What resources are available to find the positions of planets, stars, moons, artifical satellites, asteroids, and other heavenly bodies?
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There are many resources online, so I'm making this a community wiki answer. Please feel free to add to it!
If you want to visualize the stars/planets/etc (as viewed from Earth or another location), you are looking for planetarium software: https://en.wikipedia.org/wiki/Planetarium_software
If you want accurate positions for stars/planets/etc, you are looking for HORIZONS:
https://en.wikipedia.org/wiki/JPL_Horizons_On-Line_Ephemeris_System
http://ssd.jpl.nasa.gov/?horizons
If you want to compute star/planet positions yourself, you have several options:
SPICE (http://naif.jpl.nasa.gov/naif/tutorials.html) will give you results that match HORIZONS very closely. You can also use some of SPICE's functionality online at http://wgc.jpl.nasa.gov:8080/webgeocalc/#NewCalculation
SPK (Spice kernel) files are available at https://naif.jpl.nasa.gov/pub/naif/generic_kernels/spk/ -- while these are primarily intended for use with CSPICE, the format is documented and you can use it directly. It's described at https://github.com/skyfielders/python-skyfield/issues/19 and implemented in Python at https://github.com/brandon-rhodes/python-jplephem/tree/master/jplephem
If you want to do the SPICE computations yourself (numerically solving the differential equations), see https://astronomy.stackexchange.com/a/13491/21
You might also want to use an n-body simulator to do the computations yourself: https://physics.stackexchange.com/questions/25241/what-open-source-n-body-codes-are-available-and-what-are-their-features
IAU SOFA (http://www.iausofa.org/) will give you the International Astronomical Union's "official" libraries to compute positions.
VSOP theory (https://en.wikipedia.org/wiki/VSOP_%28planets%29) is another option.
PyEphem (https://en.wikipedia.org/wiki/PyEphem), which derives from MIT's libastro library (https://stuff.mit.edu/afs/athena/project/xephem/src/xephem-3.5.2/libastro/)
libnova (http://libnova.sourceforge.net/)
If you want to treat planetary orbits as simple ellipses and ignore perturbations, you can find orbital elements at https://ssd.jpl.nasa.gov/txt/p_elem_t1.txt but you'll probably want to visit https://ssd.jpl.nasa.gov/?bodies#elem first
Note that these computational libraries don't always agree with each other or with planetarium software:
https://stackoverflow.com/questions/16293146
If you want a catalog of stars/etc, NOMAD (http://www.usno.navy.mil/USNO/astrometry/optical-IR-prod/nomad) and GAIA (http://gea.esac.esa.int/archive/) are the largest such catalogs (about 1 billion entries each, lots of overlap). The NOMAD link also provides references to smaller catalogs.
For high-resolution astrophotography try ALADIN (http://aladin.u-strasbg.fr/#AladinLite)
If you have pictures of stars you can't identify, try http://astrometry.net/use.html
If you want to find artificial satellites and spacecraft, start at https://en.wikipedia.org/wiki/Two-line_element_set
Sun/moon rise and set times? http://aa.usno.navy.mil/data/docs/RS_OneYear.php
Solar and lunar eclipses? http://eclipse.gsfc.nasa.gov/eclipse.html
Transits of Mercury and Venus across the Sun? http://eclipse.gsfc.nasa.gov/transit/transit.html
Planetary conjunctions?
http://laps.noaa.gov/albers/ast/conjun/conjun.htmlOccultations of stars by asteroids? The International Occultation Timing Association (IOTA) at http://occultations.org/
Saw something really bright that didn't look like a regular star or planet? Look over the observation logs at http://www.amsmeteors.org/fireballs/fireball-report/ (if this occurred fairly recently, be sure to look at the pending reports too).
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$\begingroup$ I also use SIMBAD (simbad.u-strasbg.fr/simbad/sim-fid) if I want to do a quick search for a star or galaxy based on its identifier, and you can also do the reverse and searching for coordinates too. $\endgroup$ – Dean Feb 3 '16 at 10:09
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$\begingroup$ I like this one for seeing what's overhead ucolick.org/~bolte/AY4_00/weblab/project/chart.html . $\endgroup$ – Jack R. Woods Mar 25 '16 at 1:37
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$\begingroup$ That appears to be a framed verison of fourmilab.ch/yoursky -- feel free to add the latter link above. $\endgroup$ – user21 Mar 25 '16 at 13:37
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1$\begingroup$ @uhoh This is a community wiki answer, so please feel free to add them yourself. But, if you run into trouble, let me know and I'll add them. $\endgroup$ – user21 Aug 14 '19 at 15:51
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To add to the excellent answer by barrycarter, there are 2 planetarium-like codes, that I know of, that run on a mac and would make excellent tools for viewing certain astronomical objects. The codes are Stellarium and Celestia. Both turn your computer into your own planetarium where you can search and view objects in space.
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$\begingroup$ It's actually a public wiki answer (not all my content), so you can add to it directly if you'd like. I had considered listing individual planetarium software (Stellarium is my favorite), but thought linking to the wikipedia list page may be more useful and neutral. $\endgroup$ – user21 Oct 1 '17 at 17:04
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2$\begingroup$ The second program is called "Celestia" not "Celestial". $\endgroup$ – posfan12 Oct 8 '18 at 21:25
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$\begingroup$ Celestia even lets you move around different from Stellarium which is more like a planetarium-theatre $\endgroup$ – Alchimista Feb 19 '19 at 10:21
