to describe galactic rotation

## AbstractKeywords: galaxy rotation curve, spiral galaxy, Keplerian
dynamics, homeoidThe dynamics of spiral galaxies is often modeled with Keplerian mechanics to show that the measured flat rotation curves imply that the cumulative mass increases linearly with the radius. The argument is as follows: consider the equation of motion GM/r ^{2}
= V^{2}/r
of a small mass in the gravitational field of a small central object of
mass M, where r is the radius of the
circular orbit, V the tangential
velocity, and G the gravitational constant. This equation is a
limiting case of Kepler's third law for circular orbits, such as those
observed for planets orbiting
the sun. Consider next two properties of spherical mass
distributions: 1-a uniform
spherical shell does not exert any gravitational force on a mass inside
the shell, and 2-the gravitational field outside the shell is as if the
mass of the shell were concentrated at its center. Using both
properties, the equation of motion can be simplified to GM_{r}
= rV^{2}, where M_{r} is the cumulative mass interior
to r. It is known experimentally, after the
work pioneered by Rubin[1],
that most spiral galaxies exhibit nearly constant rotation curves, that
is, V is independent of r. The simplified
equation leads to the conclusion that M_{r}
is proportional to r.Such a mass distribution is unusual because it is much larger than what is detected optically. There is also a problem of divergence if one wants to calculate the total mass of a galaxy. In an attempt to better describe the geometry of a galaxy, the problem has been studied more precisely using flattened shells of uniform density between similar, concentric ellipsoids, called homeoids[2]. A mass located inside a homeoid experiences no net gravitational force from that homeoid. As a result, the gravitational force at a point is
proportional to the mass of all homeoids interior
to r, as is the case in the spherically symmetric
problem. It is
therefore tempting to use this property to obtain the simplified
formula given above. However, the simplification cannot be done
since spherical shells do not share their second property with
homeoids. The gravitational force outside a non-spherical homeoid
is not described by an inverse square law. The equation
describing the motion of a star in a galaxy must be of the form Gf(M(r),
r) = Vr^{2}/r,
where f() is a function involving elliptic
integrals over the mass density M(). As a result,
the cumulative mass of an elliptical galaxy does not rise linearly with
radius.rAlthough homeoids describe the mass distribution of a spiral galaxy more precisely than spherical distributions, they are still inaccurate because of at least two reasons. The shape of a homeoid is the result of a rigid rotation at the angular velocity W, which gives the velocity V(r) = Wr. This does not correspond to the rotation of a spiral galaxy. Also, if the mass density is not constant, a homeoid is the result from the balance of the centrifugal acceleration W ^{2}r
with a centrally symmetric gravitational acceleration Gm/r^{2}.
This is not the gravitational acceleration of a spiral galaxy.
Since homeoids cannot generally be used to describe the shape of
galaxies, numerical methods are the best alternative to replace
analytical functions. The gravitational field of disk galaxies is
relatively easy to calculate using ring shaped distributions[3].
In this case, even the first property of a spherical mass
distribution does not hold: a ring distribution can exert a
gravitational force on a mass localized inside the ring[2]. This
implies that for spiral galaxies, the cumulative mass does not rise
linearly with radius.The Keplerian model is frequently used in the scientific literature as an argument to support the hypothesis of dark matter in spiral galaxies[4, 5]. However, an appropriate calculation method must be used instead of the simple and inaccurate model. The excellent data on rotation curves of galaxies and high mass-to-luminosity ratios provide the main arguments in favour of “matter that has no light”. ## Acknowledgement## References
Copyright September 30th, 2007. Version dated November 26th, 2007. |