How would you go about determining which regions of the sky are observable year-round, i.e. they're observable at some point every night of the year. This question, of all things, kept me up last night, but I was not able to quite work it out.
To simplify the question, I first tried to think about which regions of the sky would be above the horizon when the sun was not. Obviously an observer's circumpolar region (for those of us outside of the arctic) would fall into this category, but there are points at lower declination that are below the horizon for less time than the length of the shortest night, so they must also be visible at some point every night. This region, if my thinking is correct, would be everywhere north of the Tropic of Cancer, roughly speaking (for a northern observer, and ignoring for now the fact that stars aren't visible immediately after sunset).
But it's only a small area around the Summer Solstice that would rise and set with the sun on the shortest night of the year. A point on the Tropic of Cancer at RA 18h would be visible before and after sunset even when the sun was at the same RA, and this would hold true for points south, all the way down to the Winter Solstice or just north of it?
Does this thinking hold up? How would you determine whether a point falls into this category, and how would it depend on latitude? How would the calculation of this region change if you considered only the region that was above the horizon when the sun was at least e.g. 12° below the horizon?
From an observing standpoint, an interesting challenge (if you were generous with the definition of interesting) would be to see what would be the most southerly (northerly) object you could observe year-round. Has anyone attempted this?