Horizon coordinate system

An observer on the earth's surface at low latitudes notes that the sun rises in the east and sets in the west. During the night, the same observer would note that the stars and planets also rise in the east, move across the sky, and set in the west. These motions are simply an effect of the earth's daily rotation about its axis. For an observer at the earth's north pole, the north celestial pole (NCP) of the celestial sphere is directly overhead (at the zenith). For this same observer, the stars just above the horizon move all around the horizon at a fixed elevation once per day, never dipping below it. At other northern latitudes, the pole is not directly overhead, but stars sufficiently near the NCP will still appear to rotate around the NCP without setting. Stars further south do rise and set (e.g., the sun), and stars even farther south are never seen by the northern observer.

The motion of stars in an earth-based horizon system is illustrated in Fig. 1. The view is from the west looking east. The figure is drawn for an observer at an intermediate northern latitude (e.g., Kitt Peak, Arizona, USA at Latitude 32 N; #iat = +32°). The horizontal base line is tangent to the earth's surface at the observer's position, and the half circle is the celestial sphere. The observer's latitude 0lat represents the angle from the earth's equator to the observer's position. On the celestial sphere, this is the angle from the celestial equator to the observer's zenith, or equivalently the angle between the horizon and the polar axis.

As the earth rotates about its axis, the earth-based observer sees the celestial sphere rotating about the polar axis (Fig. 1). The tracks of two stars in the horizon system are shown as heavy lines A and B. Consider star B. As the earth rotates, it follows a path parallel to the celestial equator since, on the celestial sphere, it lies at a fixed angle 8 (declination) from the celestial equator. On the figure, the star rises somewhat north of east, moves south as it rises further, crosses directly south of the observer's zenith (i.e., crosses the observer's meridian), then descends as it moves westward, and finally sets north of west. This path is actually a circle around the NCP, but part of the circle is below the horizon; the star sets for part of each 24-hour day, as does the sun. If the star is very near the NCP (star A in Fig. 1), it never sets, and the complete circle could be observed if one could see the star during daylight.

Angle equal to latitude of observer

Plane of earth's surface local to northern observer.

2 northern stars: Star A: near cel. pole Star B: near cel. equator

Rotation axis of celestial sphere (or earth)

Figure 4.1. Sketches of stars rising and setting in the horizon coordinate system; the view is from west to east. The horizontal line is the horizon, the plane tangent to the earth at the observer's position. The half circle is the longitudinal meridian (on the celestial sphere) that passes through the observer's zenith at the time shown. The tracks of two stars A and B, both with northern declinations, are shown.

Plane of earth's surface local to northern observer.

2 northern stars: Star A: near cel. pole Star B: near cel. equator

Rotation axis of celestial sphere (or earth)

Figure 4.1. Sketches of stars rising and setting in the horizon coordinate system; the view is from west to east. The horizontal line is the horizon, the plane tangent to the earth at the observer's position. The half circle is the longitudinal meridian (on the celestial sphere) that passes through the observer's zenith at the time shown. The tracks of two stars A and B, both with northern declinations, are shown.

The altitude 9B of star B shown in Fig. 1 is the maximum altitude reached by the star, which occurs when it transits the observer's meridian. At this point, 9B is directly related to the star's declination 8 and the observer's latitude 0lat,

Thus, given the declination of the star and the latitude of the observer, one can deduce the elevation angle at the time it reaches its maximum elevation.

Navigators at sea can observe the sun to determine their latitude with the aid of (3). The altitude of the sun is measured repeatedly with a sextant at about the time of local apparent noon when the sun is near its highest point. The highest elevation reached by the sun yields, through (3), the latitude of the ship. The declination of the sun changes negligibly due to its annual motion during the observations. Thus it is well known at the time of the observation, even in the absence of a precise clock. A navigator can obtain latitude accurate to ~1' or, by definition, 1 nautical mile in this manner (1 nautical mile = 1' along a meridian, or 1.852 km). Unfortunately, the longitude is completely undetermined. Star sightings at sunset and sunrise may be used for a precise (~ 1') fix of both longitude and latitude if the times of the sightings (Greenwich time or UTC time, see below) are known to within a few seconds.

Annual motion

The annual motion of the earth about the sun has important consequences for astronomers who view the sky from the earth.

Sun and the ecliptic

The earth's orbital motion yields an apparent motion of the sun relative to the celestial sphere (Fig. 3.1). At some date, e.g., March 21, an observer on the earth would find the sun to be in front of a given constellation, if the sun weren't so bright as to obscure our view of the constituent stars.

At some later date, say June 21, the earth will have moved 90° in its orbit about the sun, and according to an earth-observer, the sun would appear to have moved 90° around the celestial sphere as shown in Fig. 3.1. The observer would then see a different group of stars near (and beyond) the sun, such as the large star in Fig. 3.1. Throughout the year, the sun would thus appear to move steadily through the stars at the rate of 360°/365 days ~ 1 deg/day, passing sequentially through the constellations that are well known as the signs of the zodiac and returning to its original position relative to the stars (not to the slowly moving vernal equinox) after 1.00 sidereal year,

1.00 sidereal year = 365.2564 days (4.4)

The track of the sun on the sky (or the celestial sphere) is called the ecliptic. It is a great circle tilted 23.45° relative to the celestial equator. A portion of it is shown in Fig. 3.1. See Table A7 in the Appendix for the several types of "years".

Sun and dark-sky observations

More relevant to astronomers than the signs of the zodiac are the months when certain regions of the sky are accessible for study. Ground-based optical astronomers can not make observations when the sun is above the horizon because the sun is very bright at optical wavelengths, and the atmosphere scatters the light severely. Because of this, the regions that can be observed are those more or less opposite to the sun on the celestial sphere.

A star directly opposed to the sun on the celestial sphere, 180° away, can typically be observed all night (7-11 hours at mid to low latitudes), depending on the season of the year and the latitude of the observatory. When the sun is below the horizon (night time), the star will be above the horizon, and vice versa. Three months later, the sun will be roughly 90° from the star, and it can be observed for only about 1/2 the night or 3 to 5 hours. The star is above the horizon for the usual ~12 hours but for half this time the sun is too. Observers can look up the times of sunset and the end of astronomical twilight. The latter is the time when the sky is quite dark (sun 18° below the horizon); observations usually can begin somewhat before this.

The accessibility of stars for observation also depends upon a number of other factors, including an absence of clouds. Stars usually can not be observed if they are more than —60° from the zenith. At this angle there are about two atmospheric thicknesses (air masses) of atmosphere along the line of sight, with its refractive properties, its absorbing and scattering dust, etc. By definition, there is 1.0 air mass along the line of sight from sea level toward the zenith.

Since the celestial sphere rotates past the observer at 360°/24h = 15°/h, a star that passes more or less overhead would be within the 60° angle of the zenith for a full 8 hours, from hour angle —l-hto +4 h. (The hour angle of a star at a given time is the right ascension of the star less the right ascension of the observatory zenith at that time.) For a northern telescope, stars that pass south of the zenith may be observed for lesser times than those to the north. Stars far toward the north do not set at all and thus may be observed for even longer times, if not all night; see Fig. 1.

Parallax of star positions As noted earlier (Section 3.2), parallax is due to the changing position of the earth as it moves about the sun. A star relatively close to the solar system will appear to move relative to the background stars (Fig. 2). Parallax is the same phenomenon that can be observed while riding in an automobile; the foreground objects (trees and telephone poles) appear to be moving relative to the distant mountains.

The earth's orbit about the sun is only slightly elliptical, so we will approximate it as a circle here. If a foreground star lies normal to the ecliptic plane (plane of earth's orbit), its apparent motion on the celestial sphere throughout the year is a circle as shown with star A in Fig. 2. Define the parallax angle 9par to be the radius of the circular track on the sky. If the distance to the source is r and the radius of the earth's orbit is a,the parallax angle 9par is, from Fig. 2, tan 9par = a/r (4.5)

The radius a can be set to the semimajor axis of the earth's orbit, known as the astronomical unit (AU), a = 1.00 AU = 1.496 x 1011 m (Astronomical unit) (4.7)

One can use (6) to solve for the distance r to the star if 9par is measured.

If the foreground star lies in the ecliptic plane, the path of the star on the sky is a "straight line" (actually a portion of a great circle) of length 29par (Star B in Fig. 2).

Rarth — Apparent track

Figure 4.2. The geometry of trigonometric parallax. The angular position (celestial position) of a relatively nearby star changes relative to the distant background stars (taken to be at infinite distance) as the earth proceeds around the sun. The apparent tracks of two stars are shown, star A at the ecliptic pole with a circular track, and star B in the ecliptic plane with straight-line track. The parallax angle of 0par = 1" occurs if the star is at a distance of 3.09 x 1016 m = 3.26 LY (light years); astronomers call this distance 1.0 parsec (pc).

Rarth — Apparent track

Figure 4.2. The geometry of trigonometric parallax. The angular position (celestial position) of a relatively nearby star changes relative to the distant background stars (taken to be at infinite distance) as the earth proceeds around the sun. The apparent tracks of two stars are shown, star A at the ecliptic pole with a circular track, and star B in the ecliptic plane with straight-line track. The parallax angle of 0par = 1" occurs if the star is at a distance of 3.09 x 1016 m = 3.26 LY (light years); astronomers call this distance 1.0 parsec (pc).

A star at an intermediate position tracks out an ellipse; the semimajor axis of which is 0par. At distances beyond ~1000 light years (LY), the angular motion becomes too small to measure from ground-based telescopes. The Hipparcos satellite (1989— 93) measured parallax angles with precisions of about 1 milliarcsecond (1 mas) for stars brighter than about 9th magnitude and thus could measure distances to about

The definition of the unit of distance used traditionally by astronomers, the par-sec (pc), derives from the geometry of Fig. 2. If the angle is precisely 0par = 1.00", then the distance to the star is specified to be 1.0 parsec (pc), from "parallax" and "arcsec". This defines the parsec which turns out to be 3.26 LY = 3.09 x 1016 m. The parsec is another of those unhappy historical units; however astronomers use it extensively so it is worth knowing. Think: 1 LY ~ 1016 m and 1 pc ~

Telescopes Mastery

Telescopes Mastery

Through this ebook, you are going to learn what you will need to know all about the telescopes that can provide a fun and rewarding hobby for you and your family!

Get My Free Ebook


Post a comment