Mass  Distribution Model

When, in the late 1960s, spectrographic analysis of spiral galaxy rotation first started, it was generally expected this rotation would still comply to the gravitational laws that so successfully describe the motion of the planets in the solar system. However, even a cursory review of the distribution of the mass within our own local solar system shows that 98.8% of total mass is centrally located within the Sun. As such, this highly centralized system is a good fit to the classical predictions of Newtonian gravity based on the inverse square law. So the obvious question to start with is:

Can this system model be applied to a galaxy?

Clearly, based on observation of the rotational velocity within galaxies, the answer seems to be an emphatic – no, which then leads to the question – why not? In part, we might refer to a modification of an earlier diagram that outlined what might be called the inner and outer laws of Newtonian gravity on to which some of the basic galactic structure is superimposed.

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While the planets are collocated with a star within the galaxy, the planetary orbits are dominated by the central mass of the star and therefore can be described by the outer solution, i.e. an inverse square law. However, the situation for a star is more complicated when considered in terms of the actual galactic rotation curve for the spiral galaxy NGC 2841 shown below, where the initial linear rise of the curve is still possibly reflective of the inner solution, but then flattens out suggesting a mass distribution that cannot be explained by either the inner or outer Newtonian solutions. However, as the colour coding of the various regions within a galaxy in the diagram above is trying to illustrate, the mass density of a galaxy is far from homogeneous.

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So the flatten region of this rotation curve appears to require some other explanation, such as being proposed by the dark matter or MOND hypotheses, although other possibilities may still be entertained. For example, examination of the general diagram below reflects the fact that most galaxies are far more disk-shaped than spherical, such that the geometry assumed in Newton’s shell theorem has to be re-evaluated.

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So before jumping to the conclusion that there must be some form of missing mass that cannot be observed or a need to modify the Newtonian laws of gravity, outside general relativity, it might not be unreasonable to table the next question:

Has the mass distribution within the galactic model been fully considered?

As outlined, Newton’s shell theorem allows us to examine some of the implications of gravity based on a spherical geometry. In this context, the external solution shows why the inverse square law is defined by the distance between the centre of the masses [M,m] and not the distance between the surfaces. In contrast, the internal solution suggests that a mass [m] within a spherical homogeneous density would only be subjected to a gravitational force defined by the mass inside its radius [r]. In other words, the mass implied by larger radii can be ignored. By looking at the rotation curve for NGC-2841 above, it is clear that neither the inner or outer solutions can dominate at all radii. As such, the linear relationship between the gravitational mass [M=ρV] with radius [r] does not hold true for all radii [r], as per the inner solution, while the failure to comply with the inverse square law of the outer solution suggests that the gravitation mass [M] associated with radii larger than [r] may also be having an effect.

So is there Newtonian solution that matches this mass-geometry of a galaxy?

It would seem that quite a few people have considered this problem and come to the conclusion that based on a re-assessment of the mass density distributed within various galaxies, basic Newtonian gravity formulations can explain the observed rotation curves without modification or dark matter. The following references and extracts are just a few examples, which the reader might like to review in more detail.

  •  Disk-Galaxy Density Distribution from Orbital Speeds using Newton's Law
    This paper was published in 2000 by Kenneth Nicholson in which he outlines his assumptions and method for determining the mass distribution within a galaxy and the resulting rotation curves. The following extract is taken from his conclusion:

  • “Newton's law needs no correction, and dark-matter halos are not needed, to compute galaxy density distributions from orbital speeds. Using measured speeds and defined dimensions, the results for the density and mass distributions using the methods shown here will be as good as the inputs. The use of spherical shells or Poisson's equation are not suitable for this purpose. There is no mystery about orbital speeds being constant to large radii, or rising near galaxy rims”.

  •  Galactic mass distribution without dark matter or modified Newtonian mechanics
    In an updated 2007 version of Nicholson’s earlier work in which he appears to remain convinced of his original conclusion, e.g.

  • “and so far there has been no evidence of the spherical shells, called dark matter, except that they can be used mathematically to get an approximate match for the measured rotation speeds. The statement often made that the shells are necessary to satisfy the gravitational effects for galaxies is also false”
     
  • Fully model disk galaxy rotation curves without dark matter
    This paper was published in 2008 by Dilip Banhatti. Again, this author outlines his assumptions and methods for determining the mass distribution and rotation curves of galaxies, while acknowledging the work of Kenneth Nicholson:

  • Researchers interested in rotation curves need to answer certain questions afresh with an open mind. Instead of discounting Kenneth Nicholson’s model just because it is different from the current fashion, it needs to be judged on its own merits.

  • A Newtonian Model for Spiral Galaxy Rotation Curves
    This paper was written by Geoffrey Williams, in 2008 or shortly after, and while appearing to use a different methodology to determine the mass distribution, he still reaches the same conclusion as the previous authors.
     
  • A spiral galaxy is modelled as a thin flat, axisymmetric disk comprising a series of concentric, coplanar rings. Using conventional Newtonian kinematics, it is shown that relatively flat velocity curves are produced by a variety of possible mass distributions in the disk. No halo of ‘dark matter’ is needed to produce these rotation curves. Compared with the point mass at the centre, the disk gravitational force grows with increasing distance from the disk  centre, crests and the slowly subsides beyond the  disk  perimeter.

  • Rotating thin-disk galaxies through the eyes of Newton
    This paper was written by James Feng and C. Gallo and published in 2011. This paper appears to use quite sophisticated computational techniques to underpin their calculations. The first extract is taken from the paper’s abstract.

  • “By numerically solving the mass distribution in a rotating disk based on Newton's laws of motion and gravitation, we demonstrate that the observed rotation curves for most spiral galaxies correspond to exponentially decreasing mass density from galactic center for the most of the part except within the central core and near periphery edge. Hence, we believe the galaxies described with our model are consistent with that seen through the eyes of Newton. Although Newton's laws and Kepler's laws seem to yield the same results when they are applied to the planets in the solar system, they are shown to lead to quite different results when describing the stellar dynamics in disk galaxies. This is because that Keplerian dynamics may be equivalent to Newtonian dynamics for only special circumstances, but not generally for all the cases. Thus, the conclusions drawn from calculations based on Keplerian dynamics are often likely to be erroneous when used to describe rotating disk galaxies.”

    The following extract taken from the conclusion and appears to support the general assumptions of a Newtonian mass distribution model.

    “As demonstrated, substituting the computed mass density distribution based on Newtonian dynamics into the Keplerian force-balance equation would yield a rotation curve with orbital velocity decreasing toward galactic periphery. This had led many authors to believe that the visible mass in a galaxy cannot explain the observed rotation curve. Therefore, some authors have speculated that some kind of invisible matter called dark matter must exist in the galaxy. Other authors believed modification of Newton's laws to be needed. The fundamental problem here is that astronomers tend to determine the `visible' mass in a galaxy from the measured brightness based on an over-simplified mass-to-light ratio. When the mass distribution so estimated did not generate the observed at rotation curve, especially when the Keplerian dynamics was used for another over-simplification, it is often referred to as the `galactic rotation problem' suggesting that there is a discrepancy between the observed rotation speed of matter in the disk and the predictions of Newtonian dynamics. But we believe that Newtonian dynamics can adequately explain the stellar dynamics in disk galaxies when applied correctly; neither the introduction of dark matter nor modification of Newtonian dynamics is needed for explaining the observed rotation curves.”

Now it is not being suggested that these papers provide conclusive proof that the model being described is ‘the’ explanation of the observed galactic rotation curves, but they seem to suggest an explanation worthy of some consideration, especially in terms of the somewhat esoteric alternatives that are also being reviewed.

Note: What is not clear in the mass distribution models is whether there is any explicit mechanism that explains the orbital velocity within the galaxy, i.e. how did the spiral arms acquire the necessary angular momentum to maintain an orbital velocity to counteract the inward pull of gravity? In part, the next model may offer an alternative explanation.