As have spent alot of time getting stuck on white dwarves and trying to escape, my findings are you are screwed if you end up in the cones....
As for a quick and information information on how to do this check this out, have a nice picture to show it.
https://www.reddit.com/r/EliteDangerous/comments/5738sk/slingshotting_a_comprehensive_guide_to/
OK the above tells you how to do it, and how to not do it.
One things that makes it worse, are that the exclusion zone is not visible when you go to slow, so you have to go by distance and the impact indicator.
And as long as you do not drop out of super cruise in the cone, you should be "safe", so always approach the star between the cones as depicted. So you will learn how close you can get, I think it was 0.13-0.15 LS, now the tricky part, how to fly around the star without hitting the exclusion zone. use flashing the impact light to the top right, as it will alert you if your current course will impact anything soon... like the exclusion zone. Practice in a cheap ship.... and White dwarves are the worst to practice at, but are present in the bubble, so doable in cheap ships to practice on. If can do white dwarves, then most Neutron stars will be a breeze in comparison.
Ok, if you get stuck there are a few things that I have noticed....
1. Your ship will ALWAYS try to point towards the star!
2. Your controls will sometimes be reversed, very frustrating, as it took me quite some time to figure this one out, as some tests the ship moved as expected, and in others, every things just did not work.
3. You have to swing the ship back and forth, like a pendulum to swing the nose away from the star, as the exit vector will be away from the star. Boosting can improve the pendulum effect to swing around the ship
4. You need to have the FSD to spin up and sort of time it so you are facing the exit vector when the FSD is ready,
5. Timed boosting to keep the exit vector in the general center to allow the FSD to lock on to it and let you escape.
6. You are still in the jet cone and you will in 50% of the time drop out again, due to the extreme turbulence (as it seems to get worse the close to the you are), you have in most cases very little time to get your bearing and keep going away from the star, you are obviously not in the clear when the countdown begins...so keep an eye on the navigation orb and make the best use to have it behind you in super cruise to avoid dropping back into normal space.
What ultimately will kill your ship are two things
1. Hull/Module damage from the extreme gravitation -> module mal functions and the two that ultimate gets your will be FSD and Thruster, and then it just a waiting game before the ship is crushed..
2. Heat damage from FSD 100% chraged and not jumping away -> to increased module damage -> and all the bad stuff from 1 happens even faster.
I have not tried out if you can use repair limpets and AFM to repair your ship to get more goes at escaping.
A more cheaty way of practicing this is to combat log, this will allow you to retry this many, many times, as it in most cases pointless to make several tries to fire up FSD and swing out, as the damage to your ship will be to great, so after first failed attempt, combat and when you login back in, you will in most cases be back to when you first dropped in, giving you the possibility to a repeat and try again. Ihave only tested escape on White Dwarves, so I do not know if they differs much from Neutron star in difficulty.
So how many times have I done this? well, I have lost 10 or so Anacondas to this, and I have only successfully made escapade 2 times, been locked on to escapde vector etc, when FSD/Thrusters cuts out due to malfunction, or dropping right back to the the star.
I have done it 5 or times in a Combat Vulture, no escapes, I have tried a D-rated Eagle, it disintegrated almost instantly... So in my experience you need a beefy hull to even have chance at this.
I have spent countless of hours swinging my Anaconda back and forth and do the combat logging trick... and based on all my tries at this (many, many hours), my success rate at this is really bad. And even if you do get the escape vector to lock, you have that 50% chance to get sucked back into the exclusion zone again. So if you are stubborn as I, go for it and do the combat trick, if you have a ship that can last long enough, otherwise, watch the destruction and try to avoid it in the future...
Fun trivia, every Neutron boost damages your FSD, and it is possible to totally break your FSD to the point that it crease to work, when this happens, you will be thrown out of super cruise. If you want to experience this without doing all flying throught the cones, simply disable your FSD and this is what happens. Another fun trivia to this, it is possible to survive this when bringn your FSD to a true 0% at a neutron star. Done that, and I did not end up in the danger zone of the the start inside the cone.
Also I have encountered a FSD failure that affected super cruise except in two cases, total failure of FSD, 0%, and you manually turn it off, like trying to repair and forgetting to drop to normal space.
Thrusters are used for Supercruise, so turning those off also drops you to normal space.
