Celestial Coordinates

On Earth, one way to describe a location is with a coordinate system which is fixed to the Earth's surface.

The system is oriented by the spin axis of the Earth, and has special points at the North and South Poles. We use lines of latitude and longitude to demarcate the surface. It's obvious that latitude is measured away from the equator. But where is the starting point for longitude? There is no "obvious" choice. After a lot of dickering, European nations finally decided to use the location of the Greenwich Observatory in England as the starting point for longitude.

There are several ways to specify a location -- for example, that of the RIT Observatory. One can use degrees:

 latitude 43.0758 degrees North, longitude 77.6647 degrees West of Greenwich

Or degrees, minutes and seconds:

 latitude 43:04:33 North, longitude 77:39:53 West 

Or, in the case of longitude, one can measure in time zones. The sun will set at the RIT Observatory about 5 hours and 11 minutes later than it does at Greenwich, so one could say

 latitude 43:04:33 North, longitude 05 hours 11 minutes West

The celestial coordinates

On can make a similar coordinate system which is "fixed to the sky":

Once again, we use the Earth's rotation axis to orient the coordinates. There are two special places, the North and South Celestial Poles. As the Earth rotates (to the East), the celestial sphere appears to rotate (to the West). Stars appear to move in circles: small ones near the celestial poles, and large ones close to the celestial equator:

• Altitude is the angular distance of an object above the local horizon. It ranges from 0 degrees at the horizon to 90 degrees at the zenith, the spot directly overhead.
• Azimuth is the angular distance of an object from the local North, measured along the horizon. An object which is due North has azimuth = 0 degrees; due East is azimuth = 90 degrees; due South is azimuth = 180 degrees; due West is azimuth = 270 degrees.

These two angles specify uniquely the direction of any object in the sky. Some telescopes have alt-az mounts which swivel in these two perpendicular axes; camera tripods and tank turrets are other examples of alt-az devices.

The altitude of an object is especially important from an practical point of view: any object which has an altitude less than zero is below the horizon, and hence inaccessible. Moreover, the altitude of an object is related to its airmass, a measure of how much air the light from that object must traverse to reach the observer. The larger the airmass, the more light is scattered or absorbed by the atmosphere, and hence the fainter an object will appear. We'll deal with airmass at greater length a bit later.

However, note that two observers at different locations on Earth will not agree on the (alt, az) position of an object. Moreover, as the Earth rotates, an object in the sky appears to move from East to West, so its (alt, az) position changes from moment to moment.

It is possible to convert from (RA, Dec) to (alt, az), or vice versa. One needs to know two factors:

• the location of the observer on Earth
• the time of the observation

• Textbook on Spherical Astronomy by W. M. Smart
• Computational Spherical Astronomy by L. G. Taff
• Spherical Astronomy by R. M. Green
• Practical Astronomy with your Calculator by P. J. Duffet-Smith
• Astronomical Formulae for Calculators by J. Meeus

In these modern times, it's usually easiest to use one of the many fine planetarium programs on a computer to do this work.

You've learned trigonometry in high school: sines, cosines, the Pythagorean Theorem, and all that jazz. However, unless you went to a really good high school, you probably restricted your calculations to planar geometry. Unfortunately, the sky is not a plane. We measure positions and coordinates on the inner surface of an imaginary sphere. That means that the old rules don't always work anymore. The subject of spherical trigonometry is not a simple one, but, in this course, we will only peek into it.

Given two vectors, a and b, what is the distance between them? On a plane, we can break up each vector into its components and use the Pythagorean theorem:

Along the surface of the celestial sphere, if we want to find the angular separation between two points p1 and p2, we need to use the law of cosines. In the usual case, the two points are expressed in Right Ascension (α) and Declination (δ), like so:

In this case, the law of cosines becomes

which gives us the cosine of the desired angular distance γ.

If we are interested in very small angular distances on the sky -- the separation between the two components of a binary star, for example, or the distance between two of the moons of Jupiter -- then there are two common approximations. First, if we start with the RA and Dec coordinates of the two points, we can make a pseudo-Pythagorean formula; all we have to do is correct the difference in Right Ascension with the cosine of the Declination.

Second, if we start with a picture of some very small region of the sky, together with an indication of the scale in arcseconds, like this:

then we can

• pick any two orthogonal directions on the picture
• measure the separation (in arcseconds) in each direction
• use the good old Pythagorean formula