# Acceleration due to gravity

the Schwerebeschleunigung (acceleration due to gravity, acceleration due to gravity) indicates, bodies are subject to which acceleration with the free case in the gravitational field of the earth. At the earth's surface its average value g = 9.81 amounts to m/s ², varies however because of centrifugal energy, earth flattening and elevator profile regionally around some parts per thousand. Standard acceleration due to gravity is defined as 9.80665 m/s ². Generally the Schwerebeschleunigung depends on the mass of the heavenly body.

The formula v (t) = g · t indicates, which falling speed reaches v an article in the vacuum after the Fallzeit t. It increases theoretically in each second by the value g. Outside of the vacuum air resistance reduces acceleration depending upon body form and leads to a maximum falling speed (see also trajectory parabola).

## problems of the terminology

the designations Schwerebeschleunigung, surface and/or. Acceleration due to gravity, Earth's gravity acceleration and recently also local factor are usually synonymously used.

Against in the Internet “acceleration due to gravity” usually-used is objected that it can designate also (angle) the acceleration, to which the earth on its orbit is subject around the sun.

Misleading are also gravitational pull of the earth and in particular Earth's gravity. Under first understands one usually a Kraft (the so-called. The force of gravity), while “Earth's gravity” leaves open, which is exactly meant. Generally the terminology should consider whether it itself around a Kraft (F = m · g) or around an acceleration (g) acts, whereby the mass m constitutes the difference.

## derivation

the Schwerebeschleunigung determines Kraft F, with which a body m is tightened by a heavenly body:

[itex] F = m \ cdot g [/itex]

Equating the force of inertia F with the Newton's gravitation strength supplies acceleration due to gravity g:

[itex] g (r) = \ frac {GM} {r^2}< /math>

For the values of the earth:

• Earth mass: [itex] M=5 {,} 972 \ cdot 10^ {24} \, \ mathrm {kg}< /math>
• Erdradius: [itex] r=6 {.}371 \, \ mathrm {km}< /math> (spherically, resting)

and with

• the gravitation constant: [itex] G=6 {,} 674 \ cdot 10^ {- 11} \, {\ mathrm {m^3} \ more over \ mathrm {kg \, s^2}}< /math>

g = 9.82 results m/s ².

Another method is based on the measurement of the oscillation duration T of a thread pendulum with thread length L:

[itex] g = \ frac {4 \ pi^2 L} {T^2}< /math>

The so-called. Second pendulum has a length of for instance 1m, but is such a pendulum clock on the geographical width to be calibrated.

## units

SI - unit of the Schwerebeschleunigung is m/s ². The Millionste part of it is 1 µm/s ², which corresponds for instance to the average measuring accuracy.

In the old CGS - system is called the unit Gal (after Galileo Galilei or γ, which is often divided into gravimetry and applied geophysics in 1000 Milligal:

1 Gal = 1γ = 1 cm/s ² = 0.01 m/s ²
1 mGal = 10 -5 m/s ² = 10 µm/s ²

(see below). Geophycisists use γ however usually as symbols for theoretical weight (down as g N designates).

Sometimes acceleration due to gravity serves g also as unit. On the average for the earth then approached 1

g = for 9.81 applies m/s ² = 981 Gal = 981,000 mGal.

## place-dependentness of acceleration due to gravity

the earth no ball, but an ellipsoid is approximate there and besides rotates, depends acceleration due to gravity on the geographical width and additionally on the height over the sea level .

Standard acceleration due to gravity is defined as middle acceleration due to gravity g N with the value:

• 9.82306 m/s ² on that 45. Degree of latitude in sea level.
• 9.745 m/s ² at the equator.
• 9.832 m/s ² to Poland.

Per meter height (h) decreases g by approximately 3 µm/s ², as long as h is small even against the Erdradius and the area. The theoretical gradient of a completely smooth earth would be 3.086 µm/s ².

Further deviations are to be due to the structures of different density in the underground. From the exact measurement of acceleration due to gravity one can draw therefore conclusions on structures in the earth's crust as well as their changes.

A formula for dependence on the degree of latitude φ is the weight formula for the geodetic reference system 1980 (GRS 80) in sea level:

[itex] g (\ phi)

=g_ \ mathrm {A} \ frac {1+0 {,} 001 \, 931 \, 851 \, 353 \ sin^2 \ phi} {\ {1-0 {,} 006 \, 694 \, 380 \, 022 \, 90 \ sin^2 \ phi}} /math< sqrt>, whereby

[itex] g_ \ mathrm {A} =9 {,} 7803267715 \, \ mathrm \ frac {m} {s^2}< /math>

the Schwerebeschleunigung at the equator in sea level is.

A very good (recommended) weight formula for dependence on the degree of latitude φ is given through:

[itex] g (\ phi) =g_ \ mathrm {A} (1+c_1 \ sin^2 \ phi+c_2 \ sin^4 \ phi+c_3 \ sin^6 \ phi+c_4 \ sin^8 \ phi)

< /math>, also C 1 = 0.005 279 0414 C 2 = 0.000 023 2718 C 3 = 0.000 000 1262 C 4 = 0.000 000 0007.

This approximation is exact on for instance ±10 nm/s ². An often mentioned simpler formula with for instance ±10 µm/s ² accuracy is

[itex]

g (\ phi) =9 {,} 780327 \, \ mathrm \ frac {m} {s^2} \, (1+0 {,} 005 \, 3024 \ sin^2 \ phi+0 {,} 000 \, 005 \, 8 \ sin^2 (2 \ phi))[/itex].

A correction for elevator dependence reads:

[itex]

g (\ phi, h) =g (\ phi) \ cdot \ left (1-k_1 \ left (1-k_2 \ sin^2 {\ phi} \ right) {h \ more over \ mathrm {m}} +k_3 \ left ({h \ more over \ mathrm {m}} \ right) ^2 \ right) [/itex] with

[itex] k_1=3 {,} 15704 \ cdot10^ {- 7} \; \ quad k_2=0 {,} 00666031 \; \ quad k_3=7 {,} 37452 \ cdot10^ {- 14}

< /math> This correction is quite exact for aeronautical heights; for space (over approx. 100 kilometers) it diverges however.

A simple Näherungsformel in dependence the geographical width φ and height of h reads:

[itex] g=9 {,} 780327 \, (1+0 {,} 0053024 \, (\ sin \ varphi) ^2) \, \ frac {\ rm m} {\ rm s^2} - 0 {,} 00000308 \, h \, \ frac {\ rm 1} {\ rm s^2}< /math>

In Germany place-dependent acceleration due to gravity is held in the German main weight net 1996 (DHSN96 ). It is beside the Gauss Krüger - coordinate system for the place and the German main leveling net for the height the third size for the clear definition of a geodetic reference system. The German weight net supports itself by approx. 16,000 measuring points, the weight fixed points (SFP) .

## measuring accuracy

a modern Gravimeter is able acceleration due to gravity with an accuracy of 0,01 µm/s ² (0.001 mGal, approx. to measure 10 -9 g). One could registering thereby an elevator shift of less than a centimeter. Fluctuations of the air pressure cause changes in the same order of magnitude.

If one however gravity measurements to raw material - search or for the determination of the Geoids uses, one can be content with 0,1 mGal. Because the irregularities of the area can constitute and leave 30 mGal because of uncertain rocks close hardly more exactly than on 0,5 mGal or 5 µm/s ² to compute itself. With difference measurements (for instance to the determination of underground cavities) however 10 times measuring accuracy are meaningful.

The influence of the tidal forces is with 0,005 µm/s ², because of the sea with large moved Wassermassen with 0,1 µm/s ². Changes of the ground-water level can affect the measured values around 0,2 µm/s ².

From the observation of satellite orbits fluctuations of the Earth's gravity field in the order of magnitude of 200 show themselves µm/s ²; the most modern Gradiometrie can seize also still substantially smaller orbit/trajectory disturbances (see GRACE and GOCE).

## elevator dependence of acceleration due to gravity

acceptance on g with increasing height

at ground level decreases g by approximately 3.1 µm/s ² per meter. For larger heights the acceptance of g (r) with the Newton's gravitation law is measured (see diagram).

In low satellite heights of 300-400 km acceleration due to gravity decreases by 10 15%, in 5000 km (laser satellite Lageos) approx. 70 %. In large heights it does not become under any circumstances zero, otherwise high-flying satellites would fly away straight-lined. Their characteristic is the continued free case, which never impacts without air resistance on the earth's surface, because it follows a Keplerellipse.

## Schwerebeschleunigung of other heavenly bodies

the table compares the Schwerebeschleunigung of the earth with the heavenly bodies of our planet system in units of g:

Sky
body
relative
weight
acceleration
in m/s ²
earth 1.00 9.81
Jupiter 2.50 24.53
Mars 0.39 3.83
Merkur 0.39 3.83
moon 0.16 1.57
Neptun 1.20 11.77
Pluto 0.059 0.58
Saturn 1.10 10.79
sun 27.80 272.72
Uranus 0.89 8.73
Venus 0.89 8.73

to the comparison: Briefly humans survive 15 g, some minutes of long about 6 g, see G-Kraft.

## linguistic usage

under Earth's gravity acceleration or Schwerebeschleunigung one generally understands the acceleration, which a Gravimeter on the surface of a heavenly body measures (gravimetry). The centrifugal energy of a rotary planet is contained in it. However gravitation acceleration refers to the pure attraction of mass. In the handling linguistic usage the differences remain having often unconsidered and from the connection to be opened.