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Unit 2 Basic Concepts of Positional Astronomy

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Unit 2 Basic Concepts of Positional Astronomy Basic Concepts of UNIT 2 BASIC CONCEPTS OF POSITIONAL Positional Astronomy ASTRONOMY Structure 2.1 Introduction Objectives 2.2 Celestial Sphere Geometry of a Sphere Spherical Triangle 2.3 Astronomical Coordinate Systems Geographical Coordinates Horizon System Equatorial System Diurnal Motion of the Stars Conversion of Coordinates 2.4 Measurement of Time Sidereal Time Apparent Solar Time Mean Solar Time Equation of Time Calendar 2.5 Summary 2.6 Terminal Questions 2.7 Solutions and Answers 2.1 INTRODUCTION In Unit 1, you have studied about the physical parameters and measurements that are relevant in astronomy and astrophysics. You have learnt that astronomical scales are very different from the ones that we encounter in our day-to-day lives. In astronomy, we are also interested in the motion and structure of planets, stars, galaxies and other celestial objects. For this purpose, it is essential that the position of these objects is precisely defined. In this unit, we describe some coordinate systems (horizon , local equatorial and universal equatorial ) used to define the positions of these objects. We also discuss the effect of Earth’s daily and annual motion on the positions of these objects. Finally, we explain how time is measured in astronomy. In the next unit, you will learn about various techniques and instruments used to make astronomical measurements. Study Guide In this unit, you will encounter the coordinate systems used in astronomy for the first time. In order to understand them, it would do you good to draw each diagram yourself. Try to visualise the fundamental circles, reference points and coordinates as you make each drawing. You could use clay/plastecene balls or spherical potatoes to model various planes on the celestial sphere. You could also mark the meridians and the coordinates on them for a visual understanding of the coordinates we describe in this unit. 29 Basics of Astronomy Objectives After studying this unit, you should be able to: • identify the fundamental great circles such as horizon, celestial equator, ecliptic, observer’s/local meridian; • assign the horizon and equatorial coordinates of a celestial object; and • calculate the apparent solar time at a given mean solar time on any day. 2.2 CELESTIAL SPHERE When we look at the clear sky at night, the stars appear to be distributed on the inside surface of a vast sphere centred on the observer . This sphere is called the celestial sphere (Fig. 2.1). Fig.2.1: The celestial sphere centred on the observer. Note that the celestial sphere is extremely large compared with the Earth; for all practical purposes, the observer and the centre of the Earth coincide Actually, the stars are at different distances from us. But since the distances of the stars are very large, they appear to us as if they are at the same distance from us. So it is sufficient if only the directions of stars are defined on the sphere. Since the distance is not involved, it is usual to take the radius of the celestial sphere as unity . You know that on the surface of a sphere , we need just two coordinates to describe the position of a point. We shall study below some ways used to specify these two coordinates for stars and other celestial objects. But before that you should familiarise yourself with the geometry of a sphere. 30 2.2.1 Geometry of a Sphere Basic Concepts of Positional Astronomy Let us consider a sphere of radius R = 1 unit (Fig. 2.2). We know that a plane intersects the sphere in a circle. If the plane passes through the centre of the sphere we get a great circle , and if it does not pass through the centre then it is a small circle . Fig.2.2: Great and small circles on a sphere. A great circle is the intersection of a sphere and a plane passing through its centre. In Fig.2.3, FG is a great circle whose plane passes through the centre O, and EH is a small circle. Points P and Q are called the poles of the great circle FG . Notice that PQ is the diameter of the sphere perpendicular to the plane of the great circle FG . The distance between two points on a sphere is measured by the length of the arc of the great circle passing through them. It is the shortest path on the surface of the sphere between these points. Thus, in Fig. 2.3, the distance between points C and D on the great circle FG is the arc length CD . It is equal to R (= OD ) times the ∠COD in radians. Since R = 1, arc CD = ∠COD . In astronomy, it is convenient to denote distances on a sphere in angles (radians or degrees) in this way . An angle between two great circles is called a spherical angle . It is equal to the angle between the tangents to the great circles at the points of their intersection. It is also the angle between the planes of the two great circles. In Fig. 2.3, PACQ and PBDQ are two great circles intersecting at the points P and Q. ∠APB and ∠CPD are spherical angles between these great circles. These angles are equal. They are also equal to ∠COD , since the arc length CD is common to both great circles. P E H A B F O G D C Q Fig.2.3: Spherical angle is the angle between two great circles 31 Basics of Astronomy 2.2.2 Spherical Triangle A closed figure formed by three mutually intersecting great circles on a sphere is called a spherical triangle . For example, in Fig. 2.3, PCD is a spherical triangle. All the 6 elements of the triangle, namely, 3 sides and 3 angles are expressed in angular units. The elements of a spherical triangle ABC in Fig. 2.4 are the 3 spherical angles denoted by A, B and C and the 3 sides denoted by a, b and c. B a c C b A Fig.2.4: Spherical triangle is bounded by three arcs of great circles: AB, BC and CA . The spherical angles are A, B, C and the corresponding sides are a, b and c Note that a spherical triangle is not just any three-cornered figure lying on a sphere. Its sides must be arcs of great circles. The essential properties of a spherical triangle are: i) The sum of the three angles is greater than 180 °, ii) Any angle or side is less than 180 °, iii) Sum of any two sides/angles is greater than the third side/angle. Note: Remember that the sides are being measured in degrees or radians. The four basic formulae relating the angles and sides of a spherical triangle (Fig. 2.4) are given below. These are for reference only. You need not memorise them . Sine formula sin A sin B sin C = = (2.1) sin a sin b sin c Cosine formula cos a = cos b cos c + sin b sin c cos A cos b = cos c cos a + sin c sin a cos B (2.2) cos c = cos a cos b + sin a sin b cos C Analogous cosine formulas are: sin a cos B = cos b sin c − sin b cos c cos A 32 sin a cos C = cos c sin b − sin c cos b cos A (2.3) Four parts formula Basic Concepts of Positional Astronomy cos a cos B = sin a cot c − sin B cot C (2.4) Now that you have become familiar with the geometry of the celestial sphere, you can learn about the different coordinate systems used to locate a celestial object. 2.3 ASTRONOMICAL COORDINATE SYSTEMS The method of fixing the position of a star or celestial object on the celestial sphere is the same as that of fixing the position of a place on the surface of the Earth. We shall, therefore, first consider geographical coordinates of a place on the surface of the Earth. 2.3.1 Geographical Coordinates As you know, any place on the Earth is specified by two coordinates: geographical latitude and geographical longitude (see Fig.2.5). P Longitude Prime meridian Latitude Fig.2.5: The geographical coordinates (longitude and latitude) uniquely define any position on the Earth’s surface Note that the line joining the poles is always perpendicular to the equator . The circles parallel to the equator are the circles of latitude . The great circles drawn through the north and South Poles are called the circles of longitude . How do we specify the coordinates of the place P on the Earth’s surface? The great circle passing through P is called its meridian . By international agreement, the geographical coordinates (longitude and latitude) of the place P are specified with respect to the equator and the meridian through Greenwich , which is also called the prime meridian . 33 Basics of Astronomy The general definitions are given in the box ahead. Definitions of longitude and latitude The longitude of a place is defined by the angle between Greenwich meridian and meridian of the place. It is measured from 0 to 180 °°° east or west of Greenwich. The latitude is defined by the angular distance of the place from the equator. It is measured from 0 to 90 °°° along the meridian of the place north or south of the equator. In Table 2.1, we give the longitudes and latitudes of some cities in India. Table 2.1: Longitudes and latitudes of some cities in India Cities Longitude Latitude Allahabad 81 ° 50 ′E 25 ° 26 ′N Ahmedabad 72 ° 44 ′E 23 ° 08 ′N Chennai 80 ° 10 ′E 13 ° 06 ′N Kanyakumari 77 ° 32 ′E 8° 05 ′N Kolkata 88 ° 21 ′E 22 ° 34′N Mumbai 72 ° 51 ′E 19 ° 07 ′N New Delhi 77 ° 12 ′E 28 ° 46 ′N Shillong 91 ° 52 ′E 25 ° 34 ′N Srinagar 74 ° 48 ′E 34 ° 05 ′N Ujjain 75 ° 46 ′E 23 ° 10 ′N We shall now extend the concept of geographical coordinates on the Earth to define the coordinates of a celestial body on the celestial sphere.
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