Can someone show me the math behind an Oberth maneuver around the sun, and acceleration to Jupiter?

I'm writing about a spaceship that drops past L-5 from Mars to the sun, performs a tight Oberth maneuver (@ 10 million K from sun?), has 1.5g acceleration at perihelion, but loses thrust before finishing its maneuver and swings wide. It needs a Jupiter gravity assist to change orbital planes, and I'm thinking that orbit to Jupiter will be a long one--13 AU? 15? the ship needs to "chase" Jupiter. It needs two or three months to get to Jupiter, time for the crew to repair systems that will be needed there, damaged in the CME that knocked out their propulsion. They have limited, intermittent propulsion (1/2 time) 2-3 days after leaving closest approach; I'm thinking I should limit that to .1 or .2 g acceleration. Several weeks later they get their badly-damaged main engine (a fusion-powered VASIMR) back online; that gives them ~.1 g (?) acceleration until almost Jupiter, when they finally get their inertial-reduction coils rebuilt and can thrust at 1g.

I can do simple acceleration-time-distance etc. calculations, and when I remember to add in all of the conversions, my answers are close to those produced by online relativistic spaceship calculators--and .1 or .2 g seems plenty.. But nothing I find online tells me how to account for the "gravity tax," the need to subtract out solar escape velocity from 10 diameters out.

Anybody here know that math?

• I dont think this will work. It requires incredible amounts of energy to slow down enough to get to the sun, then you gain a little (the amount you gain is related to how close you can get, which isn't usefully close in the case of the sun). In the real world the jupiter trajectories commonly use multiple earth and venus flybys. – Innovine Dec 19 '16 at 17:46
• @Innovine: OTOH maths that show exactly why this isn't practical - in numbers - would be quite informative. – SF. Dec 20 '16 at 10:29
• @SF. If the ship is already at Mars, why would it use up tons of energy to get to the sun, then tons more to get back to where it was, when it could have simply used that energy to go directly to Jupiter? The gain by doing a solar flyby is utterly insignificant compared to the massive delta V required to lower perihelion to the sun and then raise it again (required to enter orbit around Jupiter). Maths is not required unles syou want to show exactly how impractical this idea is. And that I can leave to someone more interested. – Innovine Dec 20 '16 at 12:39
• @SF. The maneuver being talked about by the OP is ridiculous, and he has bigger problems than being able to accurately calculate the Oberth effects. Why is he changing orbital planes at Jupiter? Why is he heading towards the sun in the first place? The question could be much improved by formulating what the current course is, what the destination is, what the ephemeris of the relevant bodies are and then asking for suggested trajectories. – Innovine Dec 20 '16 at 12:47
• @Innovine: it seems to me they are not transferring TO jupiter, but performing solar system ejection for interstellar travel; They use Jupiter for course adjustment, plane change and gravity assist. In this case - providing the temperature is somehow dealt with - Oberth maneuver against the Sun makes sense - although not if we start from Mars! Start from Mars, use Earth or Venus for gravity assist to Neptune, then perform a slowing gravity assist+Oberth Maneuver to the Sun, gain most during solar flyby, then sling by Jupiter on the escape trajectory. It can be done in KSP ;) – SF. Dec 20 '16 at 15:38