The definition of the Roche limit is: "The distance within which a celestial body, held together only by its own gravity, will disintegrate due to a second celestial body's tidal forces exceeding the first body's gravitational self-attraction". I can't say that I understand that perfectly, but let me try to explain why I am confused.
It is stated here that the Roche limit for the Sun-Earth system is 556,397 kilometers, and ignoring the fact that this distance is actually inside the Sun, I understand that this is the distance at which a test object placed on the surface of the Earth will be pulled equally between the Earth and the Sun, and decreasing this distance would lead that test object to start accelerating towards the Sun.
The other thing I thought of, and which is why I am confused, is the gravitational acceleration. I have calculated through $a=GM/r^2$ that an object on the surface of the Earth will have the same gravitational acceleration of 9.8 $m/s^2$ towards the Sun if the Earth is 3,680,000 kilometers away. Which is more than 6 times the Roche limit.
Now in both cases, it means, or at least how I understand it, that an object on the surface of the Earth is pulled by the same amount of force towards the Earth and the Sun, and getting closer or further will determine towards whom the object will accelerate.
So, why aren't the two numbers the same if they mean the same thing ?