In calculation of the Roche limit between the two celestial body, tidal force on small mass $u$, $F_t$ is expressed approximately as $$F_t=\frac{2GMur}{d^3}$$ While deriving $F_t$, what's the reason behind finding the difference in the gravitational pull due to primary body of mass $M$ on the center of satellite of mass $m$ and on edge of the satellite close to the primary one? $$F_t=\frac{GMu}{(d-r)^2}-\frac{GMu}{d^2}$$
The Roche limit expresses a balance between tidal forces imposed on $m$ by $M$ and self-gravity of body $m$.
This is an estimate of the tidal force pulling the object apart.
If you imagine no self-gravity and only tidal acceleration, then the center of the satellite will feel some acceleration $a$ and the edge closer to the primary body will feel some greater acceleration $a+a_{t}$. Thus, the edge will be pulled away from the center with acceleration $a_{t}$ unless self-gravity is able to keep it together.
