> nudging the dry casks sunward

[Note: I realize you might already get this. But for the sake of having the discussion in public, where this is a very common misunderstanding of space travel, and also because I enjoyed writing it:]

It's true that, in space, without air resistance, Newton's first law reaches its fullest expression: when you push something, it's going to keep going.

In a frame of reference with no gravity happening in it -- like a totally empty universe, or if you're restricting the area you're looking at to the interior or immediate vicinity of a space ship over a short period of time -- then it's going to keep going in a straight line. ie: if you give something a push in what looks like the direction of something else, it'll get there. And it wouldn't matter how hard the push was, it would still get there eventually.

But if you're working in frame of reference with a lot of gravity happening in it, straight lines stop being the usual mode of movement. If you start at rest and push straight towards something, you're going to curve away from your apparent initial trajectory, towards the source of the gravity. You can't just gently push directly towards something and eventually end up there.

"Well", you say, "we can still just push directly towards the source of the gravity -- which is our destination anyway." Absolutely! We'd successfully end up there, and it would even still be a straight line. If we were starting at rest.

But we're not starting at rest: we're starting out with a velocity of about 100,000 km/h, in a direction perpendicular to a straight line towards the sun. The velocity of the orbit of the earth around the sun.

Starting from there, if we were to give a really big push, directly towards the sun, say a ~15,000 km/h push (the magnitude of the push that moved the Apollo missions from low-earth orbit onto a collision-course with the moon)... that would just leave us in a slightly off-center orbit, which only gets very slightly closer to the sun at its closest point. And actually moving just a little faster than we were (~101,000 kh/h), at an angle less than 10 degrees closer to "towards the sun" than our previous direction of travel.

Think hypotenuse of a triangle. Pythagorean theorem and stuff. Really big width (our initial orbital velocity around the sun), relatively small height (our big push towards the sun). If we want our hypoteneuse (our actual velocity vector) to get really steep (point directly towards the sun), and changing the height of the triangle is our only move (again, pushing towards the sun), we'd basically need our triangle to achieve infinite height (infinitely big push towards the sun).

We could actually do significantly better by making our push in the direction opposite to our initial orbit, rather than directly inwards towards the sun. With a push of the same magnitude, now we're orbiting at only 85,000 km/h, making our orbit dip lower at its closest point.

In fact, that reveals what we actually need to do to dump something into the sun: stop orbiting it.

Cancel (almost) all of our initial 100,000 km/h of orbital velocity. By accelerating 100,000 km/h in the opposite direction to our orbit. When that's done, we fall into the Sun.

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No extensive comment on the logistics of this. Your [2] link discusses that more fully.

My point is that "nudge something in the right direction in space, and it'll get there eventually" is not actually a thing that can happen in any practical sense.

tldr: play Kerbal Space Program.