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
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:
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