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The surface gravity on Mimas is $≈ 0.063\text{ m}/\text{s}^2$ and Saturn's gravitational acceleration at the distance of Mimas' orbit is:
$$\frac{{GM}}{{r}^2} = \frac{{6.674 \times 10^{-11} \times 568.34 \times 10^{24}}}{{(185.52 \times 10^{6})}^2} ≈ 1.102 \text{ m}/\text{s}^2$$
How can this be? An object on Mimas' surface would be much more attracted to Saturn than it is to Mimas. Shouldn't Mimas itself get ripped apart or is my math wrong?
An object on Mimas' surface would be much more attracted to Saturn than it is to Mimas.
You are missing that Mimas as a whole accelerates gravitationally toward Saturn. What this means is that a point on the surface of the Mimas will feel the acceleration at that point toward Saturn minus the acceleration of Mimas as a whole toward Saturn. This is the tidal acceleration. It is equal to $$a_\text{tidal} = \left|\frac{GM}{(R\pm r)^2}-\frac{GM}{r^2}\right| \approx 2 \frac{GMr}{R^3} = 2\frac{GM}{R^2}\frac{r}{R}$$ where $R$ is Mimas' semi major axis length and $r$ is Mimas' mean radius. The approximation assumes that $r\ll R$, which most certainly is the case given that Mimas's radius is about 1/1000 of the semi-major axis length of its orbit about Saturn. The result is rather small, about 0.002355 m/s2.
Since Mimas is in orbit around Saturn, it is in freefall; just as an astronaut in a space station appears to not experience the Earth's gravity because that gravity is acting equally on the space station and the astronaut, the outside of Mimas will appear to not experience Saturn's gravity, as the center is also experience Saturn's gravity and thus they are moving together. The only effect Saturn will have on the integrity of Mimas is Saturn's tidal force. Also, for the tidal force to rip apart a satellite, it has to overcome not only the satellite's gravity, but also any intermolecular forces. For instance, for a space station to be ripped apart by Earth's gravity, the tidal forces would have to overcome the tensile strength of whatever the station is made out of.
How is it possible that Saturn's gravitational acceleration felt by Mimas is stronger than Mimas' own surface gravity?
That's just the way it is. An apple hanging from a tree is more strongly attracted to the Earth than to the tree. A worm crawling on it is more attracted to the Earth than the apple. Yet they retain some forces keeping them from falling to the ground.
Since Mimas and any object on it's surface are orbiting Saturn and in free fall, Saturn's force of gravity mainly curves their paths and they don't get tugged straight down to Saturn. The gravity of Mimas itself is enough to keep things from flying off it's surface. There are also cohesive forces tending to keep it together.
How can this be? An object on Mimas' surface would be much more attracted to Saturn than it is to Mimas. Shouldn't Mimas itself get ripped apart or is my math wrong?
You don't go flying off Mimas because Mimas is being affected by Saturn's gravity as well, and Mimas exerts a large enough pull to keep you in place. And since you would be travelling in orbit with Mimas you would both be experiencing Saturns's gravity.
What would make Mimas tend to break apart is tidal forces, except it's dense enough and far enough away from Saturn to avoid that fate. There's a calculation to tell you if an object in orbit will break apart called the Roche Limit. Part of the calculation is the ratio of the density of the primary to the density of the secondary, and Saturn's low density helps keep it small in this case. Calculating it myself I get 61,826 kilometers for a rigid body. That fits well with what this page says given that the density of Mimas is about 2/3 higher than that of Saturn. So Mimas orbits about 3 times the Roche limit and will not disintegrate due to Saturn's gravity. Even for the other extreme of a fluid body, the Roche Limit is just under twice that for a rigid body so Mimas still wouldn't come apart.
Using your calculation for gravity and plugging in an extra 414km for the diameter of Mimas shows that the difference in Saturn's gravity on the near side to Saturn and the far side to Saturn is just 0.005 m/s^2, less than 1/12th the surface gravity of Mimas (0.063 m/s^2)
Some thought experiments:
If you were on Mimas and it suddenly disappeared, leaving you in space, you wouldn't be sucked to Saturn. You would continue in basically the same orbit as Mimas would have. You are going fast with respect to Saturn's surface, and Saturn's gravity is just enough to curve your path to maintain your orbit so you don't go flying off into space, and so you don't crash into Saturn.
If you could somehow stop Mimas (and you) in it's tracks with respect to Saturn, Mimas and you would still be in free fall, but would both be pulled towards Saturn. The gravity of Mimas would still tend to pull you towards its center as well so you wouldn't fly off the surface.
If you could somehow stop Mimas, create an adamantite shell around it to keep it's form, and suspend it at a point above Saturn the same distance as it's orbit, you would fly off the surface towards Saturn if you were on the Saturn side. This is because you are preventing Mimas from falling with you. You would accelerate at roughly (1.102 - 0.063) m/s^2 because Saturn is pulling you down and Mimas is pulling you up.
If you could somehow stop Mimas and create an adamantite platform under it at the same distance from Saturn, it should collapse and form huge pile of ice on the platform. The Roche Limit works for orbiting bodies.
