# Theoretical gravity

In geodesy and geophysics, theoretical gravity or normal gravity is an approximation of the true gravity on Earth's surface by means of a mathematical model representing (a physically smoothed) Earth. The most common model of a smoothed Earth is a rotating Earth ellipsoid of revolution (i.e., a spheroid).

## Basic formulas

Various, successively more refined, formulas for computing the theoretical gravity are referred to as the International Gravity Formula, the first of which was proposed in 1930 by the International Association of Geodesy. The general shape of that formula is:

${\displaystyle g(\phi )=g_{e}\left(1+A\sin ^{2}(\phi )-B\sin ^{2}(2\phi )\right),}$

in which g(φ) is the gravity as a function of the geographic latitude φ of the position whose gravity is to be determined, ${\displaystyle g_{e}}$ denotes the gravity at the equator (as determined by measurement), and the coefficients A and B are parameters that must be selected to produce a good global fit to true gravity.[1]

Using the values of the GRS80 reference system, a commonly used specific instantiation of the formula above is given by:

${\displaystyle g(\phi )=9.780327\left(1+0.0053024\sin ^{2}(\phi )-0.0000058\sin ^{2}(2\phi )\right)\,\mathrm {ms} ^{-2}.}$[1]

Using the appropriate double-angle formula in combination with the Pythagorean identity, this can be rewritten in the equivalent forms

{\displaystyle {\begin{aligned}g(\phi )&=9.780327\left(1+0.0052792\sin ^{2}(\phi )+0.0000232\sin ^{4}(\phi )\right)\,\mathrm {ms} ^{-2},\\&=9.780327\left(1.0053024-.0053256\cos ^{2}(\phi )+.0000232\cos ^{4}(\phi )\right)\,\mathrm {ms} ^{-2},\\&=9.780327\left(1.0026454-0.0026512\cos(2\phi )+.0000058\cos ^{2}(2\phi )\right)\,\mathrm {ms} ^{-2}.\end{aligned}}\,\!}

Up to the 1960s, formulas based on the Hayford ellipsoid (1924) and of the famous German geodesist Helmert (1906) were often used.[citation needed] The difference between the semi-major axis (equatorial radius) of the Hayford ellipsoid and that of the modern WGS84 ellipsoid is 251 m; for Helmert's ellipsoid it is only 63 m.

## Somigliana equation

A more recent theoretical formula for gravity as a function of latitude is the International Gravity Formula 1980 (IGF80), also based on the WGS80 ellipsoid but now using the Somigliana equation:

${\displaystyle g(\phi )=g_{e}\left[{\frac {1+k\sin ^{2}(\phi )}{\sqrt {1-e^{2}\sin ^{2}(\phi )}}}\right],\,\!}$

where,[2]

• ${\displaystyle k={\frac {bg_{p}-ag_{e}}{ag_{e}}}}$ (formula constant);
• ${\displaystyle g_{e},g_{p}}$ is the defined gravity at the equator and poles, respectively;
• ${\displaystyle a,b}$ are the equatorial and polar semi-axes, respectively;
• ${\displaystyle e^{2}={\frac {a^{2}-b^{2}}{a^{2}}}}$ is the spheroid's squared eccentricity;

providing,

${\displaystyle g(\phi )=9.7803267715\left[{\frac {1+0.001931851353\sin ^{2}(\phi )}{\sqrt {1-0.0066943800229\sin ^{2}(\phi )}}}\right]\,\mathrm {ms} ^{-2}.}$[1]

A later refinement, based on the WGS84 ellipsoid, is the WGS (World Geodetic System) 1984 Ellipsoidal Gravity Formula:[2]

${\displaystyle g(\phi )=9.7803253359\left[{\frac {1+0.00193185265241\sin ^{2}(\phi )}{\sqrt {1-0.00669437999013\sin ^{2}(\phi )}}}\right]\,\mathrm {ms} ^{-2}.}$

(where ${\displaystyle g_{p}}$ = 9.8321849378 ms−2)

The difference with IGF80 is insignificant when used for geophysical purposes,[1] but may be significant for other uses.

#### Further details

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:

${\displaystyle a=6\,378\,137\,\mathrm {m} \quad \quad \quad \quad b=6\,356\,752{.}314\,1\,\mathrm {m} }$
${\displaystyle \gamma _{a}=9{.}780\,326\,771\,5\,\mathrm {\frac {m}{s^{2}}} \quad \gamma _{b}=9{.}832\,186\,368\,5\,\mathrm {\frac {m}{s^{2}}} }$

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

## Approximation formula from series expansions

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

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

#### International gravity formula 1930

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:

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

#### International gravity formula 1967

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

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

#### International gravity formula 1980

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.

## Height dependence

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.

## WELMEC formula

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]

## Example

Free-fall acceleration in Schweinfurt:

Data:

• Latitude: 50° 3′ 24″ = 50.0567°
• Height above sea level: 229.7 m
• Density of the rock plates: ca. 2.6 g/cm3
• Measured free-fall acceleration: g = 9.8100 ± 0.0001 m/s2

Free-fall acceleration, calculated through normal gravity formulas:

• Cassinis: g = 9.81038 m/s2
• Jeffreys: g = 9.81027 m/s2
• WELMEC: g = 9.81004 m/s2