Explaining Spiral Galaxy Rotation Speed with or without Dark Matter

A spiral galaxy is flat and so may not need dark matter to explain constant rotational speed.

This is a continuation of previous posts on the gravitational potential  $\phi (x)$ as primordial giving mass density $\rho (x)$ to matter by the specification $\rho =\Delta\phi$ where $\Delta$ is the Laplacian differential operator with respect to a space coordinate $x$ of dimension $d$. 

If $d=3$ then $\phi (x)=\frac{1}{4\pi\vert x\vert}$ generates a unit point mass at $x$. The corresponding gravitational force is given by $\nabla\phi (x)\sim -\frac{1}{\vert x\vert^2}$. Balancing inertial and gravitational force for a body moving around the unit point mass in a circle with radius $R$ with speed $V$ gives the following relation:

• $V \sim \frac{1}{\sqrt{R}}$    (1)

If $d=2$ then $\phi (x)=-\frac{1}{2\pi}\log{\vert x\vert}$ generates a unit point mass at $x$. The corresponding gravitational force is given by $\nabla\phi (x)\sim -\frac{1}{\vert x\vert}$. Balancing inertial and gravitational force for a body moving around the point mass in a circle with radius $R$ with speed $V$ gives the following relation:

• $V \sim 1$      (2)

We see a speed decay with increasing $R$ of $V\sim\frac{1}{\sqrt{R}}$ if $d=3$, while $V=constant$ independent of $R$ if $d=2$. 

Observations of the velocities of spiral galaxies agree with (2), which has a logic because spiral galaxies are flat disc-like connecting to $d=2$, see figure above. 

The reason planets and stars are spherical is that gravitation is balanced by pressure acting in all directions. The reason galaxies are flat is that gravitation is balanced by rotational inertial forces.  

The motivation to introduce dark matter is that observations do not agree with (3), asking for a spherical halo of dark matter to give extra gravitational force to motivate non-decay of speed. But no dark matter has not been found. 

The explanation of agreement with observations with $d=2$ is elementary, and is also made here. But how is it possible that this has escaped the minds of modern physicists? Or has it been discovered as being too simplistic? Is a spiral galaxy really a 2d object? Does the specification of mass density according to $\rho =\Delta\phi$ really take place in 2d? The creation process of mass density from gravitational potential may carry an answer.

Note that $d=2$ gives a form of MOND as Modified Newtonian Dynamics, which with the gravitational potential being primordial perhaps can be motivated as a geometrical effect (to be returned to). Or it is simply that if $\Delta\phi (x)$ is not intense enough, then $\rho (x)$ is not visible as dark matter.  More in the next post.

Also recall that with the gravitational potential being primordial, there is neither instant action at distance to explain, only local action.

Newtonian gravity may be seen as a form a priori knowledge in the sense of Kant as knowledge which can be deduced without experimental input. It lives up to Einstein's ideal of a mathematical model without parameter to determine by experiment. 

Compare with Euler CFD as parameter-free model of fluid flow.