Normal gravity formula

Normal gravity formulas are mathematical expressions that can be used to estimate the acceleration of gravity at a point near the surface of the earth . If the mass of a body is known or given, the weight on it can also be estimated.

Normal severity

The acceleration due to gravity depends on the gravity of the mass that is below the object; the gravitation decreases with the distance from the center of mass. In addition, the gravitational acceleration is influenced by the earth's rotation : the centrifugal force increases with the distance from the earth's axis , it is therefore greatest at the equator and least at the poles .

The formulas for calculating normal gravity are based on the assumption of a rotationally symmetrical earth ellipsoid whose surface is also an equipotential surface ( normal ellipsoid ). Therefore they only depend on the geographical latitude and altitude .

Formula from Somigliana

For normal gravity on the level ellipsoid, i. H. in height , the formula by Somigliana (1929) applies (after Carlo Somigliana (1860–1955) :) ${\displaystyle \gamma _{0}}$ ${\displaystyle h=0}$

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

With

• ${\displaystyle \gamma _{a}}$ = Normal gravity at the equator
• ${\displaystyle \gamma _{b}}$ = Normal gravity at the pole
• ${\displaystyle a}$= major semi-axis (equator radius)
• ${\displaystyle b}$ = small semi-axis (pole radius)
• ${\displaystyle \varphi }$= latitude

For numerical purposes this formula can be transformed to:

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

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:

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

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

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

Approximation formulas from series developments

The Somigliana formula was approximated by various series developments . These follow the scheme:

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

International weight formula 1930

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

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

In the course of time, the values ​​have been continuously improved through newer knowledge and more precise measurement methods.

Jeffreys improved the values ​​in 1948 to:

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

International weight formula 1967

The normal gravity field of the geodetic reference system from 1967 is defined by 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 weight formula 1980

The parameters of the GRS 80 result in classic row development:

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

The accuracy is about ± 10 ⁻⁶ m / s ² .

The following series development was also introduced with the GRS 80:

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

The parameters for this are:

• c ₁  = 5.279 0414 · 10 ⁻³
• c ₂  = 2.327 18 · 10 ⁻⁵
• c ₃  = 1.262 · 10 ⁻⁷
• c ₄  = 7 · 10 ⁻¹⁰

This approximation is accurate to about ± 10 ⁻⁹ m / s ² . If this precision is not needed, the trailing terms can be omitted. However, it is recommended to use the Somigliana closed formula.

Altitude dependence

Cassinis determined the altitude dependency to:

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

The mean rock density  ρ is no longer taken into account today.

Since the GRS 1967 the following applies to the dependence on the ellipsoidal height  h :

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

Another representation is:

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

derived with the parameters 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 correction is quite accurate for altitudes commonly used in aviation ; however, it diverges for space (over approx. 100 kilometers) .

WELMEC formula

In all German calibration offices , the reference value for the acceleration due to gravity  g in relation to the mean geographical latitude φ and the mean height above sea level  h is calculated using the WELMEC formula:

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

The formula is based on the 1967 International Gravity Formula.

The knowledge of the gravitational acceleration present at the measurement location is essential for precision measurements of many mechanical quantities. Scales , which usually measure mass using weight, are based on the acceleration of gravity, so they must be prepared for use at their place of use. Thanks to the concept of so-called gravitational zones, which are divided with the help of normal gravity, a scale can already be finally adjusted for use at the manufacturer .

example

Fall acceleration in Schweinfurt :

Data:

• Latitude: 50 ° 3 ′ 24 ″ = 50.0567 °
• Height above sea level: 229.7 m
• Density of the stone slab: approx. 2.6 g / cm³
• Measured gravitational acceleration: g = (9.8100 ± 0.0001) m / s²

Calculated gravitational acceleration using normal gravity formulas:

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

literature

• Wolfgang Torge : Geodesy . 2nd Edition. Walter de Gruyter, Berlin a. a. 2003. ISBN 3-11-017545-2
• Wolfgang Torge : Geodesy . Walter de Gruyter, Berlin a. a. 1975 ISBN 3-11-004394-7

Individual evidence

1. ↑ Somigliana's biography ( Memento of the original dated December 7, 2010 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (ital.) 
2. ^ Roman Schwartz, Andreas Lindau: The European gravitational zone concept according to WELMEC. (pdf) Accessed February 26, 2011 (700kB). 

Web links

• Definition of the Geodetic Reference System 1980 (GRS80) (pdf, English; 70 kB)
• Gravity Information System of the Physikalisch-Technische Bundesanstalt , engl.
• Online calculation of normal severity with various normal severity formulas