Horizon and Equatorial Coordinate Systems

Home , Capricorn (astrology), Hour circle, Meridian (astronomy)

Module 02

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

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.

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)

퐴

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.

휙 Z 푎

N 휙

T Equator

Observer’s Meridian Z’

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

Equator 휙퐴 O P Q 휙퐵

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

Observer’s

Meridian X

훿

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. For this reason, the hour angle is expressed in terms of hours rather than degrees, 1 hour corresponding to 15°. In this system, too, one of the coordinates, namely the hour angle, keeps changing with

time as the object moves across the sky. That is the reason the system is called local equatorial system. This shortcoming is rectified by choosing as origin a point on the celestial equator

훿 R

T 퐻 Equator 훼 L

ϒ Observer’s Ecliptic Meridian

Fig. 2.9. Universal equatorial coordinates of X.

which is not stationary but moves with the same speed and in the same direction as the object. This ensures that there is no relative motion between the two.

4.2. Universal Equatorial System

The point, or origin, chosen for this purpose is the first Point of Aries, or the vernal equinox (ϒ). First Point of Aries is the point of intersection of the apparent annual path of the Sun and the celestial equator. The arc length measured from the vernal equinox eastwards (counterclockwise when viewed from the north celestial pole) to the hour circle of the object, the arc length ϒL, is known as the right ascension (휶) (Fig. 2.9). It is also measured in hours because the vernal equinox completes one round of the sky in 24 hours as does any other object

in the sky. Thus, the right ascension remains unchanged as the object appears to move across the sky. We have already noted that the declination of an object is unaffected by the motion of the object across the sky. Since both the coordinates remain unchanged as the object changes its position in the sky, this system is called the universal equatorial system. This system is suitable for making catalogues of stars.

5. Annual Motion of the Sun

Let us briefly look at the apparent annual motion of the Sun. The plane of this motion is called the ecliptic. It makes an angle of 23.5° with the plane of the celestial equator. The ecliptic plane intersects the plane of the equator at two points, called the vernal equinox (ϒ) and the autumn equinox (흮 ). Around 21 June the Sun is at its northern most point in its apparent annual orbit.

Celestial Sun 21 June Equator 흮

23.5° W E

ϒ 23 Dec Ecliptic

Fig. 2.10. The apparent annual motion of the Sun in the sky is called the ecliptic. Its plane intersects the plane of the celestial equator at two points ϒ and , the vernal and the autumn equinoxes respectively.

Its declination is 23.5°. The event is called the summer solstice. It then moves southwards; in India we call it Dakshinayan (towards south). Around 23 September, it crosses the equator and moves into the southern hemisphere. On this day the duration of day and night is equal, hence it is called autumn equinox. For those living on the equator, the Sun rises exactly in the east on this day. Around 23 December, the Sun reaches its southern most point, the winter solstice. Its declination then is -23.5°. Then it reverses its direction and begins moving northwards; in India it is called Uttarayan. Around 21 March, the Sun crosses from the southern hemisphere into the northern hemisphere. This point of intersection between the equator and the ecliptic is called the first point of Aries, or the vernal equinox. It is this point that we use as the origin for the universal equatorial system of coordinates. There is no celestial object at this point in the sky; it is just a reference point. On the day of vernal equinox, the duration of day and night is equal, and for those living on the equator, the Sun rises exactly in the east. It must be noted that the Sun rises exactly in the east only for two days in the whole year, and that too for people living in the tropics, the region between the Tropic of Cancer and the Tropic of Capricorn. On all other days of the year the Sun rises either in the north of east or the south of east. A few week’s observations of the rising Sun should confirm this. Throughout the year the rising point of the Sun keeps oscillating between 23.5° north and 23.5° south.

6. Local Sidereal Time

Since the coordinates of an object on the universal equatorial system do not change with time or location, this system is widely used for preparing catalogues of celestial objects. The hour angle of ϒ is called the local sidereal time (LST), the time measured with reference to the distant stars. The relation between 퐻 and 훼 is through the sidereal time (Fig. 2.9):

LST = 퐻 + 훼. (2.2)

We shall discuss the sidereal time a little later.

7. Summary  Essential requirements for an astronomical coordinate system are a reference great circle and a reference point.  For Horizon System the reference great circle is the horizon and the reference point is the intersection of the horizon and half great circle ZXZ´.  The arc length along ZXZ´ from the horizon to X is called the altitude of X.  Either the North point or the South point of the horizon is used the reference point for measuring the azimuth of X.  Horizon System is a very convenient system and most small telescopes use alt – azimuth mounting.  Horizon for each observer is different. Moreover, the horizon for the same observer changes as the earth rotates on its axis.  The two drawbacks of the horizon system are that the coordinates change with the observer and with time for the same observer.  These drawbacks are sought to be rectified in the Equatorial coordinate system.  Equatorial system uses celestial equator as the reference great circle and its intersection with the observer’s meridian (great circle through the poles and the zenith) as the reference point for measuring the hour angle of X.  In hour angle – declination system, declination of an object is fixed, while the hour angle of the object changes as the earth rotates on its axis.  This shortcoming, a coordinate changing with time, is rectified by choosing the first Point of Aries as the reference point.  The first Point of Aries moves just like the object X, so the coordinate measured from this point to the hour circle of X stays fixed. This coordinate is called the right ascension.  The right ascension – declination system is called the universal equatorial system.  Since both the coordinates of a star in this system remain unchanged as the star moves about in the sky, this system is suitable for making catalogues of stars.  The annual path of the Sun (ecliptic) is inclined to the equator at an angle of 23.5°.

 At summer solstice the Sun is at 23.5° N and at winter solstice it is at 23.5° S. This motion of the Sun is responsible for the shifting point of Sunrise during the year.  Sidereal time is the time reckoned with respect to the distant fixed stars.  Right ascension and hour angle are related through the local sidereal time.