I parked a space station in Jool's equatorial orbit and inclinated a little to avoid collisions with Tylo or Laythe.
After a 5 year tourist contract mission near Duna, I opened the mission control to find my space station "in an escape trajectory away from the Sun". I know that either Tylo or Laythe must have slingshoted my station.
Question : How to be sure that my ship/station will not meet the SoI of a nearby moon ?
Vanilla/stock answer prefered but I have KER and MechJeb installed.
The most practical approach is to simply park in an orbit which does not intersect with the orbit of any of the moons.
A reasonable rule of thumb I use is to simply park halfway between them, maybe adjusting a bit towards the smaller moon.
If you want to be sure, though, you can look up the moon characteristics and calculate the safe altitudes. For example, taking Tylo and Vall, we can see that Vall has an apoapsis of 43.2Mm and a sphere of influence of 2.4Mm, and Tylo has a periapsis of 68.5Mm and a SoI of 10.9Mm. So an orbit around Jool with an Ap under 68.5-10.9=57.6Mm and Pe above 43.2+2.4=45.6Mm will not intersect any of the moons' orbits.
Alternatively, you can manage your orbital periods to make sure you don't encounter the moon.
The simplest example is to simply have your orbit period match the orbit period of the moon. E.g. if you parked on the other side of Jool from Tylo, and you took exactly the same amount of time to orbit Jool as Tylo, then neither of you would catch each other, and no encounters would happen.
There are more complicated alternatives, e.g. if you had an elliptic orbit with a period of exactly 1.5 (3/2) times Tylo's orbit, you would get into the same position every 2 orbits (2 ship orbits, 3 Tylo orbits), so as long as the match is exact and your positions do not drift, if you did not encounter Tylo in 2 orbits, you will not encounter Tylo.
If you want to get fancy, you can even exploit the fact that Laythe-Vall-Tylo have an orbital resonance of 1:2:4, and set up a safe elliptic orbit floating among all of them.
Both of these approaches are safe to use on the same orbital plane as the moon, but if your orbit is significantly inclined, you only need to worry about these things at the time your orbital plane crosses the orbital plane of the moon. Inclination alone will only slightly reduce your chances of an encounter, and does not offer significant protection, though.
In general, as other answers have noted, you simply want to make sure that your orbital path doesn't come too close (in 3D; remember to rotate the map!) to that of any of the moons.
In particular, as each moon orbits the planet, its sphere of influence will sweep a region of space that is shaped like a (possibly somewhat elliptical) torus. KSP's orbital map won't show you this torus by default (although I suspect such a feature could probably be modded in), but you can imagine it as a tube that surrounds the orbit line of the moon. A sufficient (but not necessary) condition for a parking orbit to be stable is that it stays out of this "torus of potential influence" of any moons of the planet (or any planets of the star) that you're orbiting (and doesn't hit the planet or leave its SOI entirely, of course).
While it's technically possible to have an indefinitely stable parking orbit that does pass through a moon's "TOPI" (just coined that acronym), that requires the orbit to be resonant with the moon, i.e. to have an orbital period that is equal to the moon's orbital period, or a simple fraction or multiple of it, so that the moon is always at some other part of its orbit when the intersection occurs. In practice, though, you'll never get the period exactly right, so the orbits will slowly drift out of phase until an intersection occurs. (In real life, some orbital resonances can be self-stabilizing, but KSP doesn't model that.)
That said, if you're careful (and especially if you make use of mods like KER that directly show your orbit period), you can get close enough to a perfect resonance for your orbit to stay safe for years or even centuries. But that's still a very fiddly process compared to just avoiding the moons' orbits entirely.
A practical way to check that your orbit will not (soon) intersect any other body's SOI is to use a dummy maneuver node:
Create a maneuver node anywhere along your (planned) parking orbit. Don't adjust any of the burn vectors, but leave the node Δv at 0 m/s.
Instead, right-click the center of the node to activate the +/- orbit buttons.
Click the + button and check if any unwanted encounters appear. If not, keep clicking the button several more times. If you don't see any after a few dozen clicks, you can be fairly sure that your orbit is likely to be stable.
Optionally, remove the node by clicking the X button when you're done.
What you're doing by clicking the + button is telling KSP that you want to stay in your parking orbit for one more full orbit before performing the (dummy) maneuver. Normally, KSP's maneuver planner only shows you encounters that happen less than one full orbit after the most recent planned maneuver, but by moving the maneuver forward, you can get it to show later encounters.
Incidentally, this trick of using zero Δv nodes is also quite useful when you do want to get an encounter. For example, I used it in this challenge mission to (partially) plan a ballistic trajectory that encounters all of Jool's moons.
If you circularize your orbit, the chance of intercept is much lower. The more eccentric your orbit, the harder it is for you to tell if an intercept is likely.
