Radius of the Earth - Radii Used in Geodesy James R. Clynch February 2006

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1 dius of the Erth - dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte. The single rdius is used for two min purposes: 1. To find the distnce between the surfce nd the center of there sphere. (In ddition to giving the size of the erth, this vlue cn be subtrcted from the locl rel rdius to get the "height" of the point.). To determine the distnce on the surfce between two points given the end point ltitudes nd longitudes. The first use clls for the physicl rdius of the erth. The second use cll for the "rdius of curvture" of the erth. For sphere, these re identicl. In relity the erth is "flttened" t the poles due to the rottion of the erth. The polr rdius is 3 km (14 miles) shorter thn the equtoril rdius. A better pproximtion is tht of n ellipse tht is rotted bout the polr xis to form solid. This model is ccurte to bout 100 m (300 ft). 1 In this cse t ech point on the erth there re three independent "rdii" tht re used. The distortion is smll frction of the totl, bout 1/300. In generl you cn use sphericl model if computtions t the 1% level re dequte. (Some errors hve mximum of 0.33 %, some twice this.) 1 This does not men the erth's surfce is modeled t the 100 m level by n ellipsoid. The geoid, which is essentilly men se level, differs from n ellipsoid by less thn 100 m. (Erth Grvity odel 1996). If the erth were homogeneous the fit would be exct. 1

2 II. The Three dii of Erth nd Their Use There re three rdii tht come into use in geodesy. Ech is function of ltitude in the ellipsoidl model of the erth. The physicl rdius, the distnce from the center of the erth to the ellipsoid is the lest used. The other two re clled the "rdii of curvture". If you wish to convert smll difference of ltitude or longitude into the liner distnce on the surfce of the erth, then the sphericl erth eqution would be, d de d, cos dλ where is the rdius of the sphere nd the ngle differences re in rdins. The vlues d nd de re distnces in the surfce of the sphere. They correspond to the smll chnges in ngles, d nd dλ expressed in rdins. For n ellipsoidl erth there is different rdius for ech of these directions. The rdius used for the longitude is clled the "dius of Curvture in the prime verticl". It is denoted by,, or ν, nd few other symbols. (ote tht in mny geodesy works the symbol is used for the quntity clled the geoid undultion.) This rdius of curvture,, hs nice physicl interprettion. At the ltitude chosen go down the line perpendiculr to the ellipsoid surfce until you intersect the polr xis. In generl, this line will not end t the center of the erth. Perpendiculr is lso clled "norml" in mthemtics. Thus the subscript "" is used for this curvture,.

3 The rdius used for the ltitude chnge to orth distnce is clled the "dius of Curvture in the meridin." It is denoted by, or, or severl other symbols. The subscript "" comes from meridin, the nme of the lines tht run north-south on globe. is the rdius of circle tht is tngent to the ellipsoid t the ltitude nd hs the sme curvture s the ellipsoid in the north-south direction there. (This circle "kisses" the ellipsoid.) hs no good physicl interprettion s the termintion point chnges with ltitude. The equtions for the reltion between the differentil distnces nd ngles for n ellipsoid now use two different rdii: d de d, cos dλ The rdii of curvture nd re used in plce of the single rdius of the sphere. For the erth these three rdii re shown in the following digrm. (To be precise, these re the WGS 84 ellipsoid vlues.) 3

4 otice tht the both rdii of curvture re lrgest t the poles where the erth is flttest. The rdius of curvture for stright line is infinite. Therefore the fltter the surfce, the lrger the rdius of curvture. The two rdii of curvture re the sme t the poles. The physicl rdius is mximum t the equtor nd minimum t the poles. III. Digrms of dii of Erth A digrm of these rdii is shown below. ote tht the geodetic ltitude (, or g ), is used in geodesy. It is the ltitude used on mps. This should not be confused with the geocentric ltitude (, or c ) lso shown on the figure. The rdius of curvture in the meridin,, is shown for two different ltitudes. Two lines nd n rc of the circle tngent to the ellipse re shown to illustrte the origin of this rdius. The importnt uxiliry line, p, is included. This is the moment rm length for the rottionl ccelertions. In the right digrm, consider the two tringles ABC nd ABD tht both include the moment rm p. Two equtions for p cn be written, one using the geodetic ltitude nd one the geocentric ltitude. 4

5 p cos cos c This is one of severl equtions relting the two ltitude types. IV. Equtions for dii of Erth The equtions for these rdii re given in the following tble. dius dius of Curvture: in Prime Verticl, terminted by minor xis dius of Curvture: in eridin dius of ellipsoid dius of Curvture: At zimuth α e Formul sin ( e sin ) (1 e ) e e sin [(1 e ) 1 e 3 / sin + cos sin 1 cos α sin α α + ] A note of cution on the use of these equtions is in order. The quntity here is the geodetic ltitude. Ellipses occur in both geodesy nd stellite work. Erth stellite work involves both. But the equtions re different in the two fields, while the symbols my be the sme, but hve different menings. In geodesy the geodetic ltitude,, is used. This quntity is rrely used in stellite work, where the geocentric ltitude is common. This uses the line from the surfce to the center of the erth. It my use the sme symbol s used here for geodetic ltitude. The formul for the rdius of curvture t rbitrry zimuth points up tht the fct tht the fundmentl mthemticl quntity is the inverse of these rdii, which re simply clled curvtures. The vlue of the eccentricity, e, used in these equtions is given by e b with " " being the semi-mjor xis (equtoril rdius) nd "b" the semi-minor xis (polr rdius) of the ellipsoid. otice tht 5

6 b 1 e. The term 1 - e is common in ellipse equtions. It just is the rtio of the xes squred. The flttening, f, is defined s f b. The flttening of the erth is bout 1/98.5. otice tht 1 - f b/. It is common in geodesy to define the size nd shpe of n ellipsoid by giving nd f. However the eccentricity occurs more frequently in equtions. The flttening is relted to the eccentricity by e f f V. Other Useful Equtions Involving dii of Erth by: The re of n ellipse with semi-mjor xis of nd semi-minor xis of b is given A π b. The volume of n ellipse of revolution, revolved bout the semi-minor xis, is given by, V 4 π b. 3 If you wnt sphere of the sme volume s n ellipse of revolution you need rdius S given by, 3 S b. The verge rdius of curvture t ny ltitude is the geometric men of nd. Tht is: c π 1 π 0 α dα, where α is the zimuth. 6

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