Persic, Salucci and Stel
(1996)
and Salucci and Persic (1997)
have
analyzed a large number of rotation curves mainly catalogued by
Persic and Salucci (1995)
taking into account the *H* data published by
Mathewson, Ford and Buchhorn
(1992)
and also adopting some radio rotation
curves. They claim that rotation curves can be fitted to what they
call the "universal rotation curve" not only for any luminosity,
but also for any type of galaxy, including spirals, low-surface-brightness,
ellipticals and dwarf irregular galaxies. The existence of a
"universal" rotation curve had previously been claimed by
Rubin et al. (1980).
Let us then reproduce the
formulae of these "universal rotation curves", or PSS curves,
restricting ourselves to spirals.

Following
Persic and Salucci (1997),
rotation curves of spirals can be
fitted by a combination of two
components. The first is an exponential thin disk, whose circular
velocity can be approximated in the range
0.04*R*_{opt} < *R*2*R*_{opt} as

(3) |

where *x* is a radial variable taking *R*_{opt} as unit

(4) |

*R*_{opt} is the radius encircling 83% of the light; for an
exponential disk
*R*_{opt} = 3.2*R*_{D} where *R*_{D} is the radial scale
length. is a constant that depends on the luminosity. This
function *V*_{disk} does not give a Keplerian fall-off for
*x* , nor is it the general expression of *V*_{disk} for
exponential disks, but its application is restricted to a radial
range.

The other component is the halo, with a circular velocity expressed as

(5) |

where *a* is another constant, also depending on the
luminosity. Then, the PSS curve is given by

(6) |

the contribution of a bulge therefore being considered negligible. The
constants *a* and are functions of the galaxy's luminosity, the
best results being obtained for

(7) | |

(8) |

where
*L*_{*} = 10^{10.4}*L*_{}. Then for a galaxy with luminosity
*L*_{*}, *a* corresponds to a value of *x* of the order of
*R*_{opt}, exactly
1.5*R*_{opt}. Note that these values
provide good fits, but for
*L* > 4.33*L*_{*} give negative values of
*V*_{halo}^{2}.

Hence

(9) |

(for a spherically symmetric halo) and

(10) |

with very little in common with the NFW theoretical halos (see later). They are reasonable, because

- The x-derivative of (
*x*) vanishes at*x*= 0, which is physically satisfactory. - Then, for
*x**a*the density slowly decreases; for*x*=*a*, the density is still 1/3 of the central value, i.e. there is a "core" of radius*a*, which is therefore called the halo core radius. - The density does not vanish for any value of
*x*, i.e. there is no sharp boundary. The density always decreases. - For
*x**a*, the density decreases as*x*^{-2}(compared to*x*^{-3}in NFW halos).

This is reminiscent of the non-singular isothermal sphere, with a faster
decrease from the centre out to the core radius, both of which for large
*x* obey
*x*^{-2}.

As
*a* *L*^{1/5}, i.e. low luminosity galaxies are
much more concentrated. For a galaxy with *L*_{*} the core
is of the order of the optical radius.

A non-physical property of the PSS halo density profile
is that *M* does not converge for very large values of *x*,
but rather linearly increases with *x*, with the mass of any
halo being infinite. To surmount this difficulty the halo mass was
defined as that at
*R*_{200}, where *R*_{200} is the radius of a
sphere within which the
mean density is 200 times the mean density of the Universe, as also
defined in theoretical models. Then

(11) |

where is the critical density of the Universe. We see
therefore that
*M*_{200} *R*_{200}^{3}. Following equation (5),
*V*_{200} (the circular velocity at *R*_{200})
is a complicated
function of *R*_{200}, but according to these authors, it
can be approximated to

(12) |

(13) |

(Note, however, that from
*R*_{200} *L*^{0.2}, together with the exact relation
*M*_{200} *R*_{200}^{3}, we should obtain
*M*_{200} *L*^{0.6}. The small difference in these exponents -0.5 and 0.6-
arises from the complexity of the problem). Therefore, the brighter
galaxies have a halo that is more massive, but only slightly larger. The
mass-luminosity ratio is then

(14) |

Brighter galaxies have smaller mass-to-light ratios, hence the dark matter has more dominant effects in small or low-surface brightness galaxies. We can also calculate the luminous to dark matter ratio

(15) |

Bright galaxies have relatively smaller dark matter halos, while
the very bright galaxies nearly reach the maximum
*M*_{lum}/*M*_{dark} ratio
( 0.1) established from primordial nucleosynthesis models for
the baryonic .

For *x* *a*, a constant value of *V*_{halo} is obtained
*V*_{halo}^{2} =
*V*^{2}(*R*_{opt})(1 - )(1 + *a*^{2}). For galaxies with
*L* *L*_{*}, it is
obtained that
*V*_{200} *V*(*R*_{opt}) which is readily
interpreted: if a constant
*V*(*R*_{opt}) is observed in a region already
dominated by
dark matter, it should be related to the halo circular velocity at
large distances.

As
*V*^{2}_{200} =
*GM*_{200}/*R*_{200} *L*^{0.5}/*L*^{0.2} = *L*^{0.3}
we should have
*V*_{200} *L*^{0.15}. In binary galaxies, which
are considered later, *V*_{200} could be identified with
the orbital
velocity of the secondary galaxy, statistically related to the
difference of the two projected velocities along the
line-of-sight. *L* would be the luminosity of the primary. A
correlation between *L* and *V*_{200} has not been found
(e.g.
Zaritsky, 1997).
This is, in part, justified as the
exponent, 0.15, is so small that the orbital velocities are nearly
independent of the luminosities. Theoretical models also agree in
this respect.

Note that, if instead of
*M*_{200} *L*^{0.5} as determined by these authors, we
had taken
*M*_{200} *L*^{0.6} from the definition of
*M*_{200} as mentioned above, we would have obtained
*V*_{200} *L*^{0.2}, and
*L* *V*^{5}_{200}, closer to the
observational infrared Tully-Fisher relation
*L* *V*^{4}(*R*_{opt}), if
*V*(*R*_{opt}) were close to *V*_{200}.

White et al. (1983)
and Ashman (1992) proposed
*M*_{200}/*L* *L*^{-3/4}, in which case
*R*_{200} *L*^{0.08}, indicating a
lower dependence of *R*_{200} on *L*, and
*V*_{200} *L*^{0.08}.
We will see later that these undetected correlations have a
natural explanation in the magnetic model of the rotation curves.

The universal rotation curves also give two relations

(16) | |

(17) |

where is the critical density of the Universe. Hence, brighter galaxies have relatively large core radii and small values for the central halo density. Therefore, even the central region of low-brightness galaxies is dominated by dark matter, while bright galaxies have their internal regions dominated by visible matter. These relations are important and confirmed by the NFW theoretical halos, even if the universal rotation curves do not have much in common with those deduced by the former.

The same formulae are valid for low-surface-brightness galaxies. In this case, the dark matter would be completely dominant, with a core radius of about 5-6 kpc.

Summarizing, the most interesting fact in the fitting effort made by these authors is that such a large variety of galactic types have rotation curves which can be adjusted to a single universal rotation curve (even for ellipticals, not considered here). This fitting assumes the existence of a dark halo that does not coincide with the universal halo profiles obtained by most theoretical models but which is very reasonable (except, perhaps, in that they have an infinite mass, which in practice is not a real problem). The explanation of the puzzling behaviour of binary systems, however, is still not completely satisfactory.

There is another general comment to be made. The universal rotation curve is a fitting problem. But this fitting should be interpreted in other models in a different way. Therefore, even if terms like "dark matter" and "dark halos" are used, this fitting does not prove the existence of dark matter in galaxies.

Bosma (1998)
considered that the notion of universal rotation curves
breaks down. He observed several galaxies with a high rotation velocity,
but non-declining rotation curves. This could be due to the
inclusion in Persic and Salucci's sample of very inclined galaxies, where
opacity problems are difficult to handle when using *H*_{}
rotation curves.
Verheijen (1997)
also found 10 out of 30 galaxies in
the Ursa Major clusters for which the rotation curves do not fit
the universal rotation curves. Despite all these exceptions the
scheme introduced by Persic, Salucci and collaborators, provides a
first fit that theoretical models should take into account.