John J. Driscoll
Abstract. Presented herein is a mathematical procedure using plain old newtonian dynamics (POND) in
determining the density distribution and the mass of spiral galaxies. POND's mathematical approach is to rely
solely on newtonian dynamics in explaining the flattening of the disk's rotational velocity—rather then
assuming that "dark matter" or a change in Newton's gravitational constant (MOND) is responsible. We will
delineate the differences in applying newtonian dynamics when it comes to spherical and spiral galaxies. The
application of POND is tested by using M31 rotational curve data compiled by A. Klypin et al, 2002.
Introduction. A lengthy effort was made using the internet, books, and published articles, to find some
information pertaining to how the distribution of matter in spiral galaxies affects their rotational velocities. The
search proved fruitless—not that such information may not exist, but it is just hard to find. In any case there
seems to be a consensus view that the flattening of the velocity curves should not be happening, and, according
to Newton's law, the velocity curves, after reaching a maximum value, should fall off inversely proportional to
r1/2. Since this is not happening, we suspect that spherical and spiral galaxies are dynamically different, and
enough so, that Newton's equation,
and its more conventional form,
may not be the appropriate equation for determining the mass of spiral galaxies, although it may be quite
appropriate for spherical galaxies and star clusters.
Equation 2 fits in with Newton's first theorem that if the earth was a hollow sphere, the gravitational potential
P(r) inside the earth would be zero. His second theorem simply states that outside the sphere, the gravitational force acts as if the earth's mass is concentrated at its center. This leads to
the concept of concentric spheres where,
at a given radius, those spheres within
the radius provide the gravitational force,
while the spheres outside the radius
provide no gravity at all.
Figure 1 shows
the results of applying Equation 2 using
velocity curve data from Figure 4 to
obtain the accumulative mass of M31.
Note the negative slopes at 5 kpc and 20
kpc, implying negative density, which is
impossible considering that M31 is assumed to be surrounded by a halo of dark matter. The negative slopes give
cause for the reconsideration of Equation 2, as useful, when applied to spiral galaxies that may not have a halo
of dark matter. It would appear from the above difficulties that a different analytical approach is required.
Consider Figure 2 where, unlike Newton's hollow spheres, we have a ring (or a hoop) that represents a hollow
disk in which the gravitation potential P(x) is influenced by the mass C1 and C2 at a distance of r1 and r2,
respectively. Using proportions, we have
where, unlike the hollow sphere, P(x) is equal to zero only when r1= r2.
Combining the above equations into Equation 3, we have
Note that the negative sign indicates that the
gravity force is towards C2. Also note that the
dominant term (b-x) approaches zero as x
increases, causing P(x) to increase abruptly near
the inside of the ring. In the case of a disk, made
up of concentric rings, it is apparent from
Equation 4, that P(x) is primarily affected by the
rings immediately adjacent to it, and less so by
those that are remote. Is there a theorem in here
someplace? If there is, it would be that Newton's
first and second theorems do not apply to spiral
The idea of concentric rings (or bands) is the basis of the analytical method of being able to determine the density distribution and the mass of spiral galaxies. The number of bands involved is equal to the number of objects mob that are used, who's radius and velocity are known. The general equation for the force Fob operating on mob is the summation of all the gravitational forces that the bands contribute:
where n = equals the number of bands, and m = 1, 2, 3, etc.
Derivations. Figure 3 shows two diagrams: the one on the right is a bird’s eye view of a disk divided into two
bands of matter. At point A, all the matter to the left is pulling A towards the center, while the shaded area is the matter that is pulling A from the center.
Knowing the velocities and radii of A and B, the densities of band 1 and
band 2 can be determined, as the following will show.
The diagram on the left has the disk revolving in the x-y plane where z represents the disk thickness. The mass
element dM is located by b, z, and angle θ. The object mob is located on the x-axis, a distance r from the center.
The element dM applies a force dF acting upon mob in the direction of vector c. The intent is to sum the element
dM for each band, and to resolve the direction of the net gravity force to the x-axis.
Differentiating Equation 1 and replacing r with c, we have
To resolve the force to the x-axis, both sides are multiplied by cos α.
and since cos α=h/c, we have
and then combining Equations 8, 9, 10, and 11, we have
Finally, with the insertion of
we have the integrand:
where bi and bo are the inside and outside radii of a band; and where the subscript ob denotes an object's mass, orbiting radius, and velocity. Equation 14 is used to solve for the volumetric density ρi. On the other hand, if z is
unknown, or if only surface density is desired, then the following equation can be used:
In order to avoid the infinity problem when b = r and θ = zero, there must be a gap between the bands
wherever mob is located. To show how all this works, examples are illustrated below.
Examples. The right hand sketch of Figure 3 is the basis of the first example where VA = VB = 2, rA = 1.4,
and rB =2.5. With a clearance of .1, the band radii are 0 to 1.3 for band 1, and 1.5 to 2.4 for band 2. G is equal to
1. Since there are two unknowns in ρi, two applications of Equation 5 are required which form a matrix: each
row cantains an object's velocity and radius, and each column contains a band's area.
For object A, we have
And for object B, we have
The intermediate solutions are
a 2 x 2 matrix, the solution of which is shown in Table 1.
Once the densities have been determined, the mass is readily found. The conventional procedure of employing
Equation 2 to determine the mass of a disk results in a mass of 10, almost twice as much as in the above
Using these data points, an 8 x 8 matrix was formed and solved as in the previous example; except in this case
Equation 14 was used to obtain the volumetric density. To this end, assumptions were made in regard to the
galaxy's profile: the center spheroid was replaced with a cylinder 2000 pc thick and 3600 pc in diameter, and
the disk taper was replaced with a constant thickness of 400 pc. These fixed dimensions were necessary because
the software was unable to handle non-numerical limits of integration. A clearance of 2 pc was used between an object and a band. The 2 pc was selected primarily for mathematical convenience. The intermediate solutions
are not included in this paper. The results of the 8 x 8 matrix are tabulated in Table 2.
The tabulated data, density and mass, are shown in Figures 5 and 6. In Figure 5 the densities of the various
bands (identified by number)
show that, except for band
3, the density exponentially
decreases as the radius
increases. The low density
at band 3 shows the
sensitivity of POND where
the rotational velocity of
M31 is at its lowest. Figure
6 shows the comparison of
M31's accumulated mass:
the dark line is obtained by
using M=V²r/G, and the
dashed line is obtained by
using POND. Note that there is no suggestion of negative densities by the POND curve. Also note that the total mass, as derived by
POND, is about half of that derived by M=V²r/G.
Figure 7 shows that Equation 5
can be used to determine the
velocities of satellites orbiting
far beyond M31's galactic
visible plane. The black dots
of curve A are the data points
selected for the previous
example. The open dots are the
computed velocities of
assumed satellites at radii of
50, 75, 100 kpc. Curve B is the
computed velocity using
V=(GM/r)1/2, based on the
total mass of 230 billion
solars as derived by POND.
Note that curves A and B
mesh at about 100 kpc
indicating that the affect of
POND disappears at three
times M31's radius.
If satellites are found beyond
the visible plane whose
velocities and radii are
known to be true and are in
conflict with the assumed satellites in Figure 7, their incorporation in POND by increasing the size of the matrix would provide the
densities of the unseen mater beyond M31's visible radius of 34000 pc.
Discussion. There is not a whole lot to discuss. If critical reviews sends POND to the lost files, then so be
it; on the other hand, if POND has legs, what would be its impact? First off, it puts a huge dent in the theory of
dark matter as a halo about a spiral galaxy; however, if POND finds the existence of matter, and its density,
beyond the visual boundaries of a disk, then one could conclude that dark matter exists but only in the disk's
plane. Secondly, POND, depending upon data accuracy, can establish the density profile of a disk that would
help firm up the relationship between surface luminosity and the disk's matter. Thirdly, POND might be the
justification that MOND is looking for to firm up its equations that are empirical in their structure. And finally,
it is recognized that the mass of 230 billion solars for M31 flies in the face of others who have evidence that its mass exceeds a trillion solars!