smallest galaxies and to AGNs feedback in the largest ones.

Figure 7: Baryonic matter fraction versus stellar mass.

Moreover, it’s interesting also to show the relation between the star disc mass and the optical velocity for the five velocity bins. Fig. 8 shows a very good allignment of the logaritmic data, which are well fitted by . When we compare it to the relation for the normal Spirals (from PSS data  (Persic et al., 1996)), represented by the dashed line in Fig. 8, we note that to fixed optical velocities correspond larger star disc masses for LSB galaxies.

Figure 8: Star disc mass versus optical velocity for each velocity bin (the contribution from the bulge is also considered for the V velocity bin). The solid and dashed lines show the best fitting relations for the LSBs and normal Spirals  (Persic et al., 1996) respectively.

This can be explained if we take into account that, on average, LSB galaxies are more extend than normal galaxies and in particular at the same optical velocity they have a larger optical radius as shown in Fig. 9.

Figure 9: Optical velocity versus optical radius from the observation on LSB galaxies (red points) and normal Spirals (blue points)  (Persic et al., 1996). The solid and the dashed lines come from the fitting relation for LSBs and normal Spirals respectively.

The best linear fit for the LSBs is represented by the solid line, , while for the normal Spirals is represented by the dashed one, . Let us stress that the circular velocity can be expressed at , in any object, by the relation

(8)

where is the total mass enclosed in . Thus, if we consider the same for an LSB galaxy and a normal spiral, since the optical radius of the former is, on average, larger than of the latter, it means according to eq. 8 that also the mass within of an LSB is larger than that of a normal Spiral. Provided that if the total mass of an LSB galaxy is larger than that of a normal spiral, the same is also true for their stellar mass, then we can explain the results shown in Fig. 8.

It could be very interesting also to compare other relations between LSB and normal spiral galaxies in order to get new ideas about their properties and, maybe, some important information about the evolution history of galaxies, of their DM and baryonic content evaluated in different cosmic contexts (we remember that the LSBs are isolated with respect the normal Spirals). Anyway, we remind these intentions to our next studies.

V Denormalization of the URC mass model

We go back from a double normalized URC of the various velocity bins to the individual RCs expressed in physical units for each single galaxy.

Given the small intrinsic scatter of the fiducial double normalized co-added RCs and the extremely good fit of the URCs to them, we consider that the relations obtained among the average values related to the velocity bins are approximately true also among the values related to each single galaxy, provided one takes in consideration the difference between , and , . Therefore, we consider:

(9)

where the constant different for each velocity bin is obtained by the velocity modeling of the previous section and its values are reported in Tab. 2. After analogous consideration about eq. 9, we also write

(10)

Then, for every galaxy of the sample, the Burkert DM mass inside the optical radius can be expressed as

(11)

where is the fraction that the baryonic matter

(12)

evaluated at and function of , i.e. of each velocity bin. The increasing values of going from the I velocity bin to the last one are reported in Tab. 2.

Finally, taking into account the eq. IV, 12, 9, 10, 11, and inserting the values of and for each single galaxy, we get all the structural parameters of the dark and luminous matter. They are reported in Tab. 3.

Name
kpc km/s kpc
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
UGC4115 0.4 24.2 6.3 0.8 -23.42 1.1 1.67 0.06 -0.14
F563V1 2.4 27.3 48 9.9 -25.20 13 0.96 -0.40 -0.58
UGC11583 0.3 27.9 6.5 0.6 -23.01 1.1 1.92 0.17 0.01
UGC2684 0.8 36.7 29 2.7 -23.81 6.9 1.69 -0.00 -0.09
F574-2 4.5 40 192 23 -25.49 74 1.05 -0.45 -0.53
F565V2 2 45.2 110 7.7 -24.58 38 1.47 -0.19 -0.21
UGC5272 1.2 48.8 77 3.8 -23.99 23 1.76 -0.02 -0.03
UGC8837 1.2 49.6 79 3.8 -23.98 24 1.77 -0.02 -0.02
F561-1 3.6 50.8 250 17 -25.07 109 1.34 -0.31 -0.30
UGC3174 1 51.7 72 2.9 -23.75 21 1.89 0.04 0.05
NGC4455 0.9 53 68 2.5 -23.62 19 1.96 0.08 0.09
UGC1281 1.7 55 138 6.1 -24.25 49 1.71 -0.08 -0.05
UGC1551 2.5 55.8 211 10 -24.63 86 1.56 -0.18 -0.14
UGC9211 1.3 61.9 165 4.3 -23.98 35 1.83 0.06 0.02
F583-1 1.6 62 201 5.6 -24.17 46 1.75 0.00 -0.03
UGC5716 2 66.4 288 7.7 -24.34 74 1.71 -0.03 -0.05
UGC7178 2.3 69.9 367 9.3 -24.44 101 1.70 -0.06 -0.05
ESO400-G037 4.1 69.9 651 20.5 -25.01 217 1.47 -0.21 -0.20
NGC3274 0.5 68 75 1.1 -22.86 12 2.36 0.35 0.32
F583-4 2.7 70.5 438 12 -24.59 128 1.64 -0.10 -0.09
F571V1 3.2 72.4 549 15 -24.74 173 1.60 -0.14 -0.11
NGC5204 0.7 73.1 115 1.7 -23.10 20 2.29 0.30 0.29
UGC731 1.7 73.3 298 6.1 -24.09 75 1.87 0.04 0.05
NGC959 0.9 75.3 172 2.7 -23.44 34 2.15 0.21 0.22
NGC100 1.2 77.2 233 3.8 -23.69 52 2.06 0.15 0.17
NGC5023 0.8 78.4 160 2.2 -23.25 31 2.26 0.27 0.28
UGC5750 5.6 80 1171 32 -25.20 460 1.47 -0.26 -0.20
UGC3371 3.1 82 681 14 -24.60 224 1.72 -0.09 -0.03
NGC4395 2.3 82.3 509 9.3 -24.30 149 1.84 -0.00 0.05
UGC11557 3.1 83.7 710 14 -24.58 236 1.74 -0.08 -0.01
UGC1230 4.5 90 1278 23 -24.91 435 1.63 -0.15 -0.08
ESO206-G014 5.2 91.3 1531 29 -25.04 552 1.59 -0.19 -0.11
NGC2552 1.6 92 475 5.6 -23.86 109 2.06 0.14 0.18
UGC4278 2.3 92.6 691 9.3 -24.22 185 1.92 0.04 0.10
UGC634 3.1 95.1 984 14 -24.49 300 1.82 -0.03 0.04
ESO488-G049 4.4 95.3 1410 23 -24.84 492 1.69 -0.13 -0.04
UGC5005 4.4 95.5 1406 23 -24.83 490 1.69 -0.12 -0.04
UGC3137 2 97.7 669 7.7 -24.03 172 2.02 0.10 0.17
F574-1 4.5 99 1546 23 -24.82 552 1.72 -0.12 -0.02
F568-3 4 100.5 1416 20 -24.70 488 1.77 -0.08 0.01
ESO322-G019 2.5 100.7 878 10 -24.22 249 1.96 0.05 0.13
F563V2 2.1 101.3 755 8.2 -24.05 201 2.03 0.10 0.18
NGC 247 2.9 106.6 1156 13 -24.33 358 1.95 0.02 0.13
ESO444-G021 6.4 107.4 2603 38 -25.1 1095 1.65 -0.19 -0.06
F579V1 5.1 111.5 2223 28 -24.84 880 1.77 -0.12 0.02
Table 3: Results from the denormalization of the LSBs’ URCs mass model. Columns: (1) galaxy name; (2) disc scale length (given in literature); (3) optical velocity (given in literature); (4) disc mass; (5) core radius; (6) central DM mass density; (7) virial mass; (8) central surface density; (9) compactness of stellar matter; (10) compactness of dark matter.
Name
kpc km/s kpc
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
F568-1 5.3 130 4218 29.5 -25.05 557 1.58 -0.03 -0.11
UGC628 4.7 130 3740 25.0 -24.94 476 1.63 0.00 -0.08
UGC11616 4.9 133.2 4094 26.4 -24.96 535 1.63 -0.00 -0.07
ESO186-G055 3.6 133.2 3041 17.5 -24.67 357 1.74 0.08 0.00
ESO323-G042 4.4 138.7 4020 23.1 -24.83 518 1.70 0.04 -0.02
PGC37759 6.8 139.4 6195 41.3 -25.23 914 1.55 -0.08 -0.13
ESO234-G013 3.7 139.4 3425 18.2 -24.65 415 1.77 0.09 0.02
F571-8 5.2 139.5 4765 28.7 -24.97 650 1.65 -0.00 -0.06
F730V1 5.8 141.6 5523 33.8 -25.08 789 1.62 -0.03 -0.08
UGC11648 3.8 142.2 3620 18.6 -24.65 445 1.79 0.09 0.03
ESO215-G039 4.2 142.9 4037 21.4 -24.75 517 1.75 0.06 0.01
ESO509-G091 3.7 146.8 3735 17.8 -24.59 460 1.82 0.11 0.06
ESO444-G047 2.7 148.4 2809 11.7 -24.28 307 1.95 0.19 0.14
UGC11454 4.5 150.3 4787 23.5 -24.77 644 1.77 0.06 0.02
UGC5999 4.4 153.0 4851 22.8 -24.73 652 1.79 0.07 0.04
UGC11819 5.3 154.6 6578 29.5 -25.02 613 1.61 0.04 -0.09
ESO382-G006 2.3 160 3097 9.5 -24.18 214 1.96 0.27 0.14
ESO323-G073 2.1 165.3 2923 8.0 -24.03 194 2.04 0.32 0.19
NGC3347B 8.1 167 11760 53.1 -25.36 1288 1.53 -0.05 -0.15
ESO268-G044 1.91 175.6 3057 7.2 -23.90 201 2.12 0.36 0.24
ESO534-G020 16.7 216.6 40638 143.4 -25.81 5909 1.51 -0.17 -0.19
NGC7589 12.6 224 32831 97.4 -25.52 4830 1.63 -0.08 -0.08
UGC11748 3.1 240.7 9418 14.2 -24.12 857 2.20 0.32 0.31
UGC2936 8.4 255.0 28363 55.6 -25.03 4053 1.89 0.07 0.12
F568-6 18.3 297.0 83839 163.0 -25.62 16302 1.76 -0.10 0.01
Table 4: It continues from Tab. 3

Vi The scaling relations

In this paragraph, we work on the scaling relations among the structural properties of dark and luminous matter of each galaxy, in order to deduce crucial informations on the relations between dark and baryonic matter.

First of all, we note that the values of the central surface density, for each galaxy of the sample, is close to , with the argument expressed in . Thus, the results from our LSBs’ sample are in agreement with the relation found over 18 blue magnitudes and in objects spanning from dwarf galaxies to giant galaxies  (Gentile et al., 2009; Donato et al., 2009; Plana et al., 2010; Salucci et al., 2012). See Fig. 10.

Figure 10: Logaritmic values of obtained for our LSB galaxies sample versus their optical velocities; scaling relation from Donato et al. (2009)  (Donato et al., 2009) (yellow shadowed area); empirically inferred scaling relation: from Burkert (2015)  (Burkert, 2015)(lightblue shadowed area).

Then, we reproduce, in Fig 11, the relation between the central density core and the core radius of the DM halo, which remarks the highest mass densities in the smallest galaxies, as also found in the past by the analysis of normal spirals. The found linear fitting relation in Log-Log scale is: .

Figure 11: Central densities versus core radii with the fitting relation.
Figure 12: - , - and - with the fitting relations.


Moreover we reproduce the relationships which are necessary in order to establish the three-dimensional URC for the present sample in physical units, which is the function . This requires to express all the quantities which appear in the equations of section IV in terms of and , the virial mass, which practically encloses the whole mass of a galaxy, out to the virial radius . This can be used to describe the URC as an alternative to the luminosity or the optical velocity. We remember that it is defined as , where is the so called virial radius and is the critical density of the Universe. We derive the stellar disk mass versus the virial mass relation, , (see the first panel in Fig. 12) by fitting it with a linear function in logaritmic scale. The found relation is

(13)

The other relationships we need to establish the 3D URC are , , . The first of them is shown in the second panel of Fig. 12 and the fitting relation is given by

(14)

In the third panel of Fig. 12, we show the plot, whose fitting relation is

(15)

while for , which is reported in Fig. 15, we found eq. 19. However, all the above relations show a large scatter that was not observed in normal Spiral galaxies. This lead us to exclude the existence of the URC in physical units.

Vii Compactness

We restore the universality by evaluating a new parameter, which was also introduced for dwarf disc galaxies  (Karukes and Salucci, 2017); this is the compactness . As for dd galaxies, we observe a large scatter in the scaling relations with respect normal Spirals. This may be due to the fact that sometimes galaxies with the same stellar mass (luminosity) have a different size of their stellar disc and a new parameter is required to restore the universality: we say they have a different "stellar compactness" .

Taking into account the linear fit for the and values (see Fig.13) deduced from the denormalization of the whole sample and described by the equation

(16)

we define according to Karukes et al.  (Karukes and Salucci, 2017) the stellar compactness through the following relation:

(17)

where is measured from photometry. The values obtained from our LSBs sample are shown in Tab. 3 and span from 0.35 to 2.26.

Figure 13: versus in the LSBs sample.

It’s interesting to note that by fitting to with the additional variable , we obtain a better fit than that in Fig. 11, whith a smaller (halved) scatter. The result is shown in Fig.14, where the model function is

Figure 14: Central DM halo density versus core radius and stellar compactness, the plane projected into a line is the fitting relation 18. Along the z-axis, is expressed in .
(18)

the errors on the fitting parameters are reported in Tab. 5. A similar improvement is also obtained for the other scaling relations if we involve .

As done for stellar compactness, we analyse the dark sector in a similar way, i.e. we investigate the case in which the galaxies with the same virial dark mass exhibit different core radius. We look then for the "compactness of DM halo" . On the other hand, by considering the Log-Log linear fit between the core radius and the virial mass values described by (see Fig.15)

(19)

we define the compactness of the DM halo through the relation:

(20)

The values obtained for are reported in Tab. 3 and span from 0.26 to 2.09.

Figure 15: versus for the LSB galaxies.

Let us stress that by the above definitions, trough we measure the deviation of the observed value from the expected value for a galaxy with fixed . Obviously, if the observed is larger than its expected value, it is related to low compactness, while if smaller, it is related to high compactness. Analogously, we can say by considering and involving and .

Now, it is interesting to plot the compactness of the stellar disc versus compactness of DM halo, as illustrated in Fig. 16.

Figure 16: Compactness of the stellar disc versus compactness of DM halo. In addition to the red points representative of our LSB galaxies, the black triangles show the dwarf disc galaxies  (Karukes and Salucci, 2017). The solid and the dot-dashed lines are the fitting relations for LSBs and dwarf discs respectively.

We note that the and the are strictly related: galaxies with high , also have high . The found fitting relation between these two quantities is . The results are in agreement with those obtained in the analogous study of dwarf disc (dd) galaxies  (Karukes and Salucci, 2017), whose fitting relation is given by . Notice that and are computed in a completely different way. This is remarkable because the same relations are found for these two different families of galaxies ( LSBs and dds). The strong relationship between the two compactness certaintly indicates that the DM and stars distributions follow each other very closely. This could indicate a non-standard interaction between baryonic and dark matter. Otherwise that baryonic feedback enters the dark matter distribution in a very fine tuned way  (Di Cintio et al., 2014; Chan et al., 2015). Or likely an interaction in dark sector’s particles, followed by baryonic gravitational interaction, could exist.

Finally we underline that, as regarded to normal spirals, the RCs in physical units for the LSB galaxies depends on their disc scale length (or optical radius ), their disc mass (or optical velocity , or luminosity , or virial mass ) and a new additional quantity, i.e. the concentration .

Viii The compactness as the third parameter in the URC

In this section we establish the analytical function for the URC involving the presence of the new additional third parameter, the stellar compactness . Moreover, we express the normalized radial coordinate in terms of the optical radius instead of , to have a simple analogy with previous works on URC.

We take into account the expressions of the central density , the core radius , the stellar disc mass and the disc scale length as functions of , , that we obtain by fitting our data through a plane. The resulting relation are

and the scatter of data from these planes is always reduced (halved) than the case of the only dependence on , similarly to what happens in the 2D Fig.11 and 3D Fig. 14. The quantities in VIII are expressed in , respectively. The errors of the fitting parameters are reported in Tab. 5.

Fitted relation
(1) (2) (3) (4)
0.04 0.03 0.09
0.10 0.01 0.04
0.27 0.02 0.11
0.33 0.03 0.14
0.14 0.01 0.06
Table 5: Errors on the fitting parameters, involved in equations 18 and VIII. Columns: (1) fitted relation; (2)-(3)-(4) error bars on the fitting parameters, listed in order of appearance in each relation.

Putting the above equations (VIII) in 1, 3, IV, 6 we obtain the analytical expression for the URC:

Figure 17: Universal Rotation Curves, with the velocity axis in physical units, for fixed star compactness respectively in blue, yellow and red colors.
Figure 18: Universal Rotation Curves, with the velocity axis normalized with respect the optical velocity, for fixed star compactness respectively in blue, yellow and red colors. On the left and right side the same plot is seen from two different points of view to remark the difference among the curves for smaller and larger virial mass respectively.
(22)

where are the modified Bessel functions evaluated at , with and

Finally we plot the above URC in Fig. 17, by distinguish three layers for three different values of stellar compactness. The central layer, related to , can be identified with the URC obtained for the normal Spirals and the other two layers highlight the different kinds of curve which can be found in galaxies where the compactness seems to be important, such as LSBs and dwarf discs. Moreover, in Fig. 18, we show the URC with the normalized velocity axis from two different points of view. In the left panel the difference in the shape of the URC for different stellar compactness at smaller virial mass is particularly evident, while in the right panel this difference is highlighted at higher virial masses.

After our complete analysis, we discover the relevance of in galaxies and we think that this new quantity should be considered in the starting subdivision of galaxies sample. Anyway, it is difficult to do in our case because our sample is not so extended at the moment, while it could be realized in next works if larger samples are involved.

Ix Conclusions

We analyzed a sample of 72 low surface brightness (LSB) galaxies selected from literature, whose optical velocities span from km/s to km/s. Our studies rely on the construction of the Universal Rotation Curves (URC) and the following deduction of properties of stellar and DM distribution. For this purpose we divided them in five different velocity bins, according to their increasing , to investigate the whole family of LSBs. The result of this led to normalized URCs . By modeling these co-added RCs, we found the DM and luminous parameters. We found that the DM component is the dominant one in the outermost region of each galaxy. In detail, the fraction of DM that contribute to the RCs is more relevant in galaxies belonging to the velocity bin with lower . Then, we denormalized the previous velocity model so that we could determine the structural properties of the individual galaxies of our sample in physical units and define their scaling relations. They are in rough agreement with the relations found within the normal spiral galaxies; anyway we postpone an accurate comparison between LSBs and normal Spirals properties to further works.
A relevant fact is the not so low scatter in the LSBs scaling relations if compared to the normal spirals, which led us to introduce, as in Karukes & Salucci  (Karukes and Salucci, 2017), the concept of compactness: galaxies with the same stellar mass (luminosity) have a different size of their stellar disc. Thus, they have a different "stellar compactness" . In the same way, galaxies with the same virial mass exhibit different core radius; in this case, we define the "compactness of a DM halo" . Thus, we evaluated the compactness of dark and luminous matter distribution and compared it with those obtained from similar studies on dwarf disk (dd) galaxies  (Karukes and Salucci, 2017). We found similar results: galaxies with high , also have high , i.e. the distributions of stellar disc and its dark matter halo are entangled. This fact could be of enormous relevance when we want to argue about the nature of dark matter and the possible interactions in the dark sector, with consequences in the baryonic sector.
The introduction of the new parameter allowed us to reduce (halve) the scatter in the LSBs scaling relations and restore the universality. After that, we built the 3D URC by involving , i.e. we found an analytical expression for . We think that the new parameter , whose relevance we discovered a posteriori, should be considered in the starting binning of galaxies sample; in other words, we should subdivide galaxies not only according to their , but also to their . However, at the moment, it is hard to apply this concept to our sample since it is not sufficiently extended, while it can be postponed to next works if a larger sample is involved.

acknowledgments

We would like to acknowledge A. Erkurt for providing the LSBs rotation curves data in electronic form. We thank E. Karukes for passing dwarf discs data and A. Lapi for providing normal spiral galaxies data, which were useful for comparisons in this paper.

References

appendix

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 6: Grouping of LSB galaxies in velocity and radial bins. Columns: (1) center of the radial bin; (2) number of data in each bin; (3) average velocity in 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 , reported in Tab. 1.
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 7: It continues from Tab. 6.
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