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Galaxies Rotation Curves in the Grumiller's Modified Gravity

The effective potential in the Grumiller's modified gravity [D. Grumiller, Phys. Rev. Lett {\bf 105}, 211303 (2010)] includes the Newtonian potential and a Rindler term. The fitting to the rotation curve data of eight galaxies suggests a universal Rindler acceleration $a\approx 0.30\times 10^{-10}$ m s$^{-2}$. We do a two-parameter fit first, with the mass-to-light ratio ($Υ_*$) and the Rindler acceleration ($a$) as free parameters. It is found that the data of six out of the eight galaxies fit well with the prediction of theory in the range $0\lesssim r \lesssim 40$ kpc, although the theoretical curves show a tendency of arising beyond this range. The Rindler accelerations of the six well-fitted galaxies have the same magnitude, with an average value $\bar{a}\approx0.30\times 10^{-10}$ m s$^{-2}$. Inspired by this fact, we then carry out a one-parameter ($Υ_*$) fit to the six galaxies, with $a$ fixed at $\bar{a}$, and find that the theory can still reproduce the observation. The value of Rindler acceleration we get here is a quarter of that of Milgrom's MOND. For the rest two galaxies, NGC5055 and DDO154, the fitting results are significantly improved if the photometric scale length ($h$) is included as another free parameter.