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Horizon and Equatorial Coordinate Systems

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Horizon and Equatorial Coordinate Systems Module 02 Horizon and Equatorial Coordinate Systems TABLE OF CONTENTS 1. Learning Outcomes 2. Introduction 3. Horizon System of Astronomical Coordinates 3.1. Altitude of the Pole star 3.2. Alt - Azimuth Mounting for Small Telescopes 3.3. Drawbacks of the Horizon System 4. Equatorial System of Astronomical Coordinates 4.1. Local Equatorial System 4.2. Universal Equatorial System 5. Annual Motion of the Sun 6. Local Sidereal Time 7. Summary 1. Learning Outcomes After studying this module, you should be able to explain the essential requirement of an astronomical coordinate system describe the Horizon System of coordinates locate horizon for any observer show that the altitude of the Pole star is equal to the latitude of the observer appreciate the convenience of the Horizon System understand the shortcomings in the Horizon System describe the Equatorial System of coordinates reason out the need for both the coordinates to be independent of time and location explain the Universal Equatorial System and its importance to astronomy describe the apparent annual motion of the Sun define the local sidereal time 2. Introduction In Module 01 we discussed the essential requirement of an astronomical system of coordinates for celestial objects. These requirements are: I. A fundamental great circle, and II. A reference point, or origin. As an example of these requirements, we discussed the terrestrial coordinate system in which equator is the fundamental great circle and the point of intersection of the prime meridian with the equator is the origin. By this system we assign longitude and latitude to places on the earth. We now discuss the Horizon System for which the fundamental great circle is the horizon and the origin is either the north point or the south point of the horizon. It is a simple system and is used by most small telescopes. However, the system has two shortcomings: the coordinates differ from observer to observer at the same point of time, and for the same observer at different times. These shortcomings are sought to be rectified in the Equatorial System of coordinates. 3. The Horizon System of Astronomical Coordinates In this system the fundamental great circle is the horizon. Let us see how we can locate our horizon. The point vertically overhead the observer on the celestial sphere is called the zenith, and the point vertically below the observer is called the nadir (Fig. 2.1). The horizon is the plane perpendicular to the line joining ZZ´. Let us see how we can locate it. Suppose an observer at latitude 휙. The point exactly overhead the observer is called the zenith. Draw the line joining the zenith with the centre of the earth (Fig. 2.1). A plane perpendicular to this line is the horizon of the observer at latitude 휙. As long as an object is above the horizon, it is visible to us. When it goes below the horizon, the object is no longer visible, we say it has set. Invisible Z 흓 Visible Fig. 2.1. Locating the horizon for an observer at latitude 휙. Exercise 2.1: Locate the horizon for an observer at the North Pole. Exercise 2.2: Locate the horizon for an observer stationed at the Equator. North Pole North Pole Horizon Horizon Equator Equator Fig. 2.2. Horizon for an observer at North Pole (left) and at Equator (right). Projected on the celestial sphere with the earth as centre, the horizon is a great circle which appears as shown in Fig. 2.3. Z and Z´ are the poles of the horizon, while P and Q are the north and south celestial poles, respectively. P Z Celestial Sphere Z’ Q Fig. 2.3. Horizon and celestial equator projected on the celestial sphere. In Fig. 2.4, N and S are the north and south points of the horizon. To fix coordinates of an object X, we draw the half great circle ZXLZ´. One of the coordinates of X is then the arc length LX, called the altitude 풂. It is measured from zero to +90° from L to Z and from zero to -90° from L to Z´. To fix the other coordinate either N or S is chosen as the origin. If N is the origin, then the arc length NL, or the spherical angle NZL, is the second coordinates of X, called the azimuth A. In this case the azimuth is measured from 0 to 180 degrees east or west. If S is chosen as the origin, the azimuth is measured from 0 to 360 degrees through west. It must be remembered that the refraction of light due to the earth’s atmosphere increases slightly the altitude of a star. Necessary correction due to atmospheric refraction must, therefore, be applied to the measured altitudes of celestial objects. The complement of the altitude, the arc length ZX, is called the zenith distance of the object. Obviously, in degrees 풛 = ퟗퟎ − 풂. (2.1) Z 퐴 X E N 푎 S Horizon 퐴 L W Z’ Fig. 2.4. The horizon system of coordinates uses the horizon NWSE as the fundamental great circle. The coordinates are the azimuth (퐴) and altitude (푎). 3.1. Altitude of the Pole star is equal to the Latitude of the Observer We immediately apply the horizon system to derive an important result: The altitude of the Pole star is equal to the latitude of the observer. Fig. 2.5 shows horizon for an observer at latitude 휙. Z is the pole of the horizon and so the zenith for the observer is at 휙 to the equator. North Celestial Pole, P, is the pole of the equator and is in the direction of the Pole star. The great circle PZRQZ´T is called the observer’s meridian, or local meridian. The arc length RZ is equal to 휙 by the definition of latitude. The arc length ZP is, therefore, 90 − 흓, because the arc RP equals 90 degrees. This makes the arc length NP, which is actually the altitude of the Pole star in accordance with the horizon system, equal to the latitude 휙. Thus, the altitude of the Pole star at any place is equal to the latitude of that place. So, to locate pole star, look in the north direction at an altitude equal to your latitude. P 휙 Z 푎 N 휙 R T Equator Observer’s Meridian Z’ Q Fig. 2.5. The figure shows the Equator and the horizon for an observer at latitude 흓. Z is the zenith, P is the pole of the equator and points in the direction of the Pole star. The altitude of the Pole star is 푎 = 흓. 3.2. Convenience of the Horizon System The horizon system is a very convenient system. Most small telescopes for amateur astronomers are based on this and use alt – azimuth mounting. The azimuth of an object gives the direction in which to look for the object. The altitude then gives the angle by which the telescope is to be raised from the horizontal to locate the object (Fig. 2.6). Fig. 2.6. A small telescope based on the horizon system. 3.3. Drawbacks of the Horizon System Even if it is convenient, the horizon system suffers from two drawbacks: (i) Since the horizon at different locations on the earth is different (Fig. 2.7), the coordinates of the same object at the same time are different for observers at different places. In other words, the coordinates of an object are location specific. (ii) As the earth rotates, the celestial objects move in planes normal to the polar axis. Since these planes are generally inclined to the horizon, the coordinates of an object keep changing with time for any observer. These shortcomings are sought to be eliminated in the equatorial system of coordinates. N HA To ZA A Equator O 휙퐴 P Q 휙퐵 HB B S To ZB Fig. 2.7. Zenith of the observer at A at latitude 휙퐴 is in the direction ZA and that of observer at B at latitude 휙퐵 is in the direction ZB. The horizon of observer at A is HA while that of the observer at B is HB. 4. The Equatorial System 4.1. Local Equatorial System The fundamental great circle chosen here is the celestial equator, whose plane is parallel to the equatorial plane of the earth. The north celestial pole (NCP) P and the south celestial pole (SCP) Q are the poles of the celestial equator, (Fig. 2.8). The great circle PTQR is the observer’s meridian, or the local meridian. Any half great circle through the two celestial poles is called an hour circle as it measures the time elapsed since an object on it crossed the observer’s meridian. To get the coordinates of the object X we draw an hour circle through it. Then one coordinate of X is the arc length LX on its hour circle. The coordinate is called the declination of X, denoted by 훿. It is measure from zero to +90° north of the celestial equator P Observer’s Meridian X 훿 R T 퐻 Equator L Observer’s Q Meridian Fig. 2.8. Equatorial coordinates 퐻 and 훿 of the object X. and from zero to -90° to the south of the celestial equator. The other coordinate is the hour angle (푯) measured from the point R on the observer’s meridian westward (clockwise when viewed from the north celestial pole.) to L, the intersection of the hour circle of X with the celestial equator. It will be noticed that as the object appears to rotate from east to west, the declination remains unchanged while the hour angle increases steadily from zero to 360° in 24 hours.
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