Intro to Positional Astronomy
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Modern astronomy relies on the ability to precisely measure the location of objects on the sky. Just like on Earth's surface, a coordinate system can be used to uniquely specify every point on it. The term 'celestial sphere' refers to the apparent sphere of sky that encases the Earth. In reality, objects in space are located at different distances from the earth, so there is no physical sphere on which stars, planets and other celestial objects sit. But from the surface of the earth we cannot readily determine if objects are near or far from us in space, we only see the angular separations objects appear to have on the celestial sphere. Just as a location on the surface of the earth can be uniquely specified using latitude and longitude, so can a location on the 'surface' of the sky be uniquely specified with the analogous measurements of declination and right ascension.
Declination is a measure of how far above or below the 'celestial equator' an object is, just as latitude is a measure of how far above or below the Earth's equator an object is. The celestial equator is the imaginary line that lies directly above the Earth's equator. Thus, declination and latitude are concurrent; a celestial object that passes directly overhead has the same declination as you have latitude.
For example, Polaris, the North Star, has a declination very near 90 degrees, and if you stood at the North Pole (which has a latitude of 90 degrees), it would be directly overhead. An example that is closer to home is Omicron Persei (abbreviated o Per), a star in the constellation Perseus. o Per has a declination of +32 degrees. As you may know from caching around town, Tucson is at a latitude of +32 degrees. So, in between rising in the east and setting in the west every day, o Per passes directly over our heads*
Right ascension (abbreviated RA) is a measure of how far around the celestial sphere an object is from the 'Vernal Equinox,' just as longitude is a measure of how far around the earth on object is from the Prime Meridian. The primary difference between longitude and RA is that longitude is measured from 0 degrees (the Prime Meridian) to 180 degrees (approximately the International Date Line), with E and W or + and - denoting the direction. RA is typically measured from 0 hours to 24 hours, starting at the Vernal Equinox, proceeding eastward and ending back at the beginning.
It may be strange to think of 'hours' as anything other than a unit of time. But in the case of RA, 'hours' is no different than radians or degrees. Although there is a good reason to measure RA in this way, for those unfamiliar the concept, it's better to think of angle 'hours' and time 'hours' as completely different entities, just as temperature 'degrees' and angle 'degrees' are different.**
Much like choosing the Prime Meridian to have longitude of 0 degrees is arbitrary, so too is choosing the Vernal Equinox (just another point on the sky) to have an RA of 0 hours. But, by making this choice, the RA of all other objects is set.
Both Earth and sky coordinate systems are relatively straightforward by themselves, but what complicates matters is the fact that the Earth rotates. While the sky coordinates are fixed for someone floating in space, on Earth the system appears to rotate. The declination of the point directly above you does stay the same, but the RA is constantly changing.
With a basic knowledge of RA, declination and LST (not covered here, sadly), a person can accurately predict what objects will be visible in the sky and when. For astronomers, it is very important to calculate accurately where these objects will be. Science telescopes view only a small patch of sky at any one time (typically less than one square degree, or 0.002% of the sky). If they didn't not know the precise location of, say, a star, a lot of time would be wasted trying to find it, and it could easily be confused with another star.
*Technically, it passes 3 minutes and 18.05 seconds north from being directly overhead, since its declination isn't exactly +32 degrees. Nor, for that matter, is Tucson's latitude exactly +32 degrees.
**In fact, there's no reason we couldn't measure Tucson's location in the same units of 'hours.' If we did, we would find it to be located at 16 hours, 36 minutes and 12.208 seconds, as measured from the Prime Meridian eastward.
Special thanks to TheBlizzles for noticing an error in the longitude calculation that would place potential cachers somewhere near Bermuda.
(No hints available.)