Newtonian Fractional-Dimension Gravity and the Mass-Dimension Field Equation

We resume our analysis of Newtonian Fractional-Dimension Gravity (NFDG), an alternative gravitational model which does not require the Dark Matter (DM) paradigm. We add three more galaxies (NGC 6946, NGC 3198, NGC 2841) to the catalog of those studied with NFDG methods. Once again, NFDG can successfully reproduce the observed rotation curves by using a variable fractional dimension $D\left (R\right )$, as was done for nine other galaxies previously studied with these methods. In addition, we introduce a mass-dimension field equation for our model, which is capable of deriving the fractional mass-dimension $D_{m}\left( R \right)$ from first principles, as opposed to the previous $D\left (R\right )$ which was obtained simply by matching the experimental rotational velocity data for each galaxy. While the NFDG predictions computed with this new $D_{m}\left(R\right)$ dimension are not as accurate as those based on the original $D\left (R\right )$, they nevertheless confirm the validity of our fractional-dimension approach. Three previously studied galaxies (NGC 7814, NGC 6503, NGC 3741) were analyzed again with these new methods and their structure was confirmed to be free from any dark matter components.