Coordinates

Coordinates. You have to have them for everything you do (at least in astronomy), they’re a pain in the neck to convert, and you have to learn to read all of them. There are three that we need to discuss—the altitude azimuth coordinate system, the equatorial coordinate system, and the galactic coordinate system. All of these should be written in reference to the J2000 epoch.

First, we’ll discuss the simplest of these—the altitude-azimuth (or horizon) coordinate system. It’s a system based on you (yes, for once, the universe does indeed revolve around you). This system basically takes where you are right now and puts all the stars and whatnot in relation to you. To understand this system, you first must know what a zenith is. For any of you who are slightly literate, you know that “zenith” means that something is at its highest point. So in astronomy, the point that is directly above you is the zenith. Because the heavens are generally referred to as a sphere kind of thing, but you can only see half of it at any given time, the altitude is the angle from the flat horizon line to the height of the star. Now, we have to measure the azimuth. The azimuth is basically the angle that the star is located on in reference to North. Confused? Here’s a diagram:

For reference (although you have doubtless already figured this out), the altitude-azimuth system is measured in degrees, just like our very own latitude/longitude system. This system is difficult to use in practice, purely because it completely depends on where you are. Because of this, the coordinates of everyone in the world are different. For this reason, astronomers hate using this system, preferring instead the equatorial coordinate system, which we’ll talk about next.

The equatorial coordinate system is the one that you all have heard of—the one with the right ascension and declination stuff. (And if you haven’t, then what the heck are you doing in astronomy?!) The system, as its name implies, is based on Earth’s latitude/longitude system, but isn’t affected about the rotations and all. Declination (DEC) is the astronomy version of latitude—and it’s measured from the equator, too! It’s just the degrees north or south of the celestial equator, which has the same plane as Earth’s equator, only more expanded. Right ascension (RA) is—you guessed it—the longitude of astronomy, and this is based on the vernal equinox, which is just astronomy’s version of the Prime Meridian. Right ascension is usually measured in hours, minutes, and seconds, going all the way up to 24 hours. Every 360 degrees equals 24 hours, so 15 degrees equals one hour, and one degree equals 15 minutes. Lovely picture here:

Because this is based on something that doesn’t change as the Earth rotates, the RA/DEC coordinates don’t change on a day to day—or even year to year—basis. This means that whatever the values of RA or DEC are, they remain constant.

Again, this is the coordinate system most often used by astronomers—and it is also the system in common use in Science Olympiad. If you are competing in Astronomy/Reach for the Stars, please know this system better than the back of your hand, and you will have a chance of not failing miserably in these events.

The last system you really need to know is the galactic coordinate system. The thing about the other two coordinate systems is that it’s based off of Earth—and Earth is tilted about 62.87° away from the equator of the Milky Way (henceforth referenced as “the Galaxy”). And that’s a problem (as are our egos, but we won’t go into that). This is really a lot like the equatorial system, only it’s based off the Galaxy midplane, which is the Galaxy’s equator. (Technically, it is based off the Sun drawing a plane parallel to the Galaxy Midplane, but really, the two are so close together that it doesn’t make a difference.) Latitude runs parallel to the Galaxy midplane, and longitude is an angle in reference to the North Galactic Pole (NGP), which is perpendicular to the Galaxy midplane. We measure the latitude and longitude, however, from the Sun. (We design a system so as to not be based off of Earth, so what do we base it off of? Yes, that’s right—the Sun, despite the fact that it’s nowhere near to the centre of the galaxy.) Before we go any further, a diagram to help all of you mind-numbingly confused people.

As you can see from the diagram, there is a line that connects the galactic centre to the Sun. This is where latitude (the universal symbol of which is b) is measured from. It goes from this line until it hits the line where it passes directly beneath the star, and there you have it! Your latitude coordinates.

We measure longitude (the symbol of which is l) very similarly. Since the Sun just happens to be directly beneath the NGP (what a coinkydink, don’t you think?), we use the NGP as a reference point. When we measure the longitude, we measure the angle between the Galaxy midplane and the height of the star in reference to the distance between it and the Sun. (Use your logic and the Pythagorean theorem here—assuming the same height from the Galaxy midplane, the farther out your star is, the smaller the angle. This is why you have to know the distance.)

Or, for a slightly better description, taken from Carroll and Ostlie:

“The Galactic coordinate system exploits the natural symmetry introduced by the existence of the Galactic disk. The intersection of the midplane of the Galaxy with the celestial sphere forms what is very nearly a great circle, known as the Galactic equator. This orientation is depicted in Fig. 24.16. Galactic latitude (b) and Galactic longitude (l) are defined from a vantage point taken to be the Sun, as shown in Fig. 24.17. Galactic latitude is measured in degrees north or south of the Galactic equator along a great circle that passes through the north Galactic pole. Galactic longitude (also in degrees) is measured east along the Galactic equator, beginning near the Galactic center, to the point of intersection with the great circle used to measure Galactic latitude.

By international convention, the J2000.0 equatorial coordinates of the north Galactic pole (b=90°) are:

­a NGP: 12 h 51 m 26.28 s

­d NGP: 27°7′ 41.7”,

­and the origin of the Galactic coordinate system (l0 0°, b0 0°) corresponds to

­a0: 17h 45m 37.20s

­d0: 28°56′ 9.6”.”

Now, for conversions. Brace yourselves for some brutal math rearing its ugly head. If you do not know how to do spherical trigonometry—which is nowhere near nice, normal plane trigonometry—then stay away and keep your sanity. YOU NEED TO HAVE TAKEN PRECALC TO BE ABLE TO UNDERSTAND THIS (or at least advanced trig).

Since the altitude-azimuth system is based on, well, us, we can’t convert it to anything. But equatorial system to the galactic coordinate system and vice versa? Astronomy says yes.

The equations for converting equatorial coordinates to galactic coordinates are as follows:

\sin {b} =\sin { {\delta}_{NGP} } \sin {\delta} +\cos { {\delta}_{NGP} } \cos {\delta} \cos { (\alpha -{\alpha}_{NGP}) } \\ \cos {b} \sin { ({l}_{NCP}-l) } =\cos {\delta } \sin { (\alpha -{a}_{NGP}) } \\ \cos {\delta} \cos { ({l}_{NCP}-l) } =\cos { {\delta}_{NGP} } \sin {\delta} -\sin { {\delta}_{NGP} } \cos {\delta} \cos { (\alpha -{\alpha}_{NGP}) }

The equations for converting galactic coordinates to equatorial coordinates are as follows:

\sin {\delta} =\sin { {\delta}_{NGP} } \sin {b} +\cos { {\delta}_{NGP} } \cos {b} \cos { ({l}_{NCP}-l) } \\ \cos {\delta} \sin { (\alpha -{\alpha}_{NGP}) } =\cos {b} \sin { ({l}_{ NCP }-l) } \\ \cos {\delta} \cos { (\alpha -{\alpha}_{NGP}) } =\cos { {\delta}_{NGP} } \sin {b} -\sin { {\delta}_{NGP} } \cos {b} \cos { ({l}_{NGP}-l) }

The “l” is the same as l –I’m just not good enough with LaTeX to make fancy fonts work. I won’t work examples this time because if you saw the equations that I had to type in…well, suffice it to say you wouldn’t want to either.

And that’s a wrap, folks. Do have fun.

__________________________________________________________

TL;DR: The altitude-azimuth coordinate system, the equatorial coordinate system, and the galactic coordinate system are the three main coordinate systems seen in astronomy. The altitude-azimuth system is based on where you are standing and is measured in degrees. The equatorial system is based on the orientation of the Earth. Longitude (right ascension) is measured in hours, while latitude (declination) is measured in degrees. The galactic coordinate system is based on the orientation of the Milky Way and the Sun, and measures longitude in hours and latitude in degrees. The math for converting is awful—brutally difficult and takes absolutely forever.

Further Reading:

http://spider.seds.org/spider/ScholarX/coords.html

http://www.shodor.org/refdesk/Resources/Applications/AstronomicalCoordinates/

http://www.physics.uc.edu/~sitko/AdvancedAstro2011/1-TheSky/Sky.pdf

https://dept.astro.lsa.umich.edu/ugactivities/Labs/coords/index.html

http://astronomy.swin.edu.au/cosmos/N/North+Galactic+Pole

http://www.physics.uc.edu/~sitko/Fall2002/1-Sky/sky.html

*Credit for information also goes to Carroll/Ostlie.

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