Given the Earth's mass to be $M_E$, I know that $F_G = -\frac{GM_Em}{d^2}$, so the gravitation acceleration of a mass towards Earth is $g = -\frac{GM_E}{d^2}$. But what about the force exerted by the mass?
Dividing the gravity by the Earth's mass instead yields the Earth's acceleration, which would be $a = -\frac{Gm}{d^2}$, different from $g$. The two accelerations should thus add up and result in a greater apparent acceleration the heavier the mass is.
For everyday objects, this obviously is minute, but if it were another planet, take one identical to the Earth, for example, the acceleration between them would be $2g$.
I know this isn't the acceleration exerted by Earth, but of two masses, the heavier mass would close its distance with the Earth quicker than the lighter one. Please correct me if I'm wrong anywhere in my logic.
