This Online Calculator for Local Gravity uses WSG 84 ellipsoidal Gravity Formula from DMA TECHNICAL REPORT TR8350.2-a, Chapter 4. The formula is combined with the altitude correction formula from the book "Atmosphere-Ocean Dynamics" by A. E. Gill, 1982.
This is the formula of Somigliana that has been selected as the official WSG 84 ellipsoidal Gravity Formula:
γ = γe(1 + k*sin2θ)/(1 - e2*sin2θ)1/2 (4-3)
k = ((b * γp)/(a * γe)) - 1 = 0.00193185138639;
a, b - semi-major and semi-minor axes of the ellipsoid, respectively;
γe - theoretical gravity at the equator, γe = 9.7803267714 m/s2;
γp - theoretical gravity at the pole;
e2 - square of the first eccentricity of the ellipsoid, e2 = 0.00669437999013;
θ - geodetic latitude, degrees.
The formula (4-3) is used to calculate normal gravity on the ellipsoidal surface.
The same WGS 84 Ellipsoidal Gravity Formula, expressed numerically, is:
γ = 9.7803267714*(1 + 0.00193185138639*sin2θ)/(1 - 0.00669437999013*sin2θ)1/2 (4-4)
The formula from the book "Atmosphere-Ocean Dynamics" by A. E. Gill, 1982, Appendix Two, Useful Values, page 597:
g = (9.78032 + 0.005172*sin2θ - 0.00006*sin2(2θ))*(1 + z/a)-2 (Gill)
g - acceleration of the gravity, m/s2 as a function of latitude θ and altitude z;
z - altitude, m;
a - radius of sphere having the same volume as the earth, a = 6371000 m.
Combining the formula (4-4) and the right part of the Gill's formula ((1 + z/a)-2), we get the very precise formula for the theoretical local gravity:
γ = γe*(1 + k*sin2θ)/(1 - e2*sin2θ)1/2 * (1 + z/a)-2 (1)
And the same formula, expressed numerically:
γ = 9.7803267714*(1 + 0.00193185138639*sin2θ)/(1 - 0.00669437999013*sin2θ)1/2 * (1 + z/a)-2 (2)
This Local Gravity Calculator uses the formula shown above. All parameters (γe, a, k, e2) may be changed if needed. The Precision of the calculator also can be adjusted.
All units used in formulas above and in this online calculator are MKS (Meter-Kilogram-Second) unit system.