Flatness of the rotation curves of the galaxies
Exit the recourse to a massive halo
Here the summary, in English of a paper, presenting
a work made with D. Méra (Cral, ENS Lyon) and J.-B. Baillon (IGD):
We use a new original method to calculate
the surface density of a galaxy from its rotation curve assuming that all
the mass is in the galactic plane. We show that this method gives
accurate masses for the disk. Moreover, given a rotation curve for
r less than Rg only (Rg is the galaxy radius), we find that the surface
density is nearly uniquely determined except near the edge of the
disk. We also demonstrate that the common assumption that a flat rotation curve
leads to a density profile Sigma(R) as 1/R is false for a finite disk.
We derive for the MilkyWay
a mass of 1.4 10^11 solar mass. The
exponential profile is well reproduced with a scale length between 3.6 and
5 kpc. The local surface density is found to be 125 +/- 10 solar mass/pc^2,
compatible with other independant determinations.
Click
here
to obtain the paper (1996). A new presentation (2007)
and a chapter a book (en français) about the problem
of the dark matter (2003). Program (2014) with maple18.
Update, july 2017 new program and many references.
Presentation :
The rotation curves of galaxies,
such as they are observed, remain flat to large distances,
say from 10 to 15 kpc center of a usual spiral
galaxy which one studies the disk speeds of the spiral arms.
This fact raises a rather
crucial question, and the only explanation given for the moment, is
to admit the existence of a massive spherical halo of matter
including such a galaxy.
This problem constituted a enigma opening
the way with an active research of " hidden mass"
in the universe.
We show that:
i) Within a Newtonian framework, the phenomenon of flatness of the curves
is very simply explained
without recourse to a possible massive halo and the proof
shows why there is in fact less " hidden mass ".
ii) The Einsteinian study of these curves changes
the manner of tackling this problem of hidden mass coming from
rotation curves of the galaxies.
This can seem surprising, insofar as the gravitational field
created by a galaxy is very weak, but on such
a scale the Einsteinian correction with the usual Newtonian gravitation
proves to be non negligeable.
On the existing methods to study the rotation curves :
An excellent presentation of the three existing methods are explained
in the work of Binney and Tremaine entitled " Galactic Dynamics ".
These three methods have as names,
the method of the elliptic integrals (based on the disc seen like
rings of matter), the method of the Bessel transform
(based on asymptotic properties of the gravitational potential
created by the disc of matter) and finally the method of
the sphéroides, most used (based on the disc seen like limit of
thin ellipsoids). The common point of these three methods is the fact
that, on the basis of a surface density, the disk speed is expressed by
a double integral which is not absolutely convergent. Thus these three
methods deducing a rotation curve from a surface
density (presumedly known) do not make it possible
to solve the initial problem which is to deduce
the distribution of masses starting from
the observed rotation curve (except in academic cases by knowledge
of indefinite integrals for example).
Our direct method; comparison with the preceding ones:
It rests on the simulation of a spiral galaxy by a disc
of radius R made of N massive bodies distributed with an axial
symmetry. Taking into account that these bodies turn while following a curve
of rotation given, then the balance of the radial forces between N body
leads to a set of equations: it is a linear
system (of which the unknown factors are the masses of N body) that
one reverses and which thus provides the surface
density curve. In fact it is a " inverse method of the N-bodies
problem", which
give results very precise and easy to implement
(the curves presented rest on a simulation by 40 000
points of a galactic disc).
Our method is a riemanian approximation
of the double integral coming from the method which uses
the elliptic integrals, it is thus theoretically equivalent to it.
To establish equivalence between the method of the Bessel transform
and that of the elliptic integrals, I have permuted the integrals and
introduces principal values (the double integrals being not
absolutely convergent), then to transform Bessel functions on the one hand
and elliptic integrals on the other hand into hypergeometric series.
Our method thus gives theoretically the same results as
the method of the elliptic integrals and than that of
the Bessel transform. As this last method is
theoretically invertible, our method provides a surface
density whatever the curve of rotation given, and this
without recourse to a massive halo.
It is obviously always possible to add a light halo, our
method spreads without sorrow. But
the results are different
from those coming from the usually used method, that of the
thin ellipsoids (which implies
the need for a massive halo). Cf. "Towars a consistent
model of the galaxy : Derivation of the model";
A. & A. vol 330, 1998, from Mera, Chabrier and Schaeffer.
: this page was writtened in 1997, but now (februar 2006) two papers (using Bessel functions)
obtain the same results (without the need of a massive halo) :
Burkhard Fuchs, Asmus Boehm, Claus Moellenhoff, Bodo L. Ziegler
:
Quantitative interpretation of the
rotation curves of spiral galaxies at redshifts z~0.7 and z~1.
F.I. Cooperstock, S. Tieu :
Perspectives on Galactic
Dynamics via General Relativity.