https://quazikb.github.io/orbits/index.html

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29<h1>Orbital Mechanics</h1>
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31Orbital mechanics for simplified problems (two massive bodies) are pretty straightforward. However, I've found a lack of good visualizations to play around with so I made this page. Usually the 2-body problem is brought up as the canonical problem in orbital mechanics. This is for good reason, as it's the only type of system that can be easily analyzed. Even though the 2 body problem is simple, it is useful for a large number of real world applications with more bodies if certain assumptions are made, e.g. patched conics. Although all objects with gravity influence each other, the effect of bodies orders of magnitude more massive, or orders of magnitude closer strongly overpowers the effect of smaller and/or further bodies. This can be observed in how the moon and satellites clearly orbit the earth, the earth orbits the sun, and the sun orbits the center of the galaxy. While the sun influences a satellite around earth through gravity, generally it's effect is hard to observe from a reference frame moving with the earth. This is somewhat of an oversimplification, but demonstrates how we intuitively visualize simple orbits as a hierarchy of many separate 2-body problems.
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34In general, when thinking of orbits we tend to picture one object orbiting the other more massive object, just like a satellite around the earth. However this is not true, the earth is also orbiting the satellite! To be more precise, when two objects are in orbit, they both orbit around their center of mass. It only seems like the satellite is orbiting the earth because the earth is so much more massive. The center of mass of the satellite-earth pair is just extremely close to the center of the earth since the earth's mass is so much larger. If two objects of similar mass orbit each other it actually looks a lot more interesting:
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57	Frame of Reference :
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69<i><b>figure 1</b> : 2 bodies of orbiting each other with variable properties.</i>
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72Another point that usually conflicts with our intuition is that orbits are rarely circular. We tend to see diagrams of the planets orbiting the sun and note how circular the orbits look, however in reality the orbits usually have some <i>eccentricity</i>. This means they're elliptic, or in extreme cases hyperbolic. In general orbits can take on the form of any <i>conic section</i>, dependent on the degree of eccentricity. They're usually referred to as trajectories in cases where the "orbital" path is open (has an infinitely long period) such as a parabolic or hyperbolic "orbit".
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74	<p>
75	\begin{align}
76	e & = 0 \rightarrow \text{circular orbit}\\
77	0< e & <1 \rightarrow \text{elliptic orbit}\\
78	e & =1 \rightarrow \text{parabolic trajectory}\\
79	e & >1 \rightarrow \text{hyperbolic trajectory}\\
80	\end{align}
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83Orbits in the solar system are themselves slightly elliptic, nature isn't always as perfect as it seems. In general when working with orbits, we center the problem around one of the bodies rather than what is shown in <b>figure 1</b> where they orbit around the center of mass. <b>figure 1</b> shows what a pair of orbiting bodies look like from an inertial frame of reference, however when working through the math most derivations and problems opt to center on one of the bodies. This doesn't change too much, except that one of the bodies will then be at the "focus" of the path geometry, and the other will be the sole object moving around around. The center of mass itself then seems to orbit around the "fixed" body.
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