5.2 Galactic Kepler-like laws of motion

As a conclusion, all the apparently independent roles that the characteristic acceleration a0 plays in the unpredicted observations of Section 4.3 (see end of Section 4.3.3 for a summary), as well as Renzo’s rule (Section 4.3.4), have been elegantly unified by the single law proposed by Milgrom [293Jump To The Next Citation Point] in 1983 as a unique scaling relation between the gravitational field generated by observed baryons and the total observed gravitational force in galaxies.

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Figure 14: The mass discrepancy (as in Figure 10View Image) as a function of radius in observed spiral galaxies. The curves for individual galaxies (lines) are color-coded by their characteristic baryonic surface density (as in Figure 5View Image). In order to be completely empirical and fully independent of any assumption such as maximum disk, stellar masses have been estimated with population synthesis models [42Jump To The Next Citation Point]. The amplitude of the mass discrepancy is initially small in high–surface-density galaxies, and grows only slowly at large radii. As the baryonic surface densities of galaxies decline, the mass discrepancy becomes more severe and appears at smaller radii. This trend confirms one of the predictions of Milgrom’s law [294Jump To The Next Citation Point].
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Figure 15: The shapes of observed rotation curves depend on baryonic surface density (color coding as per Figure 14View Image). High–surface-density galaxies have rotation curves that rise steeply then become flat, or even fall somewhat to the asymptotic flat velocity. Low–surface-density galaxies have rotation curves that rise slowly to the asymptotic flat velocity. This trend confirms one of the predictions of Milgrom’s law [294Jump To The Next Citation Point].
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Figure 16: Centripetal acceleration as a function of radius and surface density (color coding as per Figure 14View Image). The critical acceleration a0 is denoted by the dotted line. Milgrom’s formula predicts that acceleration should decline with baryonic surface density, as observed. Moreover, high–surface-density galaxies transition from the Newtonian regime at small radii to the weak-field regime at large radii, whereas low–surface-density galaxies fall entirely in the regime of low acceleration a < a0, as anticipated by Milgrom [294Jump To The Next Citation Point].

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