Need help figuring acceleration out of a gravity well

I'm writing a spaceship that just performed an Oberth maneuver around the sun, 10 million km out, and is accelerating toward a gravity sling with Jupiter. It had a high--not certain how high yet--initial velocity, and 1.5 G of thrust at perihelion, but then lost its engines for about three days. It has escape velocity (where do I find that for various solar orbital distances?), but not having thrust when you need it to climb out of a gravity well has to be a problem.

When my ship's crew gets its engines back online, their thrust is limited: I haven't decided how much yet, but well under one g, possibly 0.1 g; and for a few days, at least, they can only thrust 1/2 the time.

I can do simple time-acceleration-distance calculations, or use online relativistic spaceship calculators and get ~ the same answers for a simple, continuous acceleration. But nothing I have found in weeks of surfing tells me how to figure-in climbing out of a deep gravity well, or how to handle intermittent thrust, or what a realistic curved distance for a fast transit orbit Sol to Jupiter might be.

Are there any orbital mechanics in the crowd?

• Obligatory xkcd rerence that contains real advice for you. – JDługosz Dec 29 '16 at 2:31

Gravity sling maneuvers are unpowered

No engines are necessary to perform a gravity sling maneuver. Yes, escaping a deep gravity well requires a lot of energy - but it's exactly equal to the amount of kinetic energy that you gained by going down that gravity well.

Craft gains velocity when approaching Jupiter, turns around it, then loses all that velocity when leaving Jupiter - the remaining benefit is a different direction, and the 'slingshot' effect caused by Jupiter's orbital speed around the Sun; an effect that's a bit comparable to a fully elastic "bounce" against Jupiter, if such a bounce was possible.

So as far as the gravity well goes, they would have no problems in leaving Jupiter in the direction they intended.

Missed opportunity for another Oberth maneuver

However, your spacecraft may have wanted to perform an Oberth maneuver in the lowest point of Jupiter orbit as well. If they were/are unable to do that to the planned extent, then this likely means that they can't do the planned acceleration because now it would take a lot more fuel than what was planned. As carrying significant reserve fuel is nearly impossible in realistic spaceflight (after all, you perform months-long maneuvers just to save less fuel than missing this chance wasted), this likely implies that they will be unable to perform their planned route fully.

If that was a one-way trip, then it just became a different one way trip because they're mostly committed to it now (Jupiter flyby is likely the last event in a realistic plan), don't have fuel to do what they intended, and definitely don't have enough fuel for turning around.

If that was planned as a two-way trip, well, then it may become one-way unless they manage to fulfill a quest, retrieve a macguffin, obtain favorable star alignment (possibly literally) etc.

If you have escape velocity, then you have escape velocity

You say, "It has escape velocity (where do I find that for various solar orbital distances?), but not having thrust when you need it to climb out of a gravity well has to be a problem."

Actually, its not a problem. I assume you mean escape velocity from the sun. Escape velocity from any object is calculated with $$v_e = \sqrt{\frac{2GM}{r}}$$ where $GM$ is the gravitational constant of the sun ($1.327\times10^{11} \text{km}^3\text{s}^{-2}$) and $r$ is your distance from the sun. If you are 10 million km form the sun, then the necessary escape velocity from the solar system is $$v_e = \sqrt{\frac{2\cdot1.327\times10^{11}}{1\times10^{7}}} = 162.9 \text{ km/s}.$$

You can calculate yourself for other distances. However, since you have escape velocity, despite the fact you are at less than 1/3 the orbit of Mercury, if you have 163 km/s or more of velocity you are going to leave the solar system. Now, if you are at exactly 163 km/s, you are going to leave the solar system very slowly, but you will leave it (assuming you don't run into Jupiter or something).

Regarding the statements in the first paragraph, if they lose their thrusters during a scheduled burn, they probably won't make it to their Jupiter slingshot on time, which will be a big bummer. They will have to plot some other course through the solar system to wherever they are going, depending on their destination and the alignment of planets available for gravitational boosts.

Making use of a gravity assist maneuver depends entirely upon the direction you approach and leave the target celestial body.

Their primary use is to increase the speed of the spacecraft without needing to use up propellant. You can increase your speed by up to twice the orbital speed of the target. A change in direction is a result of how a gravity assist works rather than a primary use for such.

Depending on how you approach the target, you can also use this method to "brake", shedding velocity without needing to burn the large amount of propellant that would normally be required.

All spacecraft that have been sent into the outer solar system have made use of gravity assist maneuvers to help overcome the very large delta-v budget required to travel "up" the Sun's gravity well.