(49g 50g) Theoretical Earth gravity g = g(latitude, height), WGS84, GRS80/67

[Gil]

Version 3
Improved the details for the exact "correction" when taking into account the height/altitude for the calculation of the Earth gravity g.

Now the full parameters/constants a, f and m appear explicitly in the formulae with their corresponding 12 significant digits values according to Somigliana GRS 80 and Somigliana WGS 84.

I let here, however, a simplified version for Lambert GRS 80 regarding the height "correction".

See below in bold the main changes in the code:
\<< "Version 3 2021.09.26

2 Arg
. 
lat
 [in D.mmss]
. 
alt
 [in m]

https://en.m.wikipedia.org/wiki/Theoretical_gravity
https://eu.docworkspace.com/d/sIEG9949c5qq2igY
(GRS 80 by H Moritz)
http://www.in-dubio-pro-geo.de/index.php...lip/ngrav1

\[]gWGS84(lat 90, alt 0)
effect =9.8321849378
but form\->9.83218493787
" DROP "alt [m]" \->TAG SWAP "lat D.mmss" \->TAG DUP2 RCLF \-> alt lat f
\<< DEG lat HMS\-> 'lat' STO DEG '9.780327*(1+.0053024*SIN(lat)^2-.0000058*SIN(lat*2)^2)-.000003086*alt' \->NUM "
Lambert GRS 80
" \->TAG '9.7803267715*(1+.0052790414*SIN(lat)^2+.0000232718*SIN(lat)^4+.0000001262*SIN(l​at)^6+.0000000007*SIN(lat)^8)' \->NUM lat alt 6378137 3.35281068118E-3 3.44978600308E-3 \-> lat h a f m
\<< '1-2/a*(1+f+m-2*f*SIN(lat)^2)*h+3*(h/a)^2' * \->NUM
\>> "
Somigliana GRS 80
" \->TAG '9.7803253359*((1+1.9318526464E-3*SIN(lat)^2)/\v/(1-6.69437999014E-3*SIN(lat)^2))' \->NUM lat alt 6378137 298.257223563 INV 3.44978650684E-3 \-> lat h a f m
\<< '1-2/a*(1+f+m-2*f*SIN(lat)^2)*h+3*(h/a)^2' * \->NUM
\>> "
Somigliana WGS 84
" \->TAG f STOF
\>>
\>>

Regards,
Gil