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 r^1/2. 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 the M31 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.
To expand Equation 3, set

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 galaxies.

where n = equals the number of bands, and m = 1, 2, 3, etc.
The next step is to define Fi.

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

Furthermore, since

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 example.

The second example is a more realistic test of POND: using the actual M31 rotational velocities from Figure 4. The eight encircled data points were the objects mob selected for this example:

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!

References.

1. A. Klypin and H.-S Zhao, whose compilation of M31 rotation curve that has been spread all over the internet.

2. Linda S. Sparke and John S. Gallagher, Galaxies in the Universe, Cambridge University Press 2002, pp. 96-100.


First printing: June 5, 2005

John J. Driscoll
johnjdris@sbcglobal.net





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