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Gravity g WGS84 84 at pole
09-28-2021, 08:53 AM
Post: #1
Gravity g WGS84 84 at pole
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
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09-28-2021, 10:48 AM
Post: #2
RE: Gravity g WGS84 84 at pole
(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
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09-28-2021, 11:16 AM (This post was last modified: 09-28-2021 11:17 AM by Gil.)
Post: #3
RE: Gravity g WGS84 84 at pole
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
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09-28-2021, 03:21 PM
Post: #4
RE: Gravity g WGS84 84 at pole
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
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