2. Comet orbits and math

Some points

Kepler let us know that Earth and the other planets do not orbit in perfect circles around the Sun, but in ellipses. His laws apply to comets as well, and comets actually have highly eccentric and sometimes very inclined orbits.

Interpreting orbital elements (Page 159)
Calculating from the elements
Calculating from Kepler's and Newton's

The perihelion is the point where an object comes closest to the Sun, and some comets come so close that they become known as the Sungrazers. By Kepler's, comets also travel the fastest when they swing around the Sun. The dust tails become especially curved, and by parallax error, we might see a third tails sprout towards the Sun. Known as the antitail, it is actually the end of the curved dust tail.

The brightness is typically inversely proportional by (r^4) to the distance to the Sun, but if the nucleus is especially active, its magnitude could increase tremendously (17 to 2 for 17/P Holmes).

Eccentricity: the ratio of (distance between the two focus points) to (the semimajor axis). By Kepler's 1st Law, moving bodies orbit elliptically, with the parent (in the comet's case, the Sun) at one focus. Thus, for closed orbits, the eccentricity must always be less than 1. Otherwise, the path would be parabolic and we just have a single-apparation comet.

Conclusion: Comets are sorted into their orbital periods, and for Stellarium it's probably only worthwhile to consider short-period comets, or even just observable comets, and not long-period or single-apparition comets, since keeping in the database comets that only appear once in 97,000 years at the perihelion is kind of wasteful. By the next time that comet comes, Stellarium would probably have the AI to detect and display observable comets automatically =X

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Equations

Finally, some math.

Kepler's Laws
Different formulas to calculate comet orbits
Modeling Comet Halley's rotating nucleus with dust ejection from discrete sources
Dust tail model [1]
Comet tail with radiation pressure and resistance due to orbit
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