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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. 31424517949756396659963365515679817131108549733884857165128320852208683510093940​57094507132374770633
And GM=exactly 3.986004418E14
so that m=about 0.003449786506840845307027496505078670004841648268835534929013642504025303575739​68704757009899238851678016698284958722982651012885967016569924935793184562396036​96294749064174268559478550984385788003910837612

e'²='(a*a-b*b)/(b*b)'
so that e'²=about 0.006739496742276434954782158956759376656122065858844009981905028258322797320891​4914189860809667349634

So that (1+m+3/7*e'²*m)=
1.003459750717522732340985895556610125671094437673903614541154381375001105833623​89471878838949917880661856752741338495394522919015785562390171295023898239588827​47597080445218082941664786674609515109917498958952792145985973620855023074070345​49091368688099259583762966067132573436092217775000400440672864145748571428571428​57142857142857142857142857142857142857142857142857142857142857142857142857142857​1428571

And finally g. Pole (without the dot/comma) should be exactly equal to, according to equation 1:
'3986004418*10034597507175227323409858955566101256710944376739036145411543813750​01105833623894718788389499178806618567527413384953945229190157855623901712950238​98239588827475970804452180829416647866746095151099174989589527921459859736208550​23074070345490913686880992595837629660671325734360922177750004004406728641457485​71428571428571428571428571428571428571428571428571428571428571428571428571428571​428571428571428571/(6378137*6378137)'

Or j. Pole, placing back the dot/comma at the right place=
9.832185104404435416623584524111001588952012585446633794280230591536194799286886304178567025​98679431255291172411572286558495901685221296632175331598046360918175599536950480​56826641872205769685950655291862172166302389558892330844759850013860675472423221​62331131594199840054395148802370298815134288969023537376906669191507694470181419​8886293763195146108008175411982135184646619151544169766231068819297264703012453.​ (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*ƒé²*(q0´/q0))'
9.78032533482, from the above formulae
9.7803253359 official
—> almost equal value

j.p =
'GM/(a*a)*(1+m/3*ƒé²*(q0´/q0))'
9.83218494001, from the above formulae
9.8321849378, official
—> almost equal value!

With
ƒé² = sqrt (é²) =e'

And:

q0´ =
'3*(1+1/é²)*(1-1/ƒé²*ATAN(ƒé²))-1' .00268804118

q0 =
'((1+3/é²)*ATAN(ƒé²)-3/ƒé²)/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