RELATIONSHIP BETWEEN ASTRONOMIC COORDINATES φ, λ, AND GEODETIC COORDINATES φ G, λ G,

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1 RELTIOSHIP BETWEE STROOMIC COORDITES φ, λ, D EODETIC COORDITES φ, λ, h H In geodesy it is important to know the relationships between observed quantities such as horizontal directions (or azimuths) and zenith distances - both of which are related to the astronomic meridian and the vertical (the normal to the equipotential surface at the place of observation) - and the complementary values related to the geodetic meridian and the normal to the ellipsoid. These relationships can be used to reduce observations made between points on the terrestrial surface to quasi-observations between corresponding points on the ellipsoid. These relationships are often given in a form known as Laplace's equation 1. ORTH δλ = λ λ.. P STROOMIC EODETIC E CELESTIL. Z ξ θ Z. VERTICL O θ ORML W EQUTOR. λ. λ HORIZO MERIDI MERIDI δλ = λ λ CELESTIL P S igure 1 The celestial sphere for an observer at O on the Earth's surface 1 2 Pierre Simon, Marquis de Laplace ( ), distinguished rench mathematician. Best known for his equation V = 0 (where V is the gravitational potential) published in his great work Mécanique Céleste. Other equations in geodesy also bear his name. LPLCE.DOC PE 1

2 igure 1 shows the celestial sphere for an observer on Earth at O. P and PS are the projections of north and south poles of the Earth on the celestial sphere. The polar axis (rotational axis of the Earth) is also the axis of the ellipsoid (an ellipse of revolution). The plane tangential to the equipotential surface at O intersects the celestial sphere in a line known as the celestial horizon. The tangent to the vertical at O is perpendicular to the horizon plane and intersects the celestial sphere at the astronomic zenith Z (subscript referring to astronomic), and the astronomic meridian plane passing through Z intersects the horizon at - the reference point for astronomic azimuth α. The normal to the ellipsoid which passes through the observer at O intersects the celestial sphere at Z (subscript referring to geodetic), and the geodetic meridian plane passing through Z intersects the horizon at - the reference point for geodetic azimuth α. In general, the vertical and the normal at O do not coincide but diverge by a small angle θ known as the deflection of the vertical whose components are ξ (Xi) in the plane of the astronomic meridian, and (Eta) along a great circle perpendicular to the astronomic meridian and passing through Z. Expressions for ξ and can be obtained from spherical trigonometry and igure 2. δα 1 φ δλ P CELESTIL HORIZO α α Z Z T T z z T δα 2 = Z ξ = Z β = Z δλ = λ λ S S igure 2 The celestial sphere as viewed from above. In igure 2 the astronomic and geodetic meridians are the great circles P ZS and P Z S respectively. T is an elevated target whose astronomic azimuth α is the angle between the astronomic meridian plane and the great circle plane Z TT. The geodetic azimuth α is the angle between the geodetic meridian plane and the great circle plane ZTT. The astronomic and geodetic zenith distances of T are z and z respectively. LPLCE.DOC PE 2

3 Using apier's Rules of Circular Parts in the right-angled spherical triangle P Z 90 ( 90 φ ) φ 90 δλ 90 φ + ξ Z δλ 90 φ 90 φ + ξ P Z gives sin = cos 90 δλ cos φ (i) and sin φ = cos cos( 90 φ + ξ (ii) ow, cos( 90 x) ) = sin x and sin and sin δλ δλ = λ λ since and δλ are small, hence equation (i) becomes and equation (ii) becomes and taking the sine of both sides gives The difference in azimuth is and from igure 2 hence sin = sin δλ cos φ λ λ cos φ (1) ( ) ( ) sin φ = cos sin φ ξ sin φ sin φ ξ ξ = φ φ (2) δα = α α α δα + δα = α 1 2 δα = δα + δα 1 2 (3) (4) LPLCE.DOC PE 3

4 ow δα1 can be calculated from the right-angled spherical triangle P and using apier's Rules for Circular Parts δα 1 φ 90 δλ φ δλ P δα 1 Z Z gives giving 1 sin φ = tan δα tan 90 δλ tan δα1 = tan δλ sin φ and since δα and δλ are small, then tanδα1 δα1 and tanδλ δλ = λ λ Re-arranging equation (1) and substituting into equation (5) we have δα1 = sin φ cos φ δα1 λ λ sin φ (5) and since ξ = φ φ is a small quantity then cos φ cos φ and we may write δα1 tan φ (6) LPLCE.DOC PE 4

5 ow the figure P P Z is similar to TTT TZ δα 1 φ P and using similar reasoning to above we may write δα2 β tan 90 z = β cot z (7) β Z z Z T T 90 z T δα 2 and since ξ and are very small, the figure below (extracted from igure 2) can be considered plane Z g ε g cos θ β Z ξ ξ ξ sin g sin g α cos g g sin g cos g Z Z Z Z = α 90 = cos α = sin α = = ξ = β = ε β = ξ cos g + sin g β = ξ sin α cos α (8) ε = ξ sin g + cos g ε = ξ cos α + sin α (9) Substituting equations (6) and (7) into equation (4) gives δα = δα1+ δα2 = tan φ + β cot and substituting equation (8) gives δα = δα1+ δα2 = tan φ + ξ sin α cos α ) cot z ( z (10) LPLCE.DOC PE 5

6 Equation (10) is often expressed as = + sin cot z + ( tan cos cot ) δα ξ α φ α z (11) Equations (10) and (11) give an expression for the difference between astronomic and geodetic azimuth Using similar reasoning and the diagram above we have z + ε = z and hence an expression for the difference between astronomic and geodetic zenith distance is given by δz = z z = ε = ξ cos α sin α (12) ow from igure 2, when z is approximately 90, as it often is in geodetic survey work, then δα2 0 and the correction for the elevation of the target T can be ignored. This is the second term in equation (10), and as a consequence, the correction in azimuth δα is entirely due to the elevation of the pole P. Hence a common form of the azimuth correction is written as δα = α α = tan φ (13) ow from equation (1) = ( λ λ ) cos φ which when substituted into equation (13) gives and since ξ is a small quantity then cos φ δα = α α = λ λ cos φ sin φ cos φ cos φ and we may write sin δα α α λ λ φ Equation (14) is known as Laplace's equation for azimuths. = (14) COECTIO BETWEE STROOMIC D EODETIC COORDITES rom equations (1) and (2) the following relationships between astronomic and geodetic latitude φ and φ respectively and astronomic and geodetic longitude λ and λ respectively are φ = φ ξ (15) λ = λ secφ (16) COECTIO BETWEE STROOMIC D EODETIC ZIMUTH rom equation (13) the connection between geodetic azimuth α and astronomic azimuth α is given by α = α tanφ (17) LPLCE.DOC PE 6

7 normal COECTIO BETWEE ELLIPSOIDL D ORTHOMETRIC HEIHTS vertical ε P H h plumbline equipote l t ntia er s r e l stria u r face P 0 Q Q 0 geoid ellipsoid igure 3 sectional view of the ellipsoid, geoid and the terrestrial surface. In igure 3 the point P on the Earth's terrestrial surface is projected onto the ellipsoid at Q via the normal PQ and onto the geoid at P0 via the curved plumbline PP0. P0 on the geoid is projected onto the ellipsoid at Q0 via the normal PQ 0 0. It should be noted that the plumbline is a space curve, i.e., it has tortion and twist and in general (and Q ) will not lie in the plane containing P and Q. P0 0 The ellipsoidal height h is the distance PQ measured along the normal. The orthometric height H is the distanced measured along the curved plumbline. PP 0 The geoid-ellipsoid separation is the distance between the geoid and ellipsoid measured along the normal PQ 0 0 but for all practical purposes the geoid and the ellipsoid can be considered as parallel surfaces in the vicinity of P 0 and Q and the connection between ellipsoidal and orthometric heights is given by h = H + (18) DELECTIO COMPOET ε I THE DIRECTIO O THE ZIMUTH α In igure 3, the vertical at P is tangential to the plumbline at P and perpendicular to the equipotential surface through P. The vertical pierces the celestial sphere at Z, the astronomic zenith and the normal pierces the celestial sphere at Z, the geodetic zenith. The angle between the two zeniths is the deflection of the vertical θ (see igure 1) but igure 3 is a section in a particular azimuth α and the component of the deflection, denoted by ε, is given by equation (9) as ε = ξ cos α + sin α (19) LPLCE.DOC PE 7

8 CORRECTIOS TO OBSERVED DIRECTIOS DUE TO ξ D It should be noted that the correction for azimuth δα is made up of two terms δα1 and δα2 (see equation 10)). The first term, δα1 = tanφ, is the same for every target, independent of its azimuth and zenith distance and results from the astronomical azimuth α being reckoned from astronomical north rather than geodetic north. It thus represents a shift of the zero point or the Reference Object in a set of observed directions. The second term δα2 = ( ξ sin α cos α ) cot z depends on the azimuth and zenith distance of the particular target and arises because the target T is projected from Z and Z onto different points and T of the horizon. It thus represents a correction to an observed direction and is T analogous to the correction for an inaccurately levelled theodolite. The corrected direction is given by ξ sinα + cosα corrected direction = observed direction + tan z where the corrected direction relates to a theodolite whose rotational axis is coincident with the normal to the ellipsoid. The observed direction relates to a theodolite whose rotational axis is coincident with the vertical. (20) CORRECTIOS TO OBSERVED ZEITH DISTCES DUE TO ξ D z The correction to an observed zenith distance due to deflection components ξ and is given by equation (12) as δz = z z = ε = ξ cos α sin α and the corrected zenith distance is given by z corrected zenith distance = observed zenith distance + ξ cos α + sin α (21) ( ) where the corrected zenith distance relates to a theodolite whose rotational axis is coincident with the normal to the ellipsoid. The observed zenith distance relates to a theodolite whose rotational axis is coincident with the vertical. REERECES Heiskanen, W.. and Moritz, H., Physical eodesy, W.H.reeman & Co., London, pp Vanicek, P. and Krakiwsky, E., eodesy: The Concepts, orth-holland, msterdam. LPLCE.DOC PE 8