Planetary Transfer Calculator
transfercalculator.com
transfercalculator.com
Other times you are looking to predict the velocity and position of an orbital body at a certain time. Since there are many large planets in the solar system this requires non-linear differential equations that can only be solved (at this time) using an approximate calculation using methods such as the Euler or Runge-Kutta method in a computer.
If you read further down in Phil Plait’s article, the video is based on a conspiracy-theory-style paper written by a crazy guy.
My understanding is that the solar system formed out of a volume of gas and dust. Over time, gravity caused that volume to collapse inward; this in turn led to a bulging in the plane perpendicular to the axis of rotation. This is the same reason that the Earth bulges at the equator. But since a volume of gas and dust is rather less dense than Earth, nothing opposes gravity condensing it further. So it gradually flattened into a thick disk.
Wikipedia’s “Protoplanetary disk” page explains this process with far more fidelity than I ever could. My point is, there’s a good reason the solar system is a disk: it was formed that way to begin with.
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If you'd really like to nail home an almost intuitive understanding of orbital mechanics, I recommend Kerbal Space Program. Imagine Minecraft NASA. You plan, build, launch, and fly space missions using a simplified physics model. It's insanely fun and addictive, as long as you can get past the frustrations of early failures.
I cannot recommend this game enough. I've got hundreds of hours in, and I'm at the point where I've got huge spreadsheets carefully planning out mission parameters to spec the requirements for my new heavy launch platform to colonize the solar system. I'm even assembling a long-term research station in orbit that will eventually be parked around a gas giant.