As an alternative to dark matter models , MOdified Gravity ( MOG ) theory is a covariant modification of Einstein gravity .
The theory introduces two additional scalar fields and one vector field .
The aim is to explain the dynamics of astronomical systems based only on their baryonic matter .
The effect of the vector field in the theory resembles a Lorentz force where each particle has a charge proportional to its inertial mass .
The weak field approximation of MOG is derived by perturbing the metric and the fields around Minkowski space–time .
We obtain an effective gravitational potential which yields the Newtonian attractive force plus a repulsive Yukawa force .
This potential , in addition to the Newtonian gravitational constant , G _ { N } , has two additional constant parameters \alpha and \mu .
We use the THe HI Nearby Galaxy Survey catalogue of galaxies and fix the two parameters \alpha and \mu of the theory to be \alpha = 8.89 \pm 0.34 and \mu = 0.042 \pm 0.004 ~ { } { kpc } ^ { -1 } .
We then apply the effective potential with the fixed universal parameters to the Ursa-Major catalogue of galaxies and obtain good fits to galaxy rotation curve data with an average value of \overline { \chi ^ { 2 } } = 1.07 .
In the fitting process , only the stellar mass-to-light ratio ( M / L ) of the galaxies is a free parameter .
As predictions of MOG , our derived M / L is shown to be correlated with the colour of galaxies .
We also fit the Tully-Fisher relation for galaxies .
As an alternative to dark matter , introducing an effective weak field potential for MOG opens a new window to the astrophysical applications of the theory .

keywords : gravitation , galaxies : Spiral , galaxies : Kinematics and Dynamics , cosmology : dark matter , cosmology : theory