Theoretical gravity

For the normal gravity ${\displaystyle \gamma _{0}}$ of the sea level ellipsoid, i.e. elevation  h = 0, this formula by Somigliana (1929) applies (after Carlo Somigliana (1860–1955)[3]):

${\displaystyle \gamma _{0}(\varphi )={\frac {a\cdot \gamma _{a}\cdot \cos ^{2}\varphi +b\cdot \gamma _{b}\cdot \sin ^{2}\varphi }{\sqrt {a^{2}\cdot \cos ^{2}\varphi +b^{2}\cdot \sin ^{2}\varphi }}}}$

with

• ${\displaystyle \gamma _{a}}$ = Normal gravity at Equator
• ${\displaystyle \gamma _{b}}$ = Normal gravity at Poles
• a = semi-major axis (Equator radius)
• b = semi-minor axis (Pole radius)
• ${\displaystyle \varphi }$ = latitude

Due to numerical issues, the formula is simplified into this:

${\displaystyle \gamma _{0}(\varphi )=\gamma _{a}\cdot {\frac {1+p\cdot \sin ^{2}\varphi }{\sqrt {1-e^{2}\cdot \sin ^{2}\varphi }}}}$

with

• ${\displaystyle p={\frac {b\cdot \gamma _{b}}{a\cdot \gamma _{a}}}-1}$
• ${\displaystyle e^{2}=1-{\frac {b^{2}}{a^{2}}};\quad e}$ is the eccentricity

For the Geodetic Reference System 1980 (GRS 80) the parameters are set to these values:

${\displaystyle a=6\,378\,137\,\mathrm {m} \quad \quad \quad \quad b=6\,356\,752{.}314\,1\,\mathrm {m} }$

${\displaystyle \gamma _{a}=9{.}780\,326\,771\,5\,\mathrm {\frac {m}{s^{2}}} \quad \gamma _{b}=9{.}832\,186\,368\,5\,\mathrm {\frac {m}{s^{2}}} }$

${\displaystyle \Rightarrow p=1{.}931\,851\,353\cdot 10^{-3}\quad e^{2}=6{.}694\,380\,022\,90\cdot 10^{-3}}$

The Somigliana formula was approximated through different series expansions, following this scheme:

${\displaystyle \gamma _{0}(\varphi )=\gamma _{a}\cdot (1+\beta \cdot \sin ^{2}\varphi +\beta _{1}\cdot \sin ^{2}2\varphi +\dots )}$

The normal gravity formula by Gino Cassinis was determined in 1930 by International Union of Geodesy and Geophysics as international gravity formula along with Hayford ellipsoid. The parameters are:

${\displaystyle \gamma _{a}=9{.}78049{\frac {\mathrm {m} }{\mathrm {s} ^{2}}}\quad \beta =5{.}2884\cdot 10^{-3}\quad \beta _{1}=-5{.}9\cdot 10^{-6}}$

In the course of time the values were improved again with newer knowledge and more exact measurement methods.

Harold Jeffreys improved the values in 1948 at:

${\displaystyle \gamma _{a}=9{.}780373{\frac {\mathrm {m} }{\mathrm {s} ^{2}}}\quad \beta =5{.}2891\cdot 10^{-3}\quad \beta _{1}=-5{.}9\cdot 10^{-6}}$

The normal gravity formula of Geodetic Reference System 1967 is defined with the values:

${\displaystyle \gamma _{a}=9{.}780318{\frac {\mathrm {m} }{\mathrm {s} ^{2}}}\quad \beta =5{.}3024\cdot 10^{-3}\quad \beta _{1}=-5{.}9\cdot 10^{-6}}$

From the parameters of GRS 80 comes the classic series expansion:

${\displaystyle \gamma _{a}=9{.}780327{\frac {\mathrm {m} }{\mathrm {s} ^{2}}}\quad \beta =5{.}3024\cdot 10^{-3}\quad \beta _{1}=-5{.}8\cdot 10^{-6}}$

The accuracy is about ±10⁻⁶ m/s².

With GRS 80 the following series expansion is also introduced:

${\displaystyle \gamma _{0}(\varphi )=\gamma _{a}\cdot (1+c_{1}\cdot \sin ^{2}\varphi +c_{2}\cdot \sin ^{4}\varphi +c_{3}\cdot \sin ^{6}\varphi +c_{4}\cdot \sin ^{8}\varphi +\dots )}$

As such the parameters are:

• c₁ = 5.279 0414·10⁻³
• c₂ = 2.327 18·10⁻⁵
• c₃ = 1.262·10⁻⁷
• c₄ = 7·10⁻¹⁰

The accuracy is at about ±10⁻⁹ m/s² exact. When the exactness is not required, the terms at further back can be omitted. But it is recommended to use this finalized formula.

Cassinis determined the height dependence, as:

${\displaystyle g(\varphi ,h)=g_{0}(\varphi )-\left(3{.}08\cdot 10^{-6}\,{\frac {1}{\mathrm {s} ^{2}}}-4{.}19\cdot 10^{-7}\,{\frac {\mathrm {cm} ^{3}}{\mathrm {g} \cdot \mathrm {s} ^{2}}}\cdot \rho \right)\cdot h}$

The average rock density ρ is no longer considered.

Since GRS 1967 the dependence on the ellipsoidal elevation h is:

${\displaystyle {\begin{aligned}g(\varphi ,h)&=g_{0}(\varphi )-\left(1-1{.}39\cdot 10^{-3}\cdot \sin ^{2}(\varphi )\right)\cdot 3{.}0877\cdot 10^{-6}\,{\frac {1}{\mathrm {s} ^{2}}}\cdot h+7{.}2\cdot 10^{-13}\,{\frac {1}{\mathrm {m} \cdot \mathrm {s} ^{2}}}\cdot h^{2}\\&=g_{0}(\varphi )-\left(3{.}0877\cdot 10^{-6}-4{.}3\cdot 10^{-9}\cdot \sin ^{2}(\varphi )\right)\,{\frac {1}{\mathrm {s} ^{2}}}\cdot h+7{.}2\cdot 10^{-13}\,{\frac {1}{\mathrm {m} \cdot \mathrm {s} ^{2}}}\cdot h^{2}\end{aligned}}}$

Another expression is:

${\displaystyle g(\varphi ,h)=g_{0}(\varphi )\cdot (1-(k_{1}-k_{2}\cdot \sin ^{2}\varphi )\cdot h+k_{3}\cdot h^{2})}$

with the parameters derived from GSR80:

• ${\displaystyle k_{1}=2\cdot (1+f+m)/a=3{.}157\,04\cdot 10^{-7}\,\mathrm {m^{-1}} }$
• ${\displaystyle k_{2}=4\cdot f/a=2{.}102\,69\cdot 10^{-9}\,\mathrm {m^{-1}} }$
• ${\displaystyle k_{3}=3/(a^{2})=7{.}374\,52\cdot 10^{-14}\,\mathrm {m^{-2}} }$

This adjustment is about right for common heights in Aviation; But for heights up to outer space (over ca. 100 kilometers) it is out of range.

In all German standards offices the free-fall acceleration g is calculated in respect to the average latitude φ and the average height above sea level h with the WELMEC–Formel:

${\displaystyle g(\varphi ,h)=\left(1+0{.}0053024\cdot \sin ^{2}(\varphi )-0{.}0000058\cdot \sin ^{2}(2\varphi )\right)\cdot 9{.}780318{\frac {\mathrm {m} }{\mathrm {s} ^{2}}}-0{.}000003085\,{\frac {1}{\mathrm {s} ^{2}}}\cdot h}$

The formula is based on the International gravity formula from 1967.

The scale of free-fall acceleration at a certain place must be determined with precision measurement of several mechanical magnitudes. Weighing scales, the mass of which does measurement because of the weight, relies on the free-fall acceleration, thus for use they must be prepared with different constants in different places of use. Through the concept of so-called gravity zones, which are divided with the use of normal gravity, a weighing scale can be calibrated by the manufacturer before use.[4]