Field Astronomy

Module 3-Advanced Surveying

1. Module III FIELD ASTRONOMY By Abdul Mujeeb Asst Professor Dept Civil Engineering KVGCE

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4. The Celestial Sphere

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6. To observe the positions / direction and movement of the celestial bodies, an imaginary sphere of infinite radius is conceptualized having its centre at the centre of the earth. The stars are studded over the inner surface of the sphere and the earth is represented as a point at the centre . The important terms and definitions are as follows: 6

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11. Point of view of the observer 11

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30. (11) Latitude It is the angular distance of any place on the earths surface north o r south of equator and measured on the meridian of the place. Marked as +, - or N or S Defined as angle between zenith and celestial equator. It varies from zero degree to 90° N and 0° to 90° S. 30

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32. (12) Co-latitude It angular distance from zenith to the pole. It is complement of latitude and equal to (90-ɵ). 32

33. (13) Longitude It is the angle between fixed reference meridian (prime meridian) and meridian of the place. Universally adopted meridian- Greenwich. Varies between 0° to 180°. Represented as Φ ° east or west of Greenwich. 33

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35. (14) Altitude ( α ) Altitude of celestial or heavenly bodies is its angular distance above the horizon, measured on the vertical circle passing through the body. (15) Co-altitude or zenith distance (z) It is angular distance of heavenly body from zenith. It is complement of altitude i.e z=(90- α ) 35

36. (16) Azimuth(A ) It is the angle between observers meridian and vertical circle passing through that body. 36

37. (17) Declination ( δ ) It is the angular distance from plane of equator measured along the stars meridian called declination circle Varies from 0° to 90° and marked as + or – according to north or south. 37

38. (18) Co-declination or Polar distance. It is angular distance of heavenly body from nearer pole. Compliment of declination i.e p=(90°- δ ) 38

39. (19) Hour Circle. Are great circles passing through north and south celestial poles. Ex: Declination circle of heavenly body 39

40. (20) Hour angle. Angle between observers meridian ad declination circle passing through the body. (21) Right Ascension (R.A) It is equatorial angular distance measured eastward from the first point of aries to hour circle passing through heavenly body. 40

41. (22) Equinoctial points The points of intersection of the ecliptic with the equator are known as equinoctial points. Declination os sun is zero at this point Vernal Equinox or First point of Aries is the point in which sun’s declination changes from south to north. Marks arrival of spring. Autumnal Equinox or first point of Libra is point in which sun’s declination changes from north to south, marks arrival of autumn. They are six months apart in time. 41

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43. (23) Ecliptic The great circle along which the sun appears to move round the earth in a year is called the ecliptic. The plane of ecliptic is inclined to plane of equator at angle about 23° 27ꞌ. 43

44. (24) Solastices Are the points at which north and south declination of the sun is maximum. Point at which north declination is maximum - summer solastice Point at which south declination is maximum - winter solastice . 44

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46. The Celestial co-ordinate system

47. The position of heavenly body can be specified by 2 spherical co-ordinates, i.e by two angular distance measured along arcs of two great circles which cut each other at right angles. 47

48. In practical astronomy celestial body can be specified by following system of co-ordinates. 1. H orizon system 2. Independent equatorial system 3. Dependent equatorial system 4. The celestial latitude and longitude system 48

49. 1. H orizon system (Altitude and A zimuth system) D ependent on position of observer Horizon is plane of reference & co-ordinates of heavenly body are azimuth and altitude 49

50. M is the heavenly body in eastern part of Celestial sphere, Z-zenith & P- celestial pole Pass a vertical circle through M to intersect Mꞌ. First co-ordinate of M is azimuth- angle between observers meridian and vertical circle. 50

51. It can be either angular distance along horizon measured from meridian to foot of vertical circle. It is also equal to zenith distance between meridian and vertical circle through M. Another co-ordinate of M is altitude ( α )- angular distance above horizon on vertical through the body. 51

52. 2 . Independent Equatorial system (The declination and Right Ascension system) I ndependent on position of observer. Great circle of references are equatorial circle and declination circle. First co-ordinate of body is right ascension- angular distance along equator from first point of aries towards east to declination circle passing through the body. It is also angle measured at eastward celestial pole hour circle through RA and declination circle through M. 52

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54. RA is measured in direction to opposite to motion of heavenly body, measured in degrees, minutes and seconds on in terms of time. Another co-ordinate system is declination- it is angular distance of body from equator measured along arc of declination circle. Declination is positive if body is north and negative if body is south of equator. 54

55. 3 . D ependent Equatorial system (The declination and Hour angle system) One co-ordinate is independent and other co-ordinate is dependent on position of observer. Great circle of references are horizon and declination circle. First co-ordinate of M is hour angle- angular distance along arc of horizon measured from observers meridian to declination circle. 55

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57. It is also measured as angle subtended at pole between observers meridian and declination circle. Hour angle is measured from south towards east up to declination circle. Varies from 0° to 360°. Other co-ordinate is declination. In Fig SMꞌ is hour angle M 1 M is declination. Mꞌ and M 1 are projections of M on horizon and equator 57

58. 4. Celestial latitude and longitude system Prime plane of reference- ecliptic and secondary plane- great circle passing through first point of aries and perpendicular to plane of ecliptic. Two co-ordinates are ( i ) Celestial latitude Celestial longitude Celestial latitude is arc of great circle perpendicular to ecliptic. May be + ve or – ve Celestial longitude is arc of ecliptic intercepted between great circle first point of aries and celestial latitude 58

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60. It is measured eastwards from 0° to 360°. M 1 M is celestial latitude and M 1 is celestial longitude for heavenly body 60

61. Comparison of systems. 61 Horizon system Independent Equatorial system Dependent Equatorial system Celestial latitude and longitude system Coordinate dependency Depends on position of observer Both coordinates does not depends on position of observer One coordinate depends and another does not depends on position of observer Does not depends on position of observer Reference plane Altitude and azimuth Declination and right ascension Declination and hour angle Celestial latitude and longitude Great circle Horizon Equatorial circle and declination circle. Horizon and declination circle. Ecliptic and great circle passing through first point of aries Example: Position of star Altitude-45º Azimuth-140º Declination-70º Right ascension-4h Declination- 7 0º Hour angle-5 0º Celestial lat-40º Celestial long-170 º

62. 62 Example for Horizon system

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68. 68 Example for latitude and longitude system

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71. 71 Example for Independent Equatorial system

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74. Spherical trigonometry and spherical triangle It is triangle which is formed upon surface of the sphere by intersection of three arcs of great circle. A ngles formed by arcs at vertices of triangle- spherical angles In Fig AB, BC and CA are 3 angles of great circles and intersect each other at A, B & C. Angles at A, B, C are denoted by sides opposite to them (a, b & c) 74

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76. The sides of spherical triangle are proportional to the angle subtended by them at centre of sphere and are expressed in angular measure. Sine b means sine of angle subtended at centre of arc AC. A spherical angle is angle between 2 great circles and is defined by plane angle between tangents to their circles at point of intersection. Angle A is angle between A 1 AA 2 between tangents AA 1 and AA 2 76

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78. Formulae in spherical trigonometry For computation purpose 78

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80. Napier’s rule Relationship of right angled triangle are obtained from Napier’s rule. In Fig ABC is spherical right angled triangle. Napier defines circular part as follows. These parts arranged around circle in order as they are in triangle 80

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82. Starting with side a, orders are b,90°-A, 90°-c, 90 °-B. If any part is considered as ‘middle part’, adjacent 2 parts are ‘adjacent parts’ and remaining 2 sides are ‘opposite parts’. From Napier rule sine of middle part=product of tangents of adjacent parts s ine of middle part= product of cosines of opposite parts sin b= tan a tan (90 °- A) sin b= (cos 90 °-A) cos (90°-c) 82

83. Astronomical Triangle Astronomical triangle is obtained by joining pole, zenith and any star M on the sphere by arcs of great circle. From this triangle, relation existing amongst spherical co-ordinates may be obtained. Let α - altitude of celestial body (M) δ -declination of celestial body (M) ɵ- latitude of observer 83

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85. ZP= co-latitude of observer PM= co-declination or polar distance of M=(90- δ )=p ZM= zenith distance = co-altitude of body= (90- α )=z The angle at Z=MZP= The azimuth(A) of body The angle at P=ZPM=The hour angle(H)of body The angle at M=ZMP= parallactic angle If three sides MZ, ZP and PM are known angle A and H can be computed from formulae of spherical trignometry 85

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87. 87 Important Questions Introduction and purpose of field astronomy Definitions The Celestial co-ordinate system Comparison of system Spherical triangle and properties Napier's rule Astronomical triangle

88. THANK YOU