Imagine that, through some cosmic phenomena not yet understood, the Sun 'burps' and ejects a vast amount of its mass into the cosmic void. A huge coronal discharge, perhaps. A pressure bubble inside that bursts. A mega internal explosion. The loss of mass is sudden and dramatic, but in a trajectory that does not traverse the planetary plane, and thus does not destroy any planets. Perhaps it occurs along the axis of the planetary plane. That is, the Sun does not 'burn out', go nova, or any other destructive end-of-life process, it just loses a substantial amount of mass. No other destructive radiation or other event that would immediately destroy the planets.
This loss of mass would result in a dramatic decrease in the gravitational pull of the Sun. This would affect all of the planetary orbits, as their escape velocity from the Solar System would decrease. If they kept their current velocity, I presume they would move further from the Sun.
A. How much mass would the Sun have to lose, in order for Saturn's current velocity to become its escape velocity from the Solar System? This is a tricky calculation and equation, as it has to account for the diminishing gravity of the Sun, not an increased velocity of Saturn. That is, it does not ask for the new velocity of Saturn sufficient to reach the escape velocity of Saturn from the existing Sun, but asks for the the maximum reduced mass of the Sun such that the current velocity of Saturn becomes its escape velocity.
The following are ancillary, but not essential, questions that might arise from answering A.
B. Is there any absolute principle of physics that would make this absolutely impossible?
C. Is it feasible that Saturn, along with its moons, could become an intragalaxy or even intergalaxy wanderer using this technique? The ultimate goal is to put a sentient self-sustaining colony on one or more of its moons, and then have it wander the Universe. How to give it the ability to sustain life on a moon for millions of years is another question not within the scope of this question.
D. Does it make more sense from the escape velocity perspective for my ultimate objective to consider another planet, such as Neptune or Jupiter? I need a planet with sufficient composition for it to become a source of power for the moons. Jupiter, for instance, naturally emits a very high level of radiation that could provide a source of energy for its moons as a substitute for the Sun, but again this is beyond the scope of this question.
What happens to the Sun because of the loss of this mass AFTER Saturn becomes a wanderer is not within the scope of this question.
How the Sun actually loses the mass is beyond the scope of this question. That it can somehow lose this mass is to be taken as a given assumption.