The Earth's orbital velocity around the Sun is ~30km/s. You need to cancel almost all of that out in order to shift to a trajectory that intersects the Sun, and that trajectory still has you impacting the Sun at quite some speed.
The Moon's orbital velocity around the Earth is ~1km/s. A slingshot manoeuvre around the Moon isn't going to help nearly enough here.
One of the most efficient ways to get to the Sun is to leave Earth in the same direction it is going (and yes you can use a Moon slingshot to help here if you want) which will result in your rocket going further away from the Sun than the Earth. Once you're out in the region of the outer planets, you'll be travelling a lot slower and won't need to cancel out nearly as much velocity. Neptune's orbital velocity is ~5.5km/s, and if your rocket has an apohelion around there and a perihelion around Earth then you'll be travelling slower than that. However, it still takes a huge amount of delta-v to get out to the region of Neptune and then cancel out your velocity relative to the Sun.
Compare that to the amount of delta-v required to de-orbit an object from low Earth orbit. The object's velocity around the Earth is going to be about 7.9km/s, but it only requires a delta-v of around 100m/s to put it into a nice predictable atmospheric entry. That's comparatively nothing.
How does delta-v prevent you from aiming in a direction that intersects with the sun? That's all that's required for trash disposal. You aerobrake in the solar atmosphere, if your want to think of it that way
To reach the sun, you need to shed a lot of kinetic energy - and given that you’re in a vacuum on most of the way there, the only way to do that is through a lot of deceleration, which is really the same thing as retrograde acceleration, and maybe some gravitational slingshots, but these aren’t free either (you need delta-v to execute them) and therefore have limited gains.
(Technically it’s cheaper by a factor of three to go to escape velocity out of the solar system and then plunge back into the sun, but at that point, why go through the trouble and come back if your concern is trash disposal?)
> To reach the sun, you need to shed a lot of kinetic energy - and given that you’re in a vacuum on most of the way there, the only way to do that is through a lot of deceleration, which is really the same thing as retrograde acceleration
If you can fenagle yourself into a highly elliptical solar orbit, then a relatively small retrograde burn at apoapsis could get you into the sun.
> aiming in a direction that intersects with the sun?
We are moving really fast relative to the sun. Perturb your orbit to seem to intersect with the Sun and you’ll tend to fly past it. (Loose analogy: swimming in a current and aiming for a point on shore.)
"Aiming in a direction that intersects with the sun" requires a huge momentum change when an object already has a large orbital velocity around the sun.
You don't need to just "get past Earth's orbit", you need to deorbit out of Earth's orbit around the sun if you want to reach the sun. This requires a much higher (at least double) delta-v than leaving Earth's gravity well alone.
Slingshots steal orbital momentum. When one slingshots around Jupiter, one is stealing Jupiter’s orbital momentum about the Sun. (It only works in one direction.)
Slingshotting about the moon to gain velocity relative to the Sun doesn’t work. That said, I’d love to see an orbital solution for cheap decay into a solar-impact trajectory. (Orbital mechanics are complex enough that nobody should feel comfortable entirely precluding subliminal sets of solutions.)
> Slingshotting about the moon to gain velocity relative to the Sun doesn’t work. (It only works in one direction.)
Most slingshot maneuvers do gain or lose momentum relative to the Sun! And they do work in both directions (gain and lose) – we've used them for quite a few probes, e.g. MESSENGER.
> Slingshotting about the moon to gain velocity relative to the Sun doesn’t work.
It does, as used by e.g. STEREO [2].
All that said, they're still not "free" in terms of delta-v by any definition. They provide an efficiency gain, but sending stuff to the sun is still prohibitively expensive for anything other than lightweight scientific probes.
> Why not? And you don't actually want to gain velocity, you'd want to change direction
If you fall towards an object and then fall away from it, there is no net force. You can’t bleed or gain delta-V simply by falling into an object and then falling away from it.
The reason planetary slingshots work is you’re “dragged” along with their orbital velocity about the Sun. You can be clever about using that to reduce the delta-V to the Sun. But it’s still more than system escape velocity.
This is analogous to trying to turn a plane by only yawing. Or travel in a current by turning your head.
This guy [1] is wrong. Delta-v comes from Tsiolkovsky’s rocket equation, which in turn derives from Newton’s second law and the conservation of linear momentum. It’s incredibly fundamental math that you can’t cheat by changing direction. (You’ll change orientation and keep going where the math says you will. Because you’re going sideways relative to the Sun at an incredible velocity, the inheritance of every atom in the Earth’s sphere of influence, including the Moon.)
No, the direction change is relative to the Moon, not your orbit around the Sun. The Moon, when retrograde, is still moving forward around the Sun. Your orbit will become more eccentric, but the periapsis won’t even make it past Venus’ orbit.
It's not about "gain velocity", but "change angle".
A very wide variety of angles are frequently used in slingshot maneuvers, simply by controlling how close you get to the Moon or more often the planet.
It's expensive to get close to the Sun at low speeds (e.g. if you could land on it, or if you want a close orbit).
It's trivial to slingshot around the moon and impact the Sun at high speed.