Comment on the use of Kepler's laws
to describe galactic rotation

L. Marmet, C.F. Gallo

Abstract

  When the dynamics of a spiral galaxy is modeled from Keplerian mechanics, one finds that an unusual mass distribution is required to explain that stars revolve at a constant speed over a large range of distances from the center of the galaxy. While the Kepler's laws describe the rotation of planets around the sun, they cannot be directly applied to the dynamics of spiral galaxies. The incompatibility arises because the gravitational force outside a non-spherical mass distribution is not described by an inverse square law dependence on the radius. The conclusion resulting from the Keplerian model - that the cumulative mass increases linearly with the radius - is inaccurate.

Keywords: galaxy rotation curve, spiral galaxy, Keplerian dynamics, homeoid

  The 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/r2 = V2/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 GMr = rV2, where Mr 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 Mr 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 r 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) = V2/r, where f() is a function involving elliptic integrals over the mass density M(r).  As a result, the cumulative mass of an elliptical galaxy does not rise linearly with radius.

  Although 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 W2r with a centrally symmetric gravitational acceleration Gm/r2. 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

  The authors wish to thank D. Bernier for commenting this paper.

References

[1]
  
V. Rubin, W. K. Ford Jr., and N. Thonnard, “Rotational Properties of 21 Sc Galaxies with a Large Range of Luminosities and Radii from NGC 4605 (R=4kpc) to UGC 2885 (R=122kpc),” Astrophysical Journal 238, p. 471, June 1980.
[2]
  
J. Binney and S. Tremaine, Galactic Dynamics, Princeton University Press, Princeton, 1987.
[3]
  
D. H. Eckhardt and J. L. G. Pestaña, “Technique For Modeling The Gravitational Field Of A Galactic Disk,” Astrophysical Journal 572, pp. L135–L137, May 22 2002.
[4]
  
V. Rubin, “Seeing dark matter in the andromeda galaxy,” Physics Today 12, pp. 8–9, December 2006.
[5]
  
V. Rubin, “A two-way galaxy,” Physics Today 19, pp. 8–9, September 2007.

Copyright September 30th, 2007.  Version dated November 26th, 2007.