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Gravity g WGS84 84 at pole - Gil - 09-28-2021 08:53 AM Sure that here, in that HP calculator forum, is not the normal place to place my question. But that interrogation occurred now when developing my gravity program for the HP49-HP50. And as I know that the level of some HP user is quite high, counting a lot of maths people or ingenieurs, I dare and submit my problem, hoping for your indulgence... and help. In the Wikipedia "Theoretical Gravity" article, choose the Chinese language. There is to be found a formula j.e (indeed j.p) for the gravity g at the pole. j.pole = 'GM/(a*a)*(1+m+3/7*e'²*m)' equation 1 In WGS84 a=exactly 6 378 137 m='w*w*a*a*b/GM' with w: = exactly .00007292115 b=a-a*f and f=exactly 298.257223563 so that b=about 6356752. 3142451794975639665996336551567981713110854973388485716512832085220868351009394057094507132374770633 And GM=exactly 3.986004418E14 so that m=about 0.0034497865068408453070274965050786700048416482688355349290136425040253035757396870475700989923885167801669828495872298265101288596701656992493579318456239603696294749064174268559478550984385788003910837612 e'²='(a*a-b*b)/(b*b)' so that e'²=about 0.0067394967422764349547821589567593766561220658588440099819050282583227973208914914189860809667349634 So that (1+m+3/7*e'²*m)= 1.003459750717522732340985895556610125671094437673903614541154381375001105833623894718788389499178806618567527413384953945229190157855623901712950238982395888274759708044521808294166478667460951510991749895895279214598597362085502307407034549091368688099259583762966067132573436092217775000400440672864145748571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571 And finally g. Pole (without the dot/comma) should be exactly equal to, according to equation 1: '3986004418*1003459750717522732340985895556610125671094437673903614541154381375001105833623894718788389499178806618567527413384953945229190157855623901712950238982395888274759708044521808294166478667460951510991749895895279214598597362085502307407034549091368688099259583762966067132573436092217775000400440672864145748571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571/(6378137*6378137)' Or j. Pole, placing back the dot/comma at the right place= 9.8321851044044354166235845241110015889520125854466337942802305915361947992868863041785670259867943125529117241157228655849590168522129663217533159804636091817559953695048056826641872205769685950655291862172166302389558892330844759850013860675472423221623311315941998400543951488023702988151342889690235373769066691915076944701814198886293763195146108008175411982135184646619151544169766231068819297264703012453. (Full digit calculated result) But all the formulae given for the gravity at the pole write that it is exactly? = 9.83218 49378 (General given 11 digits result) My questions Does a term, or several, miss in the equation 1? Or, on the the contrary, Full digit calculated result is correct and General given 11 digits result was I then incorrectly rounded (with the 7th digit already false)? As a curious layman, I would appreciate your precious insights. Thanks for your help. Regards, Gil Campart RE: Gravity g WGS84 84 at pole - Albert Chan - 09-28-2021 10:48 AM (09-28-2021 08:53 AM)Gil Wrote: e'²='(a*a-b*b)/(b*b)' e²= 1 - (b/a)^2 = (a*a-b*b)/(a*a) The typo may still not make up the difference ... Calculated gravity is derived from assumed standards, thus the need to keep saying WGS84 RE: Gravity g WGS84 84 at pole - Gil - 09-28-2021 11:16 AM In fact, using the reference of H. Moritz Geodesic Reference System 80, we get quite closer, better results. The differences are certainly due to rounding errors by tan^-1 and sqrt functions. Regards, Gil RE: Gravity g WGS84 84 at pole - Gil - 09-28-2021 03:21 PM RE: HP49-50G : —>g gravity calculation = g(latitude, height) with WGS84 To check, the paper variables of Geodetic Reference System 1980, by H.Moritz, should be taken, instead of the simplifications given in Wikipedia "Theorical Gravity", Chinese page. So j.e = 'GM/(a*b)*(1-m-m/6*é²*(q0´/q0))' 9.78032533482, from the above formulae 9.7803253359 official —> almost equal value j.p = 'GM/(a*a)*(1+m/3*é²*(q0´/q0))' 9.83218494001, from the above formulae 9.8321849378, official —> almost equal value! With é² = sqrt (é²) =e' And: q0´ = '3*(1+1/é²)*(1-1/é²*ATAN(é²))-1' .00268804118 q0 = '((1+3/é²)*ATAN(é²)-3/é²)/2' .00007334625 é² = '(a*a-b*b)/(b*b)' 6.73949674208E-3 GM = 3.986004418E14 m = 'w*w*a*a*b/GM' 3.44978650683E-3 w = .00007292115 a 6378137 a = 6378137 b = 'a-a/298.257223563' 6356752.31425 The differences are now quite small, due to the roundings of the calculator. Regards, Gil |