Prowling around in the Free Google e-books, I found this physics book from 1923: https://books.google.com/books?id=YZ4_gPRaEigC&printsec=frontcover&dq=Physics&hl=en&sa=X&ved=0ahUKEwiW7bn_hM7jAhWSdt8KHUPMD_wQ6AEIQzAE#v=onepage&q=Physics&f=false
Starting on page 107, there's a really nice discussion of variations in gravity on the earth's surface. It explains two of the contributors to the direction of a plumb line that keep it from pointing to the center of the earth. Later on, it gives a formula from Clairaut for approximating?ÿg,?ÿthe force of gravity at the earth's surface.
On the NGS web site, there's an API that calculates GRAV-D values at entered latitude, longitude, and ellipsoid height. https://www.ngs.noaa.gov/web_services/grav-d.shtml
Of course, no true math nerd can stop there. The value from the 18th century formula, the API value, and the modeled gravity value from a data sheet must be compared. DP7100 near UNC-Charlotte contains a modeled gravity value of 979,720.0 mgal. Using its latitude, longitude, and orthometric height (Clairaut's formula uses height above sea level), the old formula produces 979,773.8 mgal, a difference of 55 ppm. Using the same variables except substituting ellipsoid height for orthometric height, the API gets 979,725.4 mgal, a difference of 5.5 ppm.
Entering this in your browser will access the API for DP7100; modify the entry for other points:?ÿ https://geodesy.noaa.gov/api/gravd/gp?lat=35.3026596556&lon=-80.7294671944&eht=195.776
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Well, since before NAVD88 orthometric was assumed to be the same as ellipsoidal height, for lack of better data.?ÿ So using ElHt in the old formula is quite justified.
Correct me if I'm wrong on that.
Yes, that's why I used NAVD88 height in the formula. An early version of NGS' coordinate converter would calculate an elevation factor for conversions to NAD27 from NAD83. To get it approximately correct, you could enter the NAVD88 height when converting to NAD27. Converting from NAD27, you entered an approximate NAD83 ellipsoid height.
That's been fixed so that conversions to NAD27 no longer return elevation factors, so I can't prove that it was there, but I remember asking NGS how things should be and getting the correct reply.
Well, durn. Clairaut's formula uses height in centimeters and I used meters to calculate the gravity at DP7100. Correcting that makes the answer 979,706.6 mGal. and the difference between the calculated and measured values?ÿ -14 ppm.
Also, the constants in the formula were obviously not calculated from the GRS80 ellipsoid, so there's more error there. Correcting those for GRS80 is probably doable.
Math nerdiness is incurable, but errors can accumulate with age.
I learned everything I needed to know about gravity by falling out of trees when I was an invincible kid. ???? ?ÿ
Same here, reinforced later by stepping off ladders too soon!
