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How are the comet-hunters able to precisely locate/know about the required comets ? How can they distinguish between comets from such a far distance ? How are they able to estimate the exact time period of these comets like Hale-Bopp's and Halley's comets have orbital time periods 2520-2533 yrs and 75.3 yrs respectively ? Do they have to follow the comet throughout its whole orbital path to deduce that or are there any other procedures ? I've heard of using computer generated programs that help in here. Please notify me some of these.

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I can't really answer your question because calculating a comet's orbit isn't something I can describe in a few lines (actually it's not something I can describe at all, but as always Google is your friend!).

As Martin says, finding your comet is easy in principle. You photograph the same bit of sky on (at least) three occasions a few days apart, then you look for anything that has moved i.e. that is in different positions in the three pictures. This sounds easy, but there is a lot of sky and only a few comets so it's incredibly painstaking work. Professionals have automated systems for comparing images and looking for moving objects, but amateurs just have to get on with it.

Anyhow, once you've found your moving object you can use the position in the sky and the change in position to calculate the orbit. Martin casually describes this as "a bit of algebra" but actually it's rather a lot of algebra. I found this book online if you really want the gory details.

I imagine most amateur astronomers would just feed their measurements into some software rather than do all the sums themselves. I confess I know little about this area but a quick Google found several calculators e.g. this one.

Incidentally, in a previous question you asked about how to tell a hyperbolic from a parabolic (and elliptical?) orbit. There's no easy way to tell except by calculating the orbit to see what shape it is.

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You look at the sky every night and see something that wasn't there last time you looked. If it moves the next night then it's a comet, if it doesn't move it's probably a supernovae

With three position measurements on different nights and a certain amount of algebra you can work out it's orbit then you have it's period and when/where it will reappear. Now it's a little easier with telescopes and cameras that can image the entire sky in a night or two - and software that can spot changes. But a lot of comets (and supernovae) are still found by amateurs.

Although comets do change their orbit slightly with each trip into the inner solar system as they are effected by the gravity of each planet they pass. Also as they get close to the sun some of the surface ice boils off ,which gives the comet tail, but this can also provide a thrust which changes their path slightly.

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