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I understand that there are magnificent tidal forces that come with getting too close to Jupiter, but what is causing comets like Shoemaker-Levy to get torn apart?

I wouldn't expect that the gravitational forces are so strong that spaghetification occurs, is it?

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It is. Tidal disruption occurs when $\rho_{Jup}(R_{Jup}/R)^3$ > $\rho_{comet}$ where R is the separation and $\rho$ is the density. The density of Jupiter is 1.33 g/$cm^3$ and the comet was estimated to have a density of only 0.5 g/$cm^3$ based on the radius at which it disrupted. That is, disruption took place at separation R ~ $(1.33/0.5)^{1/3} R_{Jup}$.

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