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ASTRONOMICAL SURVEYING

ASTRONOMICAL SURVEYING. H.C. King, History of the Telescope. CONTENTS. Celestial sphere Astronomical terms and definitions Motion of sun and stars Apparent altitude and corrections Celestial co-ordinate systems Different time systems Use of Nautical almanac Star constellations

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ASTRONOMICAL SURVEYING

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  1. ASTRONOMICAL SURVEYING H.C. King, History of the Telescope

  2. CONTENTS Celestial sphere Astronomical terms and definitions Motion of sun and stars Apparent altitude and corrections Celestial co-ordinate systems Different time systems Use of Nautical almanac Star constellations Calculations for azimuth of a line Sivapriya Vijayasimhan

  3. Shape and Size of Earth Shape of the earth is sphere regular figure for simplified calculation Equatorial radius of earth (a) = 6378.388 km Polar radius of earth (b)= 6356.912 km Survey of India gives a =6377.3097 km and b=6356.1087 km Ellipticity factor = India : 1/300-80 Mean radius of earth is 6367.272km Sivapriya Vijayasimhan

  4. The celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with the observer •All objects in the observer's sky can be thought of as projected upon the inside surface of the celestial sphere, as if it were the underside of a dome or a hemispherical screen. •The celestial sphere is a practical tool for spherical astronomy, allowing observers to plot positions of objects in the sky when their distances are unknown or unimportant. Sivapriya Vijayasimhan Celestial sphere

  5. Visible Horizon: Sivapriya Vijayasimhan Astronomical terms and definitions Zeinth (z) : It is a point on the upper portion of celestial sphere immediately above the overhead of an observer Nadir (z’) : It is the intersection of a vertical line through the observer’s station to the lower portion of the celestial sphere Celestial or Rotational Horizon (Geocentric or true horizon): It is a great circle traced upon the celestial sphere by that plane which is perpendicular to zeinth-nadir line and which passes through the centre of the earth Sensible Horizon : It is a circle in which a plane passes through the point of observation and tangential to earth’s surface intersects with celestial sphere. The line of sight of an accurately levelled telescope lies in this plane

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  7. Sivapriya Vijayasimhan Fig 1 Fig 2

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  9. Motion of SUN and STAR Sun Located at a distance 93,005 km from earth Dia of Sun = 109 Dia of earth Mass of Sun = 3,32,000 of earth Temperature of earth = 20 million degrees Motions: Two apparent motion of earth 1. With respect to earth east to west 2. With respect to fixed stars in celestial sphere Motion of sun is along the great circle – ecliptic Obliquity of Ecliptic – angle between the plane of equator and the ecliptic ( 23027’) Equinoctial Point : Point of intersection of ecliptic with equator. Here declination of sun is zero Vernal Equinox : First point of aeries in which the sun’s declination changes from south to north Autumnal Equinox : First point of libra in which the sun’s declination changes from north to south Solstices : Sun’s declination is maximum Summer solstices : north declination is maximum at a point Winter Solstices : South declination is maximum Sivapriya Vijayasimhan

  10. Star Moon rerates the earth in elliptical orbit(average angle 508’)inclined to the plane of ecliptic, which is intersected at points called Nodes Motions: 1. With respect to earth east to west 2. With respect to fixed stars it is from west to east Moon rotates about its polar axis. New and full occur when sun, earth and moon lie in same vertical plane (not necessarily in same straight line) Conjunction : in new moon ,moon lies between sun and earth and has same latitude as sun Opposition : In full moon , earth lies between sun and moon Waxing : Illuminated limb increase in size of moon between the time interval of new and full moon period Waning : Illuminated limb decrease in size of moon between the time interval of full and new moon period Lunar Month : Time taken between two successive new moons (27 ⅓ days) Siderial Month : One complete revolution relative to stars (29.5 days) Solar Eclipse: moon passes in front of sun’s disc Lunar Eclipse: Shadow of earth passing over the moon Sivapriya Vijayasimhan

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  12. SPHERICAL TRIGONOMETRY Properties of spherical and astronomical triangle are studied 1.Sphere Every point on the surface of the sphere is equidistant from a certain point called centre of sphere. Every section of sphere is circle. The section through the centre of the sphere is called a great circle The section not passing through the centre is called a small circle CDEFG – great circle MON – diameter of sphere M and N – poles of great circle R- radius of sphere φ – angle subtended by great circle DE = R φ (if R=1), DE = φ In right angle Δ dO1O, = =sin dOO1 =sin Md (Md+ dD =90 deg) Small arc = arc DE cos dD Sivapriya Vijayasimhan

  13. 1. Spherical Triangle It is formed by surface of the sphere by interaction of three arcs of great circle The angle subtended by the axes at the vertices of the triangle is called spherical angles ABC – spherical triangle AB and AC are great circles with subtended angle BAC = A0 1.1Properties of spherical triangle Any angle is less than two right angles or π Sum of three angles is less than six right angles or 3π and greater than two right angles or π Sum of any two sides is greater than the third If the sum of any two angles, is equal to two right angles to π, the sum of the angles opposite them is equal to two right angles or π The smaller angle is opposite the smaller side and vice-versa Sivapriya Vijayasimhan

  14. 1.2 Angles and Sides of spherical triangles Sin formula : Cosine Formula: 3. = : = : = = : = : = 1.3 Area of spherical triangle Area of spherical triangle = = R- radius of sphere E =A + B +C - Sivapriya Vijayasimhan

  15. 2. Right angled Spherical Triangle The relationships of right angled spherical triangle may be obtained from “Napier’s circle of Circular Parts” The circle is divided into five parts. Consider any part as middle part, the part adjacent to it as adjacent parts, we have Napier’s rule as follows, Sine of middle part = product of tangent of adjacent parts : -A) Sine of middle part = product of cosines of opposite parts : 3. Spherical Excess Spherical excess of spherical triangle exceeds 180 deg s seconds r –radius of earth Sivapriya Vijayasimhan

  16. Astronomical Triangles Formed by joining the pole, zenith and any star M on sphere by arcs of great circles α –altitude of celestial body (M) δ – declination of the celestial body (M) θ – latitude of observer ZP = co-latitude of observer = MP = co-declination of the polar distance M = ZM = zenith distance or co – altitude of the body = Angle Z = MZP = azimuth (A) of the body Angle P = ZPM = hour angle (H) of the body Angle M = ZMP = parallactic angle If three sides of the triangles are known, A and H are computed by spherical trigonometry formula Sivapriya Vijayasimhan

  17. = = = (z + c + p) = = = Sivapriya Vijayasimhan

  18. 1.Position of Stars Star of Elongation :When it is at greater distance east or west of meridian. Under this condition azimuth of star is maximum. Eastern or western elongation of a star is at its greatest distance to west or east of meridian respectively Star at Prime Vertical: When the observer, at zenith , the angle is right angled in the astronomical triangle. A = 90 deg Star of Horizon : Its altitude is zero and the zenith distance is equal to 90 deg Star at Culmination : When the star crosses an observer meridian the star is said to be culminate or transit. In one revolution, each star crosses a meridian twice. Upper culmination : altitude is maximum Lower culmination : altitude is minimum Sivapriya Vijayasimhan

  19. Circumpolar Star : Stars which are always above the horizon and which evidently do not set. For an observer it is an circle above the pole Declination of such stars is always greater than the co-altitude of the place of observation M1 – circumpolar star having circular path A1A2 (path above horizon) M2 – circumpolar star having circular path B1B2 (path below horizon) Sivapriya Vijayasimhan

  20. Co-ordinate Systems Position of heavenly body can be located by two-spherical co-ordinates, two angular distances measured along arcs of two great circles which cut each other at right angles One of great circle passing through the heavenly body is called Primary circle of reference, whereas the other is called as Secondary circle of reference Point M represents heavenly body with reference to a plane OAB O –origin of the co-ordinates A plane passing through OM shall cut a perpendicular plane OAB in line OB Two spherical co-ordinates of the point M are angles AOB and BOM at centre O Systems :1. Altitude and Azimuth , 2.Declination and right ascension system and 3. Declination and hour angle system Sivapriya Vijayasimhan

  21. 1.Altitude and Azimuth System Also called as horizon system which is dependent on the position of the observer Horizon is a plane of reference and the co-ordinates of a heavenly body (azimuth and altitude) - It is the primary and secondary reference great circle in observer’s meridian - Horizontal and vertical angles are measured - theodolite • The heavenly body can be in eastern or western part of the celestial sphere Heavenly body in eastern part of celestial sphere. Let Z be the observer’s zenith and P be the celestial pole Great circle is passing through Z and M is drawn to cut the horizon plane at M’ The azimuth (A) angle between the observer’s meridian and the vertical circle through the body is the first co-ordinate Azimuth is equal to the angle at zenith between the meridian and the vertical circle through M. The co-ordinate of M is the altitude (α), which is measured above or below the horizon on vertical circle Sivapriya Vijayasimhan

  22. Heavenly body in western part of celestial sphere. The concerned angles NOM(azimuth) and MOM’ (altitude) In northern hemisphere, the azimuth is always measured from north to east or west In southern hemisphere, the azimuth is measured from south to east or west Zenith Distance = ZM-MM’ Sivapriya Vijayasimhan

  23. 2.Declination and Right Ascension System (Independent equatorial system) Two great circles : 1. Equatorial circle – primary circle 2. Declination circle – secondary circle The first co-ordinate of heavenly body is the right ascension It is the angle along the arc of celestial equator measured from the first point of aeries and also the angle between the hour circle through (γ) Declination (δ) is the angle of the body measured from equator along the arc of declination circle Sivapriya Vijayasimhan

  24. 3.Declination and Hour angle System (Dependent equatorial system) Two great circles : 1. Horizon – primary circle 2. Declination circle – secondary circle The first co-ordinate of M is the hour angle It is the angle subtended at the pole, between observer's meridian and the declination of the body In northern hemisphere the hour angle is measured from south towards the west up to the declination circle. It varies between 00 to 3600 . 00 to 1800 – star is in western hemisphere 180o to 3600 – star is in eastern hemisphere Sivapriya Vijayasimhan

  25. Relationships between co-ordinates 1.Relationship between altitude of the pole and latitude of the observer H-H horizon plane E-E equatorial plane O – is the centre of the earth ZO is perpendicular to HH while OP is perpendicular to EE Latitude of place Altitude of pole Equating both equation Altitude of the pole is always equal to the latitude of the observer Sivapriya Vijayasimhan

  26. 2.Relationship between latitude of observer and declination an altitude of a point on the meridian declination meridian altitude of star meridian zenith of star latitude of the observer If star is below the equator, -ve sign for δ and also if the star is to the north of zenith –ve sign for z If the star is north of zenith but above the pole as at M2 p= polar distance = M2 P If the star is north of zenith but below the pole p= polar distance = M3 P Sivapriya Vijayasimhan

  27. 3.Relationship between right ascension and hour angle M – position of the star - westerly hour angle - westerly hour angle for first position of aeries position γ - right ascension of star Hour angle of equinox = Hour angle of star + RA of star Sivapriya Vijayasimhan

  28. Correction to Apparent altitude 1.Instrumental correction 2. Observational correction 1. Instrumental Correction Corrections for altitudes i. Index and ii.Bubble error Corrections for azimuths i. Index error Small vertical angle between the line of collimation and the horizontal bubble line of the altitude or azimuthal bubble Procedure With telescope normal in face left position any well-defined object such as church spire or a chimney is bisected and angle is α1 The face is changed (Right face) and the telescope is reversed and the same object is bisected and angle is α2 Mean, If the observations are not possible to take on both sides, correction for index error is applicable It can be eliminated by taking reading on both faces Sivapriya Vijayasimhan

  29. ii.Bubble error If the bubble tube is not at the centre while taking reading, correction for bubble error is applicable Correction fro bubble error, C (seconds) - sum of readings of the object glass end of the bubble - sum of readings of the eye piece end of the bubble n – number of bubble ends read v- angular value of one division of bubble in seconds The observed altitude when corrected for index error and bubble error is called apparent altitude b. Corrections for azimuths c – correction for azimuths b – inclination of horizontal axis of the transit with respect to horizontal, sec α – vertical angle to high point Sivapriya Vijayasimhan

  30. ‘ SivapriyaVijayasimhan 2.Astronomical Correction i. Correction for parallax ii. Correction for refraction iii. Correction for dip of the horizon iv. Correction for semi diameter Correction for parallax When Sun and star are viewed from different points, change in direction of the body is observed due to parallax Parallax in altitude is called diurnal parallax O – centre of earth A – plane of observation S – position of Sun at time of observation S’ – position of sun at horizon OC – true horizon AB – sensible horizon = observed altitude = true altitude, corrected fro parallax = parallax correction = Sun’s horizontal parallax

  31. When Sun is on horizon (apparent altitude is zero), Sun’s horizontal parallax varies from 8.95” from Jan to 8.66” during early July True altitude Parallax correction From But, pa and ph are very small, Correction for parallax = (horizontal parallax) x cos (apparent altitude) = 8.8” cos - Correction is additive - Correction is maximum when the Sun is at horizon Sivapriya Vijayasimhan

  32. ii. Correction for refraction As the distance from surface increases, the layers of atmospheric air surrounding the earth becomes thinner Due to variation in atmospheric density, the ray of light passes through the atmosphere bents Because of this, body appears to be nearer to zenith than it actual Refraction angle of correction:Deviation of angle of ray from its direction on entering the earth’s atmosphere to its direction at the surface of earth At a pressure of 76 cm of mercury and 10 o C , Correction for refraction ( in sec) = 58 “ cot α = 58” tan z α – apparent altitude of heavenly body : z – apparent zenith distance of heavenly body Correction is subtractive Factors influencing 1. Density of air 2. Temperature 3. Barometric pressure and 4. Altitude Sivapriya Vijayasimhan

  33. iii. Correction for dip of the horizon Angle of Dip : angle between the true and visible horizon Due to curvature of earth, visible horizon is below the true horizon Angle of dip is angle between the two horizons and this has to be subtracted from the observed altitude of the body A – position of observer AB – h – height of observer above sea-level S – position of Sun or Star AD – visible horizon AC – true horizon - observed altitude of sun or star - true altitude of sun or star - angle of dip R - radius of earth Then, BO = R and AO = (R + h) Sivapriya Vijayasimhan

  34. Β= small, then Correction for dip is subtractive Sivapriya Vijayasimhan

  35. iv. Correction for semi-diameter Half of angle subtended at centre of earth by sun and star is the semi-diameter of earth Semi diameter of earth varies from 15’46” (July) to 16’18” (January) Mean distance value is 16’1.18” Sun’s diameter is the tangent sight of sun’s image by cross hair Sivapriya Vijayasimhan

  36. OA – ray corresponding to lower limb of Sun - observed altitude α - corrected altitude OB – ray corresponding to upper limb of Sun - observed altitude γ/2 is semi diameter, When horizontal angle is measured to Sun’s right or left limb correction is equal to sun’s semi-diameter times the second of altitude is applied. Correction for semi – diameter in azimuth = semi diameter x secant α Sivapriya Vijayasimhan

  37. Astronomers Relevant every day Time System Earth moves from west to east Measurements of time depends on the apparent motion of heavenly bodies by earth’s rotation on its axis Four kinds of time Sidereal time 3. Mean solar time Apparent solar time 4. Standard time 1.Sidereal Time Sidereal Day: Time interval between two successive upper transits of first point of aeries over same meridian Sidereal noon : instant of crossing Time : 1 Day ( 0 to 24 hrs) 1 hrs = 60 min 1 min = 60 seconds Local sidereal time (LST) : Right Ascension (RA) of meridian of place LST = RA of star + westerly hour angle of star If LST > 24 hrs, 24 hrs has to be deducted: If LST < 24 hrs, 24 hrs has to be added LST = RA of mean sun ± 12 hr + (mean time of that place) Sidereal time of transit of star = RA of star Sivapriya Vijayasimhan

  38. 2. Apparent Solar time Apparent solar Day : Time interval between two successive lower transit of center of sun over the same meridian Apparent solar day(24 hrs) 60 min 60 sec Calculated on the basis of “motion of Sun” 3.Mean Solar Time Mean sun(imaginary body) is assumed to move at a uniform rate along the equator in order to make solar day of uniform period. Mean Solar Time : Time when measured by diurnal motion of mean sun (clock time) Mean solar day or civil day : time interval between two successive lower transits of mean sun over same meridian Astronomical day : zero hr to midnight Civil day : 1. midnight to noon - anti meridian (am) 2. noon to midnight – post meridian (pm) i. Conversion ii. Relationships iii. Local mean time Sivapriya Vijayasimhan

  39. Conversions If civil time is am, the astronomical time is same If civil time is pm, the astronomical time = civil time + 12 hrs If astronomical time is less than 12 hrs, civil time is same If astronomical time is greater than 12 hrs, civil time = astronomical time+12 hrs ii. Relationships Between hour angle, right ascension and time Apparent solar time = hour angle + 12 hrs Mean solar time = hour angle of mean sun + 12 hr Local sidereal time = RA of mean sun + hour angle of mean sun Sidereal time of apparent moon(sun crosses the meridian of any place) = RA of Sun Sidereal time of mean noon = RA of mean sun iii. Local Mean Time Mean time at meridian of observer All places along the same meridian shall have same local time. Mean time - Greenwich mean time Sivapriya Vijayasimhan

  40. 4.Standard Time Mean time on meridian as the standard time for the whole of country Standard meridian Meridian passing Greenwich – Greenwich mean time (GMT) Time : 0 to 24 hrs Mean time associated with standard meridian - Standard time India : 82030’ E or 5 hrs 30 m east Standard time = LMT ± difference of longitude in time between the given place and standard meridian + sign – standard meridian to west - sign - standard meridian to east Sivapriya Vijayasimhan

  41. Equation of Time Difference between apparent solar time and mean solar time + sign – Sun after clock - Sign – Sun before clock Equation of time = RA of mean Sun – RA of Sun varies between 0 to 16 min April 15, June 14 , September 1 and December 25 - mean time and apparent time are same The difference is due to obliquity of real sun and mean sun LST = RA of mean sun + hour angle of mean sun LST = RA of sun + hour angle of sun RA of mean sun – RA of sun = hour angle of sun – hour angle of mean sun Equation of time = hour angle of sun – hour angle of mean sun Equation of time = apparent time – mean time SivapriyaVijayasimhan

  42. Azimuth of a Survey Line Angle between observer's meridian and vertical circle passing through the body Azimuth Observation Measuring the horizontal angle between a reference mark and heavenly body Determine the azimuth of the celestial body Reference mark – azimuth of star or heavenly body - Triangulation station lantern or electric light - Line of sight should be well above ground to minimum the error due to lateral deflection Azimuth of reference mark is calculated from measured angle and known azimuth of celestial body Azimuth of survey line may be obtained measuring the horizontal angle between the reference mark and line and combining with azimuth of the reference Sivapriya Vijayasimhan

  43. Determination of Azimuth of a Survey Line • Extra meridian observation of the Sun • Extra meridian observation of circumpolar star or of a star near Prime vertical • Observation of a circumpolar star at elongation 1.Extra meridian observation of Sun Astronomical triangle ZPS is used to compute azimuth Sun Azimuth OB = Observation of Sun’s time = 8 am to 10 am or between 2 and 4 pm Sivapriya Vijayasimhan

  44. 2.Extra meridian observation of circumpolar star Observation of Star is taken when it is on or near the prime vertical as it move slowly in azimuth Refraction will be greater if the star is too low 3. Observation of a circumpolar star at elongation Plane of declination and plane of vertical circle is right angles Procedure to calculate star elongation 1.Hour angle of star is calculated by knowing latitude of the place and declination of star 2.Hour angle is converted into time and added to RA of star (west elongation) or subtracted to RA of star(east elongation) 3.Time is converted into mean time Azimuth of Star, Sivapriya Vijayasimhan

  45. Nautical Almanac(NA) Astronomical data available Salient Features Greenwich hour angle of Sun and declination are given for every angle of GMT to 0.1’ Tables for increments and corrections for every minute and second 2. Equation of time(ET) is given to nearest second for intervals of 12 hours and time of meridian passage every day 3. ET is the quantity to be added to mean solar time to get apparent solar time 4 . Semi-diameter of sun is given to 0.1’ for every 3 day period 5. Sidereal hour angle and declinations are given for 173 stars including 57 selected stars (accuracy = 0.1’) 6. Polar star table are given Sivapriya Vijayasimhan

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