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As an alternative to dark matter models, MOdified Gravity (MOG) theory cancompensate for dark matter by a covariant modification of Einstein gravity. Thetheory introduces two additional scalar fields and one vector field. The aim isto explain the dynamics of astronomical systems based only on their baryonicmatter. The effect of the vector field in the theory resembles a Lorentz forcewhere each mass has a charge proportional to the inertial mass. In this work,we obtain the weak field approximation of MOG by perturbing the metric and thefields around Minkowski space-time. We derive an effective gravitationalpotential which yields the Newtonian attractive force plus a repulsive Yukawaforce. This potential, in addition to the Newtonian gravitational constant,$G_N$, has two additional constant parameters $\alpha$ and $\mu$. We use theTHINGS catalog of galaxies and fix the two parameters $\alpha$ and $\mu$ of thetheory to be $\alpha =8.89 \pm 0.34$ and $\mu =0.04 \pm 0.004 {\rm kpc}^{-1}$.We then apply the effective potential with the fixed universal parameters tothe Ursa-Major catalog of galaxies and obtain good fits to galaxy rotationcurve data with an average value of $\bar{\chi^2} = 1.07 $. In the fittingprocess, only the stellar mass-to-light ratio $(M/L)$ of the galaxies is a freeparameter. As predictions of MOG, our derived $M/L$ is shown to be correlatedwith the color of galaxies, and we fit the Tully-Fisher relation for galaxies.As an alternative to dark matter, introducing an effective weak field potentialfor MOG opens a new window to the astrophysical applications of the theory.

The MOG weak field approximation and observational test of galaxy rotation curves