Gravity field

A gravity field is a force field caused by gravity and certain inertial effects . The field strength of the gravity field is the gravity , formula symbol . As the weight of a test specimen related to the mass , it has the SI unit N / kg = m / s² and is also referred to as acceleration due to gravity or acceleration due to gravity . With this acceleration , a freely falling body starts moving. ${\ displaystyle {\ vec {g}}}$

A plumb line shows the direction of the gravitational field

${\ displaystyle {\ vec {g}}}$ is a vector quantity with magnitude and direction. The direction is called plumb direction . The amount is also called the location factor to emphasize that and thus also the weight of a body depends on the location. In Germany the acceleration due to gravity is about 9.81 m / s² = 981 Gal . The variation over the earth's surface is a few Gal. ${\ displaystyle g}$

In a narrower sense - especially in the geosciences - the gravity field of a celestial body is composed of its gravitational field ("gravity") and the centrifugal acceleration in the reference system that rotates with the body and possibly falls freely with it in the gravitational field of other celestial bodies.

In celestial mechanics , non-rotating reference systems are often used. The gravitational field of one or more celestial bodies is then based only on gravitation.

In a broader sense, one speaks of the gravitational field in arbitrarily accelerated reference systems. In the gravitational field of a centrifuge which dominates centrifugal force . In free-falling reference systems (e.g. space station ) there is weightlessness .

Measurement

In addition to the direct measurement of the acceleration of a freely falling body, the amount of gravitational acceleration can be calculated from the period of oscillation of a pendulum . A modern gravimeter is a special spring scale and achieves a precision of one microgale, approx. 10 ⁻⁹ g . You could register a height shift of less than one centimeter. However, fluctuations in air pressure also cause changes of the same order of magnitude, and tidal accelerations due to the inhomogeneity of the external gravitational fields, especially those of the moon and sun, even change g by many milligals.

Sum of gravitational and centrifugal acceleration

The gravitational acceleration is the vector sum of a gravitational and a centrifugal part:

{\ displaystyle {\ vec {g}} = {\ vec {g}} _ {\ mathrm {Gravitation}} + {\ vec {a}} _ {\ mathrm {Centrifugal}}}

The gravitational acceleration is caused by the gravitational field . If the celestial body can be viewed as spherically symmetrical , the gravitational acceleration is calculated according to the law of gravitation :

{\ displaystyle {\ vec {g}} _ {\ mathrm {Gravitation}} = - {\ frac {GM} {r ^ {2}}} \, {\ vec {e}} _ {r}}

Here is the gravitational constant , the mass of the celestial body, the distance between the center of gravity of the celestial body and the test body and a unit vector which is directed from the center of gravity of the celestial body to the test body. If the mass distribution of the celestial body is not isotropic , as is usually the case, gravity anomalies result.

{\ displaystyle G}

{\ displaystyle M}

{\ displaystyle r}

{\ displaystyle {\ vec {e}} _ {r}}

The centrifugal acceleration has an effect because one is on the surface of the celestial body in a co-rotating reference system. ${\ displaystyle {\ vec {a}} _ {\ mathrm {Centrifugal}}}$

Tidal forces arise through the influence of other celestial bodies (e.g. the moon or the sun). Whether these forces are considered part of the gravitational field is a matter of definition. In this article, they are not counted as part of the gravity field.

For the gravitational field on a planet's surface, this results in: The gravitational acceleration depends on the altitude, because according to the law of gravitation . It also follows from this relationship that due to the flattening of the planet, the distance to the center of the planet is smallest at the poles, and the gravitational effect is therefore greatest. In addition, the centrifugal acceleration at the poles of the celestial body disappears because the distance from the axis of rotation is zero. The gravitational field is weakest at the equator: there the centrifugal acceleration is maximal and directed against the gravitational effect and the distance to the center of the planet is greatest. ${\ displaystyle {\ vec {g}} _ {\ mathrm {Gravitation}} \ sim {\ frac {1} {r ^ {2}}}}$

The direction of the gravitational acceleration is called the perpendicular direction . This perpendicular direction points approximately to the center of gravity of the celestial body. Deviations ( apart from gravity anomalies ) arise from the fact that the centrifugal acceleration at medium latitudes is at an oblique angle to the gravitational acceleration. Lines that follow the plumb line are called plumb lines. They are the field lines of the gravitational field. If a body moves in the gravitational field, the direction of the effective acceleration deviates from the plumb direction with increasing speed. This can be interpreted as the effect of the Coriolis force .

Gravity potential

Since the weight of a conservative force , the gravitational acceleration as an appropriate field strength of the negative is gradient of a potential U , . In the physical geodesy but not underground , but W - = U used and W despite different in sign gravity potential (in the earth also Geopotential called). With this convention is ${\ textstyle {\ vec {g}} ({\ vec {r}}) = - {\ vec {\ nabla}} U ({\ vec {r}})}$

{\ displaystyle {\ vec {g}} ({\ vec {r}}) = + {\ vec {\ nabla}} W ({\ vec {r}})}

.

The gravitational potential is composed - similar to the acceleration due to gravity itself - from a gravitational and a centrifugal component,

{\ displaystyle W ({\ vec {r}}) = G \ iiint \! {\ frac {\ rho ({\ vec {r}} ')} {| {\ vec {r}} - {\ vec { r}} '|}} \, \ mathrm {d} ^ {3} r' ​​+ {\ frac {1} {2}} ({\ vec {\ omega}} \ times {\ vec {r}}) ^ {2}}

.

The first term is the gravitational potential in the general form for an extended body with the density distribution . For a radially symmetrical body of mass M , it is simplified to in the outer space . This contribution disappears in infinity. The second summand, the form of which assumes that the origin of the coordinate system lies on the axis of rotation, is the potential of the centrifugal acceleration . It can also be written as with the distance from the axis of rotation . This contribution disappears in the origin. Since both summands never become negative, W only takes positive values. ${\ displaystyle \ rho ({\ vec {r}} ')}$ ${\ textstyle G {\ frac {M} {r}}}$ ${\ textstyle r _ {\ perp}}$ ${\ textstyle {\ frac {1} {2}} \ omega ^ {2} r _ {\ perp} ^ {2}}$

Water in a rotating bucket. As soon as it rotates smoothly, its surface forms a potential surface.

Areas on which the gravitational potential is constant are called potential surfaces or level surfaces of the gravitational field. They are pierced at right angles by the plumb lines. When moving from one level surface to a higher one, lifting work has to be done, see also potential (physics) .

Geopotential

The earth's gravitational potential W is also called geopotential . A particularly important level surface here is the geoid , on which the gravity potential determines the value

{\ displaystyle W_ {0} \ approx W_ {0, \ mathrm {conv}} = 62 {.} 636 {.} 856 {,} 0 {\ frac {\ mathrm {m} ^ {2}} {\ mathrm {s} ^ {2}}}}

accepts. The value mentioned here is known as the "conventional geoid potential". It is used, among other things, in the definition of International Atomic Time , by the International Service for Earth Rotation and Reference Systems and by the IAU to define Terrestrial Time . It is the best measured value known in 1998. However, more recent measurements show a value for W ₀ which is about 2.6 m² / s² smaller , which corresponds to a height difference of 26 cm. Potential differences are often related to W ₀ ,

{\ displaystyle C (P) = W_ {0} -W_ {P} = - \ int _ {P_ {0}} ^ {P} {\ vec {g}} \ cdot \ mathrm {d} {\ vec { s}}}

,

and then called geopotential level (unit of geopotential meter gpm ). If the geopotential elevation is divided by the normal gravity , the result is the dynamic height . For medium latitudes, the dynamic height is roughly the same as the metric height above sea level. The distance between two equipotential surfaces depends on the local gravitational acceleration: the larger this is, the smaller the distance.

More general definition

If one does not choose the surface of a planet as the reference system, but any accelerated reference system, the "fall" acceleration effective there can also be understood as a gravitational field. The forces prevailing in this reference system are also composed of gravitational and inertial forces.

Examples

In a freely falling frame of reference, the gravitational force and the inertial force are oppositely equal. A body in the freely falling frame of reference is therefore weightless, i.e. H. force-free. So the free falling frame of reference is an inertial frame . A space station that is in an orbit around the earth is also in "free fall", since its movement is determined exclusively by gravity. Weightlessness, i.e. H. the disappearance of the gravitational field on board this space station is therefore not the result of an absence of gravity, but rather the result of an equilibrium between gravitational force and inertial force. (see weightlessness ).
A planet moves in an orbit of the sun on a circular or elliptical path. If you now choose the sun-planet axis as the reference system for the movement of a third body, e.g. B. a space probe, the gravitational field effective for this body is determined by the interaction of the gravitation of both celestial bodies and the centrifugal field due to the rotation of the reference system. (see Lagrange points ).

Earth's gravitational field

Earth's gravitational field in the Southern Ocean

Large celestial bodies take on a shape under the influence of their gravitational field, which corresponds to one of the level surfaces. In the earth's gravitational field , the level surface that roughly follows the height of sea level is called the geoid . It is slightly flattened by the centrifugal acceleration . This flattening and the decrease in the acceleration due to gravity (acceleration due to gravity on earth) with altitude are taken into account by normal gravity formulas. In addition, there are gravity anomalies, i. H. global, regional and local irregularities, as the mass is not evenly distributed both in the earth's crust (mountains, continental plates ) and deeper (in the earth's mantle and core ). The satellite geodesy determines the geoid with the help of the observation of satellite orbits , see gradiometry . The gravity anomalies reach the order of magnitude 0.01% and 0.01 ° in terms of amount or direction, see deviation from the perpendicular , gravity gradient and vertical gradient . There are up to 100 m between the geoid and the central ellipsoid.

Earth's gravity field on the earth's surface

Due to the centrifugal force, the flattening of the earth and the elevation profile, the value of the gravitational acceleration varies regionally by a few per thousand around the approximate value 9.81 m / s². The acceleration due to gravity is 9.832 m / s² at the poles and 9.780 m / s² at the equator. The attraction at the pole is about 0.5% greater than at the equator. If the gravity on a person at the equator is 800 N, it increases to 804.24 N at the earth's poles. In 2013 it was determined that the acceleration due to gravity was 9.7639 m / s² on the Nevado Huascarán mountain in the Andes (highest mountain in Peru ) is the lowest.

Standard acceleration

In 1901, at the third general conference for weights and measures, a standard value, the normal acceleration due to fall , was arbitrarily set at g _n = 9.80665 m / s², a value that had already been established in various state laws and that serves to define technical units of measurement (DIN 1305) . The basis was measurements by G. Defforges at the BIPM near Paris (outdated from today's perspective) , extrapolated to 45 degrees latitude and sea level.

German main heavy network 1996

Identification badge of the German Heavy Network 1962

In Germany, the location-dependent acceleration due to gravity is recorded in the German Main Gravity Network 1996 ( DHSN 96 ) , which is a continuation of the (West German) DHSN 82 . In addition to the German main triangular network for the location and the German main height network for the height, it is the third variable for the unambiguous definition of a geodetic reference system. The German gravity network is based on around 16,000 measuring points, the gravity fixed points.

Historically significant was the value 9.81274 m / s² in Potsdam, determined by Kühnen and Furtwänger from the Potsdam Geodetic Institute in 1906. In 1906, Potsdam became the fundamental point for determining the local acceleration due to gravity using difference determination until the International Gravity Standardization Net was introduced in 1971.

With the introduction of the integrated spatial reference in 2016 , the DHSN 96 was replaced by the DHSN 2016 .

Earth's gravity field in the interior of the earth

Gravitation in the interior of the earth according to the seismic PREM -Earth model, as well as approximations by constant and linearly increasing rock density for comparison.

If the earth were a non-rotating, homogeneous sphere, there would be a linear increase in gravitational acceleration from zero at the center of the earth to a maximum on the earth's surface. In fact, the earth is built up in layers of very different densities. Therefore, the relationship between depth and gravitational acceleration is more complicated. In the earth's core , the acceleration due to gravity initially increases uniformly with the distance from the center of the earth. At the core-mantle boundary (approx. 2900 km from the center of the earth), also known as the Wiechert-Gutenberg discontinuity after their discoverers Emil Wiechert and Beno Gutenberg , it reaches a maximum of almost 10.68 m / s². This effect is due to the fact that the predominantly metallic core of the earth is more than twice as dense as the earth's mantle and crust . From there up to approx. 4900 km it first slowly decreases again to 9.93 m / s², increases again at 5700 km to 10.01 m / s² and then decreases monotonically until it is about 9.82 m on the earth's surface / s² reached.

Earth's gravitational field outside the earth

In the vicinity of the earth's surface, g decreases by about 3.1 µm / s² for every meter increased. In meteorology, the geopotential in the atmosphere is given as equipotential surfaces . In practice, main pressure areas have been defined (1000, 500, 200 hPa, and others).

Outside the earth, the gravitational field decreases proportionally to the square of the distance from the center of the earth, while at a constant position in terms of latitude and longitude, the centrifugal acceleration increases proportionally with this distance. The earth's gravitational field is therefore (like the gravitational field of every celestial body) in principle unlimited, but it quickly becomes weaker with increasing distance. At low satellite altitudes of 300 to 400 km the acceleration due to gravity decreases by 10 to 15%, at 5000 km by approx. 70%. At an altitude of almost 36,000 km, both influences cancel each other out exactly. As a result, a satellite moves in such a geostationary orbit exactly synchronized with the rotation of the earth and remains at the same longitude.

In practice, the influence of the other celestial bodies can only be neglected in the vicinity of a heavy celestial body, since it is then very small - the influence of the nearby body is dominant.

Gravity and gravitational acceleration of celestial bodies

Stars and other gas or plasma bodies have a non-trivially defined star surface , on which their surface acceleration can be specified. This depends not only on its mass, but also on its density. A giant star has a much larger star radius , which means that its surface acceleration is smaller than that of the sun.

Since the surface acceleration of celestial bodies fluctuates over many orders of magnitude, it is often given in astrophysics in logarithmic form (log g). The surface acceleration g in the unit cm / s² is implicitly divided by the reference value 1 cm / s² (which means that it has no units) and the logarithm for base 10 is calculated from this. For example, the sun has a surface acceleration g of approx. 27,400 cm / s². This results in a value of approx. 4.44 for log g.

Examples of different celestial bodies

Heavenly bodies	log g
Sun ( yellow dwarf )	4.44
Betelgeuse ( red giant )	approx. −0.6
Sirius B ( white dwarf )	approx 8
Gliese 229 B ( brown dwarf )	approx. 5

Selected celestial bodies of the solar system

The rotation of the celestial body reduces its gravitational acceleration due to the centrifugal acceleration. The following table contains the gravitational and centrifugal of a body rotating at the equator on the surface and the resulting gravitational acceleration of the sun, the eight planets, Plutos and some moons of the solar system . The negative sign of the centrifugal acceleration is intended to make it clear that this is opposite to the gravitational acceleration.

	Heavenly bodies	Acceleration in m / s²
	Heavenly bodies	Gravity	Centrifugal	Heaviness
	Sun	274.0	−0.0057	274.0
1	Mercury	003.70	−3.75 · 10 ⁻⁶	003.70
2	Venus	008.87	−0.541 · 10 ⁻⁶	008.87
3	earth	009.80665	−0.0339	009.780
	moon	001.622	−12.3 · 10 ⁻⁶	001.622
4th	Mars	003.711	−0.0171	003.69
5	Jupiter	024.79	−2.21	023.12
	Io	001.81	−0.007	001,796
	Amalthea	000.02	−0.003	000.017
6th	Saturn	010.44	−1.67	008.96
7th	Uranus	008.87	−0.262	008.69
8th	Neptune	011.15	−0.291	011.00
	Larissa	000.0355	−0.00186	000.0336
	Pluto	000.62	−154 · 10 ⁻⁶	000.62

literature

Douglas Roy Tate: Gravity measurements and the Standards Laboratory. In: US National Bureau of Standards. Technical note, 491. Superintendent of Documents, United States Government Printing Office , Washington 1969. hdl: 2027 / mdp.39015077289141 HathiTrust Digital Library.
Christoph Reigber, Peter Schwintzer: The Earth's gravitational field . In: Physics in Our Time . tape 34 , no. 5 , September 2003, p. 206–212 , doi : 10.1002 / piuz.200301023 .

Web links

Calculation of the value of the gravitational acceleration for any location (Gravity Information System of PTB )
GGMplus - 200m-resolution maps of Earth's gravity field Gallery . Western Australian Center for Geodesy, Curtin University

Individual evidence

↑ ^a ^b Wolfgang Torge : Geodesy . 2nd Edition. de Gruyter, 2003, ISBN 3-11-017545-2 ( limited preview in the Google book search).
↑ Martin Vermeer: Physical geodesy . School of Engineering, Aalto University, 2020, ISBN 978-952-60-8940-9 , pp. 10, 88 (English, full text [PDF]): "In physical geodesy - unlike in physics - the potential is reckoned to be always positive ..."
^ Resolutions adopted at the 26th CGPM. BIPM , November 2018, accessed on February 23, 2020 (English, French, Resolution 2: On the definition of time scales ).
↑ ^a ^b A conventional value for the geoid reference potential W ₀ . (PDF) In: Unified Analysis Workshop 2017. German Geodetic Research Institute , pp. 5–7 , accessed on February 23, 2020 (English).
^ Luh: The acceleration of gravity fluctuates more than expected dradio - Forschung Aktuell , August 20, 2013.
↑ Gravity Variations Over Earth Much Bigger Than Previously Thought in Science Daily, September 4, 2013.
↑ Comptes rendus de la 3 ^e CGPM (1901), 1901, 70, there still in cm / s². BIPM - Résolution de la 3e CGPM
↑ standard DIN 1305 mass, weight value, force, weight force, weight, load; Terms ( beuth.de ).
^ Tate, 1969.
↑ State Office for Internal Administration (LAiV) Mecklenburg-Western Pomerania: Spatial reference - position, height and gravity fixed point fields ( Memento from January 14, 2014 in the Internet Archive )
^ Tate, 1969.
↑ [1]
^ ^A ^b David R. Williams: Planetary Fact Sheet - Metric. NASA, November 29, 2007, accessed on August 4, 2008 (English, including subpages).
^ German-Swiss Mathematics Commission [DMK] and German-Swiss Physics Commission [DPK] (ed.): Formulas and tables . 11th edition. Orell Füssli Verlag, Zurich 2006, ISBN 978-3-280-02162-0 , p. 188 .

[Toge03-1] Wolfgang Torge : Geodesy . 2nd Edition. de Gruyter, 2003, ISBN 3-11-017545-2 ( limited preview in the Google book search).

[Vermeer20-2] Martin Vermeer: Physical geodesy . School of Engineering, Aalto University, 2020, ISBN 978-952-60-8940-9 , pp. 10, 88 (English, full text [PDF]): "In physical geodesy - unlike in physics - the potential is reckoned to be always positive ..."

[TAI-definition-3] Resolutions adopted at the 26th CGPM. BIPM , November 2018, accessed on February 23, 2020 (English, French, Resolution 2: On the definition of time scales ).

[geopot17-4] A conventional value for the geoid reference potential W ₀ . (PDF) In: Unified Analysis Workshop 2017. German Geodetic Research Institute , pp. 5–7 , accessed on February 23, 2020 (English).

[5] Luh: The acceleration of gravity fluctuates more than expected dradio - Forschung Aktuell , August 20, 2013.

[6] Gravity Variations Over Earth Much Bigger Than Previously Thought in Science Daily, September 4, 2013.

[7] Comptes rendus de la 3 ^e CGPM (1901), 1901, 70, there still in cm / s². BIPM - Résolution de la 3e CGPM

[8] standard DIN 1305 mass, weight value, force, weight force, weight, load; Terms ( beuth.de ).

[9] Tate, 1969.

[10] State Office for Internal Administration (LAiV) Mecklenburg-Western Pomerania: Spatial reference - position, height and gravity fixed point fields ( Memento from January 14, 2014 in the Internet Archive )

[11] Tate, 1969.

[12] [1]

[nasa-13] A ^b David R. Williams: Planetary Fact Sheet - Metric. NASA, November 29, 2007, accessed on August 4, 2008 (English, including subpages).

[14] German-Swiss Mathematics Commission [DMK] and German-Swiss Physics Commission [DPK] (ed.): Formulas and tables . 11th edition. Orell Füssli Verlag, Zurich 2006, ISBN 978-3-280-02162-0 , p. 188 .