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

[Gil]

Version 6e
(Theoretical) GRAVITY of Earth g
= g(latitude [D.mmss] ; height [m]).

With latitude [D.mmss] in stack level 2
and height [m]) in stack level 1.

Returns 4 results:

-g GRS67, according to International Gravity equation;
-g GRS80, according to Somigliana's equation;
-g WGS84, according to Somigliana's equation;
- g FREE, according to closed form,
Li & GÖtze: 'Tutorial Ellips,geoid,gravity'.

Main change 1

-You can choose your "FREE" ellipsoid.
-How?
Go in FREE directory (inside main file GRAVITY Dir) and
change/save any value of the four (normally fixed) variables GM, a, f or w.
-But don't suppress any of them (modify, yes ; delete, no).
-Then EXECUTE in that FREE directory
—>gFREE,
with, as usual, latitude [D.mmss] in stack 2
and height [m] in stack level 1.
-Everything is then calculated inside this FREE Dir automatically in a CLOSED form: no need like in SOMIGLIANA to compute/have the intermediary g official values at Equator & at Pole.
-The result will appear with the label/tag "CLOSED FORM FREE"
or "CLOSED FORM WGS84" (if all the 4 values GM a f w to be found in FREE Dir are the same as GM a f w official GSM84-Values located in G84EP Dir).
-The final, CLOSED result is quite accurate.
-However generally somewhat less accurate than the Somigliana's equation for GRS80 & WGS84.

Main change 2

-When executing —>g
(in the main GRAVITY Dir),
the program —>gFREE (to be found in FREE Dir & discussed above) will be launched automatically, with the corresponding label /tag "CLOSED FORM FREE"
or "CLOSED FORM WGS84" added to the final g result.

Main change 3

-Besides the FREE Dir, a g84EP Dir was added inside the main GRAVITY Dir.
-In the NAME of the Dir g84EP, 84 stands for WGS84, E for Equator and P for Pole.
-The four variables GM a f w in that file G84EP Dir belong to the official WGS84 model and therefore should not be deleted or even modified.
-The intermediary variables/equations inside that G84EP Dir are commonly to be found in the literature.
- They show a different, instructive way of calculating g WGS84 at Equator and at Pole when having the four, official WGS84 fixed Values GM, a, f and w.
-In that g84EP Dir, the final calculated results for g WGS84 at Equator and at Pole, though quite accurate, are — unfortunately — not perfectly in adequation with the official WGS84 g values at Equator and at Pole.

Main change 4

Most explanations/references are now given inside NOTES inside the adequate directories:

GRAVITY DIR: NOTE1 NOTE2 NOTE3 NOTE4

GRAVITY FREE Dir: NOTE

GRAVITY g84EP: NOTE1 NOTE2.

But the version number of the whole program is soon to appear at the beginning of the main program —>g.

Summary & Conclusion

Some doubts remain regarding the best equations to be used when refering to (old dated) GRS67.

For GRS80 or WGS84, Somigliana's equations give here most accurate results (to almost full calculator digits capacities).

CLOSED FORM equation is best fit for theorical gravity, when modelling an ellipsoid, changing one or several of its four "fixed" parameters GM, a, f and w.

Numerical Examples
Executing in main file GRAVITY Dir
—>g

with latitude [D.mmss] in stack level 2
and height [m] in stack level 1

will result in the following outputs:

:alt [m]: 0 :lat D.mmss: 0 :

Int Grav GRS 67
: 9.780318 :

So
m
igliana GRS 80
: 9.7803267715 :
So
m
igliana WGS 84
: 9.7803253359 :
Closed Form
 
WGS 84
: 9.78032532324

:alt [m]: 1000 :lat D.mmss: 0 :

Int Grav GRS 67
: 9.777232 :

So
m
igliana GRS 80
: 9.77723980166 :
So
m
igliana WGS 84
: 9.77723836651 :
Closed Form
 
WGS 84
: 9.77723826177

:alt [m]: 0 :lat D.mmss: 90 :

Int Grav GRS 67
: 9.83217715816 :

So
m
igliana GRS 80
: 9.83218636846 :
So
m
igliana WGS 84
: 9.83218493787 :
Closed Form
 
WGS 84
: 9.83218496308

:alt [m]: 1000 :lat D.mmss: 90 :

Int Grav GRS 67
: 9.82909115816 :

So
m
igliana GRS 80
: 9.82910370419 :
So
m
igliana WGS 84
: 9.82910227404 :
Closed Form
 
WGS 84
: 9.82910231835

:alt [m]: 0 :lat D.mmss: 45 :

Int Grav GRS 67
: 9.80618987521 :

So
m
igliana GRS 80
: 9.80619920255 :
So
m
igliana WGS 84
: 9.80619776931 :
Closed Form
 
WGS 84
: 9.80619777838

:alt [m]: 1000 :lat D.mmss: 45 :

Int Grav GRS 67
: 9.80310387521 :

So
m
igliana GRS 80
: 9.80311437628 :
So
m
igliana WGS 84
: 9.80311294349 :
Closed Form
 
WGS 84
: 9.80311291004

Last example practice

-Go into FREE Dir.
-Change the value of
WGS84 1/298.257223563
into 1/298.25722356
(cut the final, right digit 3 in the expression),

' 1/298.25722356' ENTER
'f' STO

Then
0 0 —>gFREE will return the following results:

alt [in m]: 0 :lat [in D.mmss]: 0 :

Closed Form 

FREE
: 9.78032534903

and 90 0 —>gFREE will return the following results:

alt [in m]: 0 :lat [in D.mmss]: 90 :

Closed Form 

FREE
: 9.83218491167

Regards,
Gil Campart