I finally found a way to complete that challenge with combustion engines only!
Of course, as the rocket will have a limited capacity, that implies that I'll have to make a lot of maneuvers to save as much fuel as possible. But with an optimized design and a good trajectory, this is... surprisingly easy in practice
I also played with an additional constraint, which is not to left any debris in space. All debris shall at least be put on a reentry trajectory so that drag deorbits them.
@8bitCosmonaut if you want to do that mission with a realistic design, you can use this as a basis and improve the launcher so that it has a better look
So here we go:
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After the booster phase, the ship quickly reaches LEO:
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The main stage used a lot of fuel for this, as expected. That's why I made a separate fuel reserve. Now it's empty, I separate it, put it on a reentry trajectory, and redock the propeller with the payload:
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Much better now: space is kept clean, but above that, I won't have to propell that dead weight further.
Now, time to aim for Venus for a few gravity assists:
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Now the main stage is empty. It's separated and its trajectory slightly modified so that it crashes on Venus. I won't need such power from now on. I've still not left the Earth's SOI, but the lander with its cargo is now on its own.
The rest of the trajectory is precalculated and only need a few adjustments:
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I finally enter Jupiter's SOI. Time to satellize:
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That small ship can hardly brake with all that mass attached to it, but because of the massive Oberth effect from Jupiter it's far enough. My intention is only to left my ship on a very elongated ellipse, so this doesn't require so much delta-V.
To be continued...