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The Universal Rotation Curve of Spirals


The rotation curves of spirals follow, from their centers out to their virial radii, an universal profile that implies a tuned combination of their stellar disk and dark halo mass distributions. This phenomenology replaces the current paradigm of flat RC curves  and it switches the focus from the structure of a typical galaxy  to the typical systematics of the mass structure of the whole family of spirals

Dark matter and Kinematics of Spirals
Cosmology tells us that about 84% of the mass of the Universe is composed of dark matter (DM), a massive component which does not emit radiation, but it dominates the gravitational potential of galaxies and cluster of galaxies. Galaxies are baryonic condensations made by stars and gas (namely H and He) lying at the centres of much larger dark haloes made by unknown matter, affected by a gravitational instability caused by primordial density fluctuations.

The main goal in the current cosmological context is to understand the nature and the history of these ubiquitous dark haloes by investigating the properties of the galaxies they contain (i.e. their luminosities, kinematics, sizes, and morphologies). More specifically, the measurement of the kinematics of tests particle (stars and gas) i.e. the Rotation Curve (RC) of disk galaxies is a powerful tool to investigate the Nature of Dark Matter (DM), so as its content and distribution also relative to that of  the various baryonic components in Galaxies

Rotation Curves Introduction
In Spirals, rotation curves $$ {V(r)} $$ (RCs) yield their mass structure, in fact:
 * $$V(r)= (r \, d\Phi/dr)^{1/2}

$$ with $$\Phi$$ the galaxy gravitational potential. It is well known that RCs do not show a Keplerian fall-off at large distances and then do not match the distribution of luminous matter (Rubin et al. 1980; Bosma 1981 see also: ). This implies that spiral galaxies contain large amounts of dark matter (DM) or, in alternative, the existence of exotic physics in action on galactic scales. The additional invisible component becomes progressively more conspicuous in each galaxy  at outer radii and among galaxies  in the less luminous ones.



The distribution of stars in spirals
In order to investigate the dark matter in galaxies we have, first, to set the scene for their luminous matter. The stars in spirals are distributed in thin disks with surface luminosity (Freeman, 1970).

I(r) = I_0 e^{-r/R_D} $$ with $$R_D$$ being the disk scale-length; $$I_0$$ is the central value; it is useful to define: as the size of the stellar disk, whose luminosity is

$$ L_{tot} = 2\pi I_0 R^2_D $$.

The spiral's light profiles,  in terms of the coordinate $$r/R_D$$, do not depend on galaxy luminosity.

Individual and co-added rotation curves
The investigation starts with the about 200 available very high quality individual RCs. Next step is to build, from the about 3000 available individual RCs of sufficient quality, the corresponding co-added rotation curves that, by construction, have negligible observational errors. These co-added curves, so as the individual RCs of high quality, show an Universal character, according to which the galaxy magnitude (e.g. the absolute I-band magnitude $$M_I$$) identifies a specific galaxy.

To obtain $$V_{coadd} $$ we place each available individual RC into the corresponding galaxy magnitude bin, among the 11 in which we divide the full Spiral's (I-band) magnitude range $$-24.3<M_I <16.3$$. The RCs in each luminosity bin are then averaged in radial bins of width $$0.3 R_D$$. This leads to a family of 11 synthetic coadded RCs $$V_{coadd}(r/R_D, M_I )$$.

These curves, so as the individual ones result regular, smooth and with a very small intrinsic variance. In the $$(V,r/R_D,L_I)$$ 3D space the individual and the co-added RCs of all Spirals occupy a very small portion of the total volume. Additional data have extended the co-added RCs out to a good fraction of the galaxies virial radii defined as $$ R_{vir} = 259 \, (M_{vir}/10^{12} M_\odot)^{1/3} kpc $$.

The concept of URC


The Universal Rotation Curve means the existence of a function of the galaxy radius (in units of the disk length scale, $$R_D$$) and of galaxy magnitude (e.g.$$M_I$$ ) which reproduces the RC of any object (of known $$M_I$$ and $$R_D$$). This concept, implicit in Rubin et al. (1985) was pioneered by Persic and Salucci (1991) and then set in Persic, Salucci \& Stel (1996). Noticeably, the RCs of spirals are not flat: their RC slopes take all sort of values from +1, the maximum slope for a rotating body, found in RCs of dwarf spirals  to -1/2, the value corresponding to a Newtonian point-mass, found in RCs of high luminosity galaxies. The RCs that are flattish inside $$R_{opt}$$ are declining farther out. At a given fixed radius the above phenomenology can be seen in detail (metti qui il numerino di PSS)  $$ \nabla $$,   the logarithmic slope of the circular velocity at $$R_{opt}$$,  is plotted,  for a very large sample of galaxies as a function of $$V_{opt} $$  and $$ M_B $$; one finds  −0.2 ≤ $$ \nabla $$ ≤ 0.8. Noticeably spirals show an inner baryon dominance region, whose amplitude ranges between 1 to 3 $$ R_D$$, according to the galaxy luminosity (see Fig(8) of and . Inside this region, the ordinary baryonic matter fully accounts for the rotation curve. By means of progressively larger samples of spirals of different luminosities with progressively more extended RCs,  and Salucci et al (2007) a) probed the URC paradigm and b) derived the analytical curve $$V_{URC }(radius; luminosity)$$ to represent it.  In fact,  they derived  the URC representing  individual and  stacked/co-added $$ V_{coadd}(r/R_D, M_I )$$ out to their virial radii   ( Salucci et al, 2007) . With such universal analytical function $$V_{URC}(r/R_D,M_I )$$ all kinematical data are fit by a same  function; see Salucci et al.2007.  Thus, at any radius r $$V_{URC} $$ predicts the circular velocity of any  spirals of known luminosity and disk scale-length within an error margin which is one order of magnitude smaller than i) the radial variations of the RC in each object, ii) the  variance, among galaxies, of their RC amplitudes at any  same radius $$  R/R_D$$.

Although a complete assessment of the role in the URC of the minor parameters (e.g. bulge, stellar surface density) has yet to be performed, the function $$V_{URC}$$ obtained in the above studies well matches and then represents the RC in any Dark Matter study of Spirals.

The Universal Rotation Curve. The interpretation
The analytical form of the URC model is simply the sum in quadrature of the disk and halo contributions to the circular velocity. In spirals the gaseus and the bulge contributions play a role which may be  important  in the mass modelling of some particular object, but, it is negligible in a first generalization of the properties of the dark and luminous matter. So:

V^2_{URC} = V^2_{URCD} + V^2_{URCH} $$ For the stellar disk term is we have:

V^2_{URCD}(x) = {1\over 2} {GM_D\over {R_D}} (3.2 x)^2 (I_0K_0-I_1K_1) $$ where $$G $$ is the gravitational constant, $$x=r/R_{opt}$$ and $$I_n$$ and $$K_n$$ are the modified Bessel functions computed at  1.6 x.

The dark matter is described by a spherical halo with a Salucci & Burkert (2000) density profile

\rho(r)={\rho_0\, r_0^3 \over (r+r_0)\,(r^2+r_0^2)}~ $$ where $$r_0$$ is the core radius and $$\rho_0$$ the effective central density and

V_{URCH}^2 (r) = {G\, M_{H}(<r)\over r} \ = \ 6.4 \ G \ {\rho_0 r_0^3\over r} \Big\{ ln \Big( 1 + \frac{r}{r_0} \Big) - \tan^{-1} \Big( \frac{r}{r_0} \Big) +{1\over {2}} ln \Big[ 1 +\Big(\frac{r}{r_0} \Big)^2 \Big] \Big\}~. $$. The free parameters of the URC are obtained by fitting, with the above functions, co-added and individual rotation curves. They are found related with the halo mass $$ M_{vir} $$ and disk mass $$ M_D$$ by:



log {\rho_0\over g~cm^{-3}} = -23.5-0.96  M_D^{0.31} \ ; \ \ log{(r_0/kpc)} = 0.66+0.58 \, log  M_{vir} ~ $$

M_D = 2.3 \times  10^{10}  M_\odot \frac {(M_{vir}/3) ^{3.1}}{1+(M_{vir}/3 )^{2.2}} $$ All masses are measured in units of $$ 10^{11} M_\odot  $$ The above equations define the analytical form of the URC and make us able to predict the RC of a galaxy from its disk mass (or halo mass) and disk lenght-scale (see also ).The morphology of the 3-Dimensional URC surface can be easily  seen in a movie .The URC shows that the DM halos and stellar disks are both, separately,  self-similar, but the whole system is not: the collapse of baryons that formed the luminous part of spirals has affected their innermost 10\% of the halo size. The maximum value of the RC, for galaxies of different masses, occurs at very different radii, viz. at 2$$R_D$$ for the most massive objects and at 10$$R_D$$ for the least massive ones. Remarkably, just one global quantity takes into account for more than 90% of the variance of RC of Spirals. The URC phenomenology replaces the “flat RC” paradigm and switches the focus from the structure of “a typical galaxy” to the typical systematics of the mass structure of spirals. We do not see any “cosmic conspiracy”, i.e. any fine-tuning among the quantities describing the DM/LM distributions (the halo core radius and the central density and the disk mass) which creates  a flattish RC profile but  strong relationships that lead to a complex and well order family of profiles.