128 GEODETIC REFERENCE SYSTEM 1980 by H. Moritz Corrigendum: GM = 3986 005 ´ 108 m3 s)2, Due to some unfortunate error this article appeared · dynamical form factor of the Earth, excluding the wrongly in The Geodesists Handbook 1992 (Bulletin permanent tidal deformation: Geodesique, 66, 2, 1992). Among several errors a h J = 108 263 ´ 10)8, (polar distance) was interchanged with a F (geographi- 2 cal latitude) aecting the formulas for normal gravity. It · angular velocity of the Earth: is advised that you use the formulas here or in the Geodesists handbook from Bulletin Geodesique, Vol. x = 7292 115 ´ 10)11 rad s)1, 62, no. 3, 1988. b) that the same computational formulas, adopted at 1- De®nition the XV General Assembly of IUGG in Moscow 1971 and published by IAG, be used as for Geodetic Refer- The Geodetic Reference System 1980 has been ence System 1967, and adopted at the XVII General Assembly of the IUGG in c) that the minor axis of the reference ellipsoid, de- Canberra, December 1979, by means of the following: ®ned above, be parallel to the direction de®ned by the Conventional International Origin, and that the primary ``RESOLUTION N° 7 meridian be parallel to the zero meridian of the BIH adopted longitudes''. The International Union of Geodesy and Geophysics For the background of this resolution see the report recognizing that the Geodetic Reference System 1967 of IAG Special Study Group 5.39 (Moritz, 1979, sec.2).c adopted at the XIV General Assembly of IUGG, Lu- Also relevant is the following IAG resolution: cerne, 1967, no longer represents the size, shape, and gravity ®eld of the Earth to an accuracy adequate for ``RESOLUTION N° 1 many geodetic, geophysical, astronomical and hydro- graphic applications and The International Association of Geodesy considering that more appropriate values are now available, recognizing that the IUGG, at its XVII General As- sembly, has introduced a new Geodetic Reference Sys- recommends tem 1980, a) that the Geodetic Reference System 1967 be re- recommends that this system be used as an ocial placed by a new Geodetic Reference System 1980, also reference for geodetic work, and based on the theory of the geocentric equipotential el- lipsoid, de®ned by the following conventional constants: encourages computations of the gravity ®eld both on the Earth's surface and in outer space based on this · equatorial radius of the Earth: system''. a = 6378 137 m, 2- The Equipotential Ellipsoid · geocentric gravitational constant of the Earth According to the ®rst resolution, the Geodetic Refer- (including the atmosphere): ence System 1980 is based on the theory of the equipo- 129 tential ellipsoid. This theory has already been the basis of If atmospheric eects must be considered, this can be the Geodetic Reference System 1967; we shall summarize done by applying corrections to the measured values of (partly quoting literally) some principal facts from the gravity; for this purpose, a table of corrections will be relevant publication (IAG, 1971, Publ. Spe c. n° 3). given later (sec.5). An equipotential ellipsoid or level ellipsoid is an el- 3- Computational Formulas lipsoid that is de®ned to be an equipotential surface. If an ellipsoid of revolution (semimajor axis a, semiminor axis An equipotential ellipsoid of revolution is determined b) is given, then it can be made an equipotential surface by four constants. The IUGG has chosen the following ones: U U0 const: a equatorial radius, of a certain potential function U, called normal poten- GM geocentric gravitational constant, tial. This function U is uniquely determined by means of J2 dynamical form factor, the ellipsoidal surface (semiaxes a, b), the enclosed mass x angular velocity. M and the angular velocity x, according to a theorem of Stokes-Poincare , quite independently of the internal The equatorial radius a is the semimajor axis of the density distribution. Instead of the four constants a, b, meridian ellipse; the semiminor axis will be denoted by M and x, any other system of four independent pa- b. The geocentric gravitational constant GM is the rameters may be used as de®ning constants. product of the Newtonian gravitational constant, G, and the total mass of the earth, M. The constant J2 is given The theory of the equipotential ellipsoid was ®rst by: given by Pizzeti in 1894; it was further elaborated by C A Somigliana in 1929. This theory had already served as a J2 ; base for the International Gravity Formula adopted at Ma2 the General Assembly in Stockholm in 1930. where C and A are the principal moments of inertia of Normal gravity c jgrad Uj at the surface of the the level ellipsoid (C... polar, A... equatorial moment of ellipsoid is given by the closed formula of Somilgiana, inertia). We shall also use the ®rst excentricity e, de®ned by: ac cos2 U bc sin2 U c pe p ; a2 cos2 U b2 sin2 U a2 b2 e2 ; a2 where the constants ce and cp denote normal gravity at the equator and at the poles, and F denotes geograph- and the second excentricity e¢, de®ned by: ical latitude. a2 b2 e02 The equipotential ellipsoid furnishes a simple, con- b2 sistent and uniform reference system for all purposes of geodesy: the ellipsoid as a reference surface for geo- Closed computational formulas are given in sec.3 of metric use, and a normal gravity ®eld at the earth's (IAG, 1971, Pub.Spe c. n° 3); we shall here reproduce surface and in space, de®ned in terms of closed formulas, this section practically unchanged. as a reference for gravimetry and satellite geodesy. The derivation of these formulas is found in the book The standard theory of the equipotential ellipsoid (Heiskanen and Moritz, 1967) sections 2±7 to 2±10. regards the normal gravitational potential as a harmonic Reference to this book is by page number and number of function outside the ellipsoid, which implies the absence equation. of an atmosphere. (The consideration of the atmosphere Computation of e2 in the reference system would require an ad-hoc modi- ®cation of the theory, whereby it would lose its clarity and simplicity.) The fundamental derived constant is the square of the ®rst excentricity, e2, as de®ned above. Thus, in the same way as in the Geodetic Reference System 1967, the computation are based on the theory of From p. 73, equations (2-90) and (2-92¢), we ®nd: the equipotential ellipsoid without an atmosphere. The e2 2 me0 reference ellipsoid is de®ned to enclose the whole mass J 1 2 3 15 q of the earth, including the atmosphere; as a visualiza- o tion, one might, for instance, imagine the atmosphere to This equation can be written as: be condensed as a surface layer on the ellipsoid. The normal gravity ®eld at the earth's surface and in space 0 2 2 2me e can thus be computed without any need for considering e 3J2 the variation of atmospheric density. 15q0 130 with: Z !1=2 p=2 cos U 2 2 R c dU x a b 2 2 m 0 1 e02 cos2 U GM 2 26 100 7034 c1 e02 e04 e06 e08 (p. 69, eq. (2-70)) and with be¢ = ae it becomes: 3 45 189 14175 2 3 3 2 4 x a a radius of sphere of the same volume: e 3J2 15 GM 2q0 p 3 2 R3 a b: This is the basic equation which relates e2 to the 2 data a, GM, J2 and x. It is to be solved iteratively for e , Physical Constants taking into account: The reference ellipsoid is a surface of constant normal 3 3 0 potential, U = U . This constant U , the normal po- 2q0 1 arctan e 0 0 e02 e0 tential of the reference ellipsoid, is given by: X1 n1 4 1 n n1 e02 GM 1 2n 1 2n 3 0 2 2 n1 U0 arctan e x a E 3 ! GM X1 e02n 1 with 1 1n m b 2n 1 3 e n1 e0 p second excentricity 1 e2 (p. 67, eq. (2-61)). (p. 66, eq. (2-58), p. 72, second equation from top). The normal gravitational potential V (gravity po- tential U minus potential of centrifugal force) can be Geometric Constants developed into a series of zonal spherical harmonics: ! 1 Now the other geometric constants of the reference GM X a 2n V 1 J P cos h ; ellipsoid can be computed by the well-known formulas: r 2n r 2n n1 p b a 1 e2 semiminor axis; where r (radius vector) and h (polar distance) are a b spherical coordinates. The coecient J is a de®ning f flattening; 2 a constant; the other coecients are expressed in terms of p J by: E a2 b2 linear excentricity; 2 a2 3e2n J c polar radius of curvature: J 1n1 1 n 5n 2 b 2n 2n 1 2n 3 e2 The arc of meridian from equator to pole (meridian (p.73, eqs. (2-92) and (2-92¢)). quadrant) is given by: Z Normal gravity at the equator, ce, and normal gravity p=2 dU Q c at the poles, cp, are given by the expressions: 3=2 0 1 e02 cos2 U GM m e0q0 c 1 m 0 where F is the geographical latitude. This integral can be e ab 6 q 0 evaluated by a series expansion: GM m e0q0 c 1 0 p a2 3 q p 3 45 175 11025 0 Q c 1 e02 e04 e06 e08 2 4 64 256 16384 with Various mean radii of ellipsoid are de®ned by the 1 1 q0 31 1 arctan e0 1 following formulas: 0 e02 e0 arithmetic mean: and a a b f x2a2b R1 a1 m 3 3 GM radius of sphere of the same surface: (p.