This "should" be obvious, and we've discussed it before, but
here's a reminder. Most textbooks get this badly wrong. Even
Feynman gets it wrong.
There are *two* different concepts that we call "gravity".
Textbooks mercilessly switch back and forth between the
two meanings. No wonder students are confused.
There is no hope of understanding the Roche limit until
this is cleared up.
-- There is the extrinsic gravity, namely the acceleration
relative to some chosen local reference frame. I call
this /framative/ gravity since it is 100% frame-dependent.
This is commonly denoted g, but I recommend g_@F to make
explicit the dependence on the frame F.
-- Then there is the intrinsic, massogenic gravity. This
is what you calculate using the law of universal gravitation.
It is 100% frame-independent. I recommend denoting this
as δg or perhaps δg_M(r) since it depends on M and r.