KubeRoot

So, the issue is, as far as I know the calculations are dead simple - you “enter” and “exit” the planet’s influence at the same distance from the planet, which means your potential gravitational energy didn’t change, so from the orbital mechanics point of view, from the planet’s frame of reference, your velocity should stay the same.

As you “fall” in the orbit around the planet, you’re converting potential gravitational energy to kinetic energy, but as you “climb” you convert it back into potential gravitational energy, ending with the same amount of each kind of energy. The only change is that the velocity is redirected.

With that in mind, it’s why, from my knowledge, the equations are really simple, with the only complications being trigonometry (to resolve the angles) and pythagoras (squaring, adding and getting the square root make the result unintuitive).

Going back to your graph, if I were to do the math, according to my theory:

• In A, let’s say you go in with 200 speed, and 0° angle (for simplicity). That means relative to the planet you have 100 speed.
• In B, you gain some speed by converting potential energy to kinetic. We can’t say how much you gained, because we’re missing any real measure of distance and mass, but the neat thing is - it doesn’t matter, because:
• In C, you turn that kinetic energy back into potential energy, and end up with the same speed you entered at, at the same distance. This means you now again have 100 speed relative to the planet, but aimed at a 60° angle. We can now add the velocity vectors of the planet and the velocity relative to the planet to get the velocity relative to the sun, using the planet’s velocity as one axis, getting a vector of [100+100*cos(60°); 100*sin(60°)], or [150; 86.6025], with magnitude of 173.2051, which is less than the 200 we went in with.

If you want an intuitive example of what I’m referring to, consider a planet approaching you as you are stationary relative to the sun. If we assume ideal, presumably impossible, entry and exit angles of 0° and 180°, leaving the planet’s gravity field moving in the exact opposite direction than what you entered, you’ll note you’ll be gaining speed on exit either way, despite not moving towards the planet on the approach and “catching up”.

The graph doesn’t really show anything other than illustrate your thoughts - but there’s absolutely nothing backing that as being true :/

Either way, it does feel like we’re going around in circles, and I don’t want to be taking up your time unnecessarily. If you have something to disprove my math (maybe my understanding of orbital dynamics is wrong, and it’s not that simple), that’d be a starting point to try to figure out what’s wrong; if you’re interested, I could try to make diagrams, though I feel like they might kind of look the same, just with different numbers based on calculations.

I guess one last thing I can offer is a video somebody replied to me with elsewhere in the thread, explaining this idea: https://youtube.com/shorts/kD8PFhj_a8s

Well, relative motions are more intuitive to me - they make sense, and I can use calculations for them.

In the first example, you presented 101 speed - this means only 1 speed relative to the planet, and that’s all that’s getting redirected (in the planet’s frame of reference your enter and exit velocity should be the same, since that’s how orbits work). The number is just too small, but your velocity would be planet velocity + 1 on a different vector, which will be less than 101 total.

If we estimate the angle on the picture is about 60 degrees from the velocity vector, and the speed to be 100+v1, the speed from the planet’s frame of reference is v1 - so, the exit velocity will have components of (100+v1cos(60°)) and (v1sin(60°)), so the final speed relative to the sun should be

sqrt((100+x*cos(60°))^{2+(x*sin(60°))}2)

Wolfram alpha suggests this simplifies to sqrt(x(x+100)+10000), and comparing the equation by appending <x+100 gives the solution of x>0

This means, if my math is correct, with an entry angle of 0° and exit angle of 60°, you always lose speed.

I could try replacing the angle with a variable and setting a constraint of x>0 and see if the free version of wolfram alpha would spit out something, but just replacing the 60 with y is spitting out some convincing solutions, since in those x is never greater than 0.

PS: Thanks for taking the time to explain, the fact that you went out of your way to draw the diagrams is not lost on me. I unfortunately still don’t see it, but I really do appreciate the effort!

I’m sorry, but this comment thread genuinely makes me feel like I’m going insane. You seem to have explained exactly the same thing as me, with the same example, and none of it includes the “fall for longer before you catch up” bit.

As for the orbit not curving, yeah, I think you’re right - the obvious case is if you’re sitting stationary on the planet’s orbit, but the curious case is if you’re approaching from the sun, with the planet’s velocity plus velocity away from the sun. If I’m not mistaken, in that case you’d end up with the same velocity (minus what you might have lost to the sun’s gravity), but on the other side of the planet’s gravity well, which means you still gained energy.

Ayy, I’m not crazy, that sounds like exactly what I described… The only question is, is the explanation of “you spend longer falling” is bs, or if it makes sense if you conceptualize it differently?

Right, but as I explained, it’s the how that doesn’t make sense to me - the explanation that you “fall for longer” doesn’t make sense, since 1. with how orbits work, it takes the same energy and time to “fall” as it does to ascend, and 2. at these scales you can use the planet as an inertial frame of reference, so the angle of approach doesn’t matter for how “long” you “fall”, it’ll be the same regardless of whether you’re moving towards or away from the planet.

I’m confused, but this doesn’t make sense to me.

It shouldn’t matter whether you’re moving in the same direction or not for this, because ultimately it’s all relative - if you set the planet as the frame of reference, the direction you come in from doesn’t matter - just the velocity and angle.

What I can see working is calculating the in and out angles - if the exit velocity is at a sharper angle relative to the planets velocity than the entrance angle, then your exit velocity “gains” more of the planet’s velocity than the entrance velocity “loses”.

If you were completely stationary, from the planet’s point of reference, you’re moving with the velocity of the planet. If you then did half an orbit, exiting in the other direction (theoretically), from the planet’s point of reference you have the same speed, just in the other direction - but from the sun’s point of reference, you’re now moving at the planet’s speed on top of the planet’s own speed, thus gaining double the velocity of the planet.

The issue is, of course, I have no idea if I’m making sense, or missing the point.