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# Coordinate Systems

It takes little talent to see what lies under one's nose, a good deal to know in what direction to point that organ.

--W. H. Auden

## Overview

• Learn how to specify distance on the sky as angular distance
• Explore how coordinate systems function
• Understand the definition and use of the altitude-azimuth coordinate system
• Understand the definition and use of the equatorial coordinate system
• Place the apparent motion of the night sky in the equatorial coordinate framework

### Measuring Distance on the Sky

When thinking about positions and distances on the sky, think about it as the inside of a hollow, solid sphere. The apparent distance between two points on the dome of the sky is the angular distance, which is the angle between the two objects, measured with the vertex at the center of the sphere. Any angle measured with the vertex at the center is said to be measured in degrees of arc. Fractional values can be specified in arcminutes and arcseconds. Note that 1 degree of arc is always the same size on the sphere of the sky, no matter where it is measured on the "surface" (Figure 1a). This means that it doesn't matter where an equator or poles are defined.

Figure 1a (left): Angular distance or angular size (Figure 1b) can be measured in any direction on any part of the sky.
Figure 1b (right): Angular distance and angular size.   Figure 1c: Using your hands to estimate angular size.

Next, we can then define a coordinate system based on a polar axis and equator, to specify exact positions. There are several different coordinate systems for the sky, which are based on different axes. However, remember that angular distance remains independent of the coordinate system used.

### The Earth's Coordinate System

You're already familiar with the Earth's coordinate system, shown in Figure 2. We specify locations and directions on the Earth with the cardinal directions, North, East, South, West. The Earth's rotation conveniently defines the North and South poles as the points that are on its rotational axis. The equator is on the surface, where the Earth is sliced exactly in half between the poles. This natural geometry, shown in Figure 2a, defines a coordinate system with the cardinal directions and thus allows us to to specify positions and distances on the surface. Note that because the Earth is a sphere, coordinate distances are really angles. Latitude is defined as an angle in degrees, North or South of the equator, with the vertex of the angle at the center of the Earth. Longitude is an angle East or West of of the Greenwich Meridian in England, which is arbitrarily defined to be 0º longitude. Longitude is measured on a parallel, which is a plane parallel to the equator, with the vertex on the Earth's axis, as shown in Figure 2b. Note that the actual distance between lines of longitude is smaller near the poles compared to the equator, as seen in Figure 2b; whereas the distance between lines of latitude remains constant. The term "meridian"generally refers to any line that runs pole to pole, on the surface of a sphere.

Figure 2a (left): The Earth's coordinate system, latitude and longitude. The central axis runs through the North and South poles.
Figure 2b (right): The pole-to-pole lines along the surface are lines of longitude. Longitude can be measured along any parallel, for example, the longitude between A and B is the same as between C and D. However, the distance between A and B is smaller than the distance between C and D.  ### The Altitude - Azimuth Coordinate System

Now consider the sky, again as if it were the inside of a solid, hollow sphere. A simple, purely local, coordinate system for the sky has an axis defined by the points directly overhead and and underfoot of the observer. The overhead point is the zenith and the underfoot point is the nadir. The equator of this coordinate system is the circle that slices exactly between the zenith and nadir, corresponding to the observer's horizon. Altitude is measured in degrees above the horizon, so that the zenith is at 90º altitude, while the horizon is 0º altitude. Note that 0º altitude is not necessarily the same as your observed horizon, because the ground may not be flat, and buildings or trees might be in your sight line. It is important to measure altitude from the coordinate system's horizon, not the observed one. Azimuth is measured on a parallel to the horizon, in degrees East from North, like the markings on a compass: 90º is due East, 180º is due South, and 270º due West. If only approximate directions are required, the azimuthal direction can be specified with just cardinal directions, for example "Capella rises in the NNE". Figure 3 illustrates the altitude-azimuth coordinate system. On the sky, the "meridian" usually refers to the pole-to-pole line that runs overhead, through the zenith.

These coordinates are usually given with altitude first, azimuth second: in Ann Arbor, the Sun's highest position in the sky on the first day of summer is 71.2º at 180º. Figure 3: The Altitude-Azimuth coordinate system.

### The Equatorial (Celestial) Coordinate System

Equatorial coordinates, also called simply celestial coordinates, are the standard coordinate system for the sky. It is meant for the fixed sphere of the stars within which the Earth and Sun move. The equatorial system is almost the same as Earth's coordinate system, since it is also defined by the Earth's axis and equator. The celestial equator is the projection of Earth's equator onto the sky, and the North and South celestial poles (NCP and SCP) are the points on the sky directly above Earth's North and South poles.

Instead of latitude, the North-South celestial coordinate is called Declination (Dec). As on the Earth, North-South positions are measured from the celestial equator, with the vertex at the Earth's center. Like latitude, it is measured in degrees, with negative values for positions South of the celestial equator. The equator is 0º Dec, and the South celestial pole is at -90º Dec. The East-West celestial coordinate is called Right Ascension (RA). Like longitude, East-West positions are measured on a parallel, with the vertex on the Earth's axis, from an arbitrary zero-point meridian on the fixed sky. This prime meridian is set at the March equinox, which is the Sun's position on the sky, relative to the background stars, on the date that night and day are of equal length (see the Seasons Activity for details). Instead of degrees, RA is measured in hours, minutes, seconds, with 24 hours corresponding to the full circle. This is convenient because we can watch an object moving from East to West across the sky, returning to about the same position in 24 hours. As time passes, new objects rise above the horizon in the East, and so later hours correspond to objects farther East. For this reason, RA increases from West to East. Figure 4 shows the equatorial coordinate system.

Important: RA, longitude, and azimuth are measured around a coordinate system's axis, instead of central point (see Figures 4 and 2b). This means, for example, that 1 hour of RA is larger at the equator than near the pole (see Figure 4). Similarly, 1 degree of longitude also varies in size with distance from the equator. Thus, a degree of longitude is not a degree of arc, recalling that degrees of arc are constant in size, regardless of position on the sphere (Figure 1). On the other hand, Dec, latitude and altitude are all measured in degrees of arc, because they are measured from the center of the coordinate systems (Figures 4 and 2a).

Coordinates are normally given with RA first, then Dec. As usual, fractional units may be given as a decimal, or in minutes and seconds, with 60 minutes in an hour of RA, and 60 seconds in a minute. Thus, the equatorial coordinates for Vega are 18h 36m 56s and 38º 47' 01"

Figure 4: Equatorial Coordinates and the rotation of the sky. ### Apparent motion of celestial objects

Because of the Earth's rotation, we have the illusion of being on a fixed Earth with the sky turning, marking the hours of Right Ascension as the as the stars cross from East to West. As mentioned above, "meridian" is a general term that refers to any North-South line on the sphere, connecting the poles. Astronomers also refer to the "local meridian", or simply "The" meridian, which is the meridian running directly overhead, i.e., through the zenith. When an object transits (crosses) the meridian, it is also at its highest point in the sky. Circumpolar objects are an exception; these have positions close enough to the celestial pole that they never set below the horizon. Therefore, they transit the meridian twice, at the highest point above the pole and also at their lowest point below the pole (Figure 4; see the Southern Constellations activity).

## Activities

Your instructor will tell you which of the following files you will use.

Last updated: 12/15/11 by SAM and MSO