Gravitational acceleration

In physics, gravitational acceleration is the acceleration on an object caused by the force of gravitation. Neglecting friction such as air resistance, all small bodies accelerate in a gravitational field at the same rate relative to the center of mass.[1] This equality is true regardless of the masses or compositions of the bodies.

At different points on Earth, objects fall with an acceleration between 9.764 m/s2 and 9.834 m/s2[2] depending on altitude and latitude, with a conventional standard value of exactly 9.80665 m/s2 (approximately 32.17405 ft/s2). This does not take into account other effects, such as buoyancy or drag.

For point masses

Newton's law of universal gravitation states that there is a gravitational force between any two masses that is equal in magnitude for each mass, and is aligned to draw the two masses toward each other. The formula is:

${\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}\ }$

where ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$ are the two masses, ${\displaystyle G}$ is the gravitational constant, and ${\displaystyle r}$ is the distance between the two masses. The formula was derived for planetary motion where the distances between the planets and the Sun made it reasonable to consider the bodies to be point masses. (For a satellite in orbit, the 'distance' refers to the distance from the mass centers rather than, say, the altitude above a planet's surface.)

If one of the masses is much larger than the other, it is convenient to define a gravitational field around the larger mass as follows:[3]

${\displaystyle \mathbf {g} =-{GM \over r^{2}}\mathbf {\hat {r}} }$

where ${\displaystyle M}$ is the mass of the larger body, and ${\displaystyle \mathbf {\hat {r}} }$ is a unit vector directed from the large mass to the smaller mass. The negative sign indicates that the force is an attractive force.

In that way, the force acting upon the smaller mass can be calculated as:

${\displaystyle \mathbf {F} =m\mathbf {g} }$

where ${\displaystyle \mathbf {F} }$ is the force vector, ${\displaystyle m}$ is the smaller mass, and ${\displaystyle \mathbf {g} }$ is a vector pointing toward the larger body. Note that ${\displaystyle \mathbf {g} }$ has units of acceleration and is a vector function of location relative to the large body, independent of the magnitude (or even the presence) of the smaller mass.

This model represents the "far-field" gravitational acceleration associated with a massive body. When the dimensions of a body are not trivial compared to the distances of interest, the principle of superposition can be used for differential masses for an assumed density distribution throughout the body in order to get a more detailed model of the "near-field" gravitational acceleration. For satellites in orbit, the far-field model is sufficient for rough calculations of altitude versus period, but not for precision estimation of future location after multiple orbits.

The more detailed models include (among other things) the bulging at the equator for the Earth, and irregular mass concentrations (due to meteor impacts) for the Moon. The Gravity Recovery And Climate Experiment (GRACE) mission launched in 2002 consists of two probes, nicknamed "Tom" and "Jerry", in polar orbit around the Earth measuring differences in the distance between the two probes in order to more precisely determine the gravitational field around the Earth, and to track changes that occur over time. Similarly, the Gravity Recovery and Interior Laboratory (GRAIL) mission from 2011-2012 consisted of two probes ("Ebb" and "Flow") in polar orbit around the Moon to more precisely determine the gravitational field for future navigational purposes, and to infer information about the Moon's physical makeup.

Gravity model for Earth

The type of gravity model used for the Earth depends upon the degree of fidelity required for a given problem. For many problems such as aircraft simulation, it may be sufficient to consider gravity to be a constant, defined as:[4]

${\displaystyle g=}$ 9.80665 metres (32.1740 ft) per s2

based upon data from World Geodetic System 1984 (WGS-84), where ${\displaystyle g}$ is understood to be pointing 'down' in the local frame of reference.

If it is desirable to model an object's weight on Earth as a function of latitude, one could use the following ([4] p. 41):

${\displaystyle g=g_{45}-{\tfrac {1}{2}}(g_{\mathrm {poles} }-g_{\mathrm {equator} })\cos \left(2\,\varphi \cdot {\frac {\pi }{180}}\right)}$

where

• ${\displaystyle g_{\mathrm {poles} }}$ = 9.832 metres (32.26 ft) per s2
• ${\displaystyle g_{45}}$ = 9.806 metres (32.17 ft) per s2
• ${\displaystyle g_{\mathrm {equator} }}$ = 9.780 metres (32.09 ft) per s2
• ${\displaystyle \varphi }$ = latitude, between −90 and 90 degrees

Neither of these accounts for changes in gravity with changes in altitude, but the model with the cosine function does take into account the centrifugal relief that is produced by the rotation of the Earth. For the mass attraction effect by itself, the gravitational acceleration at the equator is about 0.18% less than that at the poles due to being located farther from the mass center. When the rotational component is included (as above), the gravity at the equator is about 0.53% less than that at the poles, with gravity at the poles being unaffected by the rotation. So the rotational component of change due to latitude (0.35%) is about twice as significant as the mass attraction change due to latitude (0.18%), but both reduce strength of gravity at the equator as compared to gravity at the poles.

Note that for satellites, orbits are decoupled from the rotation of the Earth so the orbital period is not necessarily one day, but also that errors can accumulate over multiple orbits so that accuracy is important. For such problems, the rotation of the Earth would be immaterial unless variations with longitude are modeled. Also, the variation in gravity with altitude becomes important, especially for highly elliptical orbits.

The Earth Gravitational Model 1996 (EGM96) contains 130,676 coefficients that refine the model of the Earth's gravitational field ([4] p. 40). The most significant correction term is about two orders of magnitude more significant than the next largest term ([4] p. 40). That coefficient is referred to as the ${\displaystyle J_{2}}$ term, and accounts for the flattening of the poles, or the oblateness, of the Earth. (A shape elongated on its axis of symmetry, like an American football, would be called prolate.) A gravitational potential function can be written for the change in potential energy for a unit mass that is brought from infinity into proximity to the Earth. Taking partial derivatives of that function with respect to a coordinate system will then resolve the directional components of the gravitational acceleration vector, as a function of location. The component due to the Earth's rotation can then be included, if appropriate, based on a sidereal day relative to the stars (≈366.24 days/year) rather than on a solar day (≈365.24 days/year). That component is perpendicular to the axis of rotation rather than to the surface of the Earth.

A similar model adjusted for the geometry and gravitational field for Mars can be found in publication NASA SP-8010.[5]

The barycentric gravitational acceleration at a point in space is given by:

${\displaystyle \mathbf {g} =-{GM \over r^{2}}\mathbf {\hat {r}} }$

where:

M is the mass of the attracting object, ${\displaystyle \scriptstyle \mathbf {\hat {r}} }$ is the unit vector from center-of-mass of the attracting object to the center-of-mass of the object being accelerated, r is the distance between the two objects, and G is the gravitational constant.

When this calculation is done for objects on the surface of the Earth, or aircraft that rotate with the Earth, one has to account for the fact that the Earth is rotating and the centrifugal acceleration has to be subtracted from this. For example, the equation above gives the acceleration at 9.820 m/s2, when GM = 3.986×1014 m3/s2, and R=6.371×106 m. The centripetal radius is r = R cos(φ), and the centripetal time unit is approximately (day / 2π), reduces this, for r = 5×106 metres, to 9.79379 m/s2, which is closer to the observed value.

General relativity

In Einstein's theory of general relativity, gravitation is an attribute of curved spacetime instead of being due to a force propagated between bodies. In Einstein's theory, masses distort spacetime in their vicinity, and other particles move in trajectories determined by the geometry of spacetime. The gravitational force is a fictitious force. There is no gravitational acceleration, in that the proper acceleration and hence four-acceleration of objects in free fall are zero. Rather than undergoing an acceleration, objects in free fall travel along straight lines (geodesics) on the curved spacetime.

References

1. ^ Gerald James Holton and Stephen G. Brush (2001). Physics, the human adventure: from Copernicus to Einstein and beyond (3rd ed.). Rutgers University Press. p. 113. ISBN 978-0-8135-2908-0.
2. ^ Hirt, C.; Claessens, S.; Fecher, T.; Kuhn, M.; Pail, R.; Rexer, M. (2013). "New ultrahigh-resolution picture of Earth's gravity field". Geophysical Research Letters. 40 (16): 4279–4283. Bibcode:2013GeoRL..40.4279H. doi:10.1002/grl.50838.
3. ^ Fredrick J. Bueche (1975). Introduction to Physics for Scientists and Engineers, 2nd Ed. USA: Von Hoffmann Press. ISBN 978-0-07-008836-8.
4. ^ a b c d Brian L. Stevens; Frank L. Lewis (2003). Aircraft Control And Simulation, 2nd Ed. Hoboken, New Jersey: John Wiley & Sons, Inc. ISBN 978-0-471-37145-8.
5. ^ Richard B. Noll; Michael B. McElroy (1974), Models of Mars' Atmosphere [1974], Greenbelt, Maryland: NASA Goddard Space Flight Center, SP-8010.
Acceleration due to gravity

Acceleration due to gravity may refer to

Gravitational acceleration, the acceleration caused by the gravitational attraction of massive bodies in general

Gravity of Earth, the acceleration caused by the combination of gravitational attraction and centrifugal force of the Earth

Standard gravity, or g, the standard value of gravitational acceleration at sea level on Earth

Archimedes number

In viscous fluid dynamics, the Archimedes number (Ar) (not to be confused with Archimedes' constant, π), named after the ancient Greek scientist Archimedes is used to determine the motion of fluids due to density differences. It is a dimensionless number defined as the ratio of external forces to internal viscous forces and has the form:

${\displaystyle \mathrm {Ar} ={\frac {gL^{3}\rho _{\ell }(\rho -\rho _{\ell })}{\mu ^{2}}}}$

where:

When analyzing potentially mixed heat convection of a liquid, the Archimedes number parametrizes the relative strength of free and forced convection. When Ar >> 1 natural convection dominates, i.e. less dense bodies rise and denser bodies sink, and when Ar << 1 forced convection dominates.

When the density difference is due to heat transfer (e.g. fluid being heated and causing a temperature difference between different parts of the fluid), then we may write

${\displaystyle {\frac {\rho -\rho _{0}}{\rho _{0}}}=\beta \left(T_{0}-T\right)}$

where:

Doing this gives the Grashof number, i.e. the Archimedes and Grashof numbers are equivalent but suited to describing situations where there is a material difference in density and heat transfer causes the density difference respectively. The Archimedes number is related to both the Richardson number and Reynolds number via

${\displaystyle \mathrm {Ar} =\mathrm {Ri} \,\mathrm {Re} ^{2}}$
Barometric formula

The barometric formula, sometimes called the exponential atmosphere or isothermal atmosphere, is a formula used to model how the pressure (or density) of the air changes with altitude. The pressure drops approximately by 11.3 Pa per meter in first 1000 meters above sea level.

Coriolis–Stokes force

In fluid dynamics, the Coriolis–Stokes force is a forcing of the mean flow in a rotating fluid due to interaction of the Coriolis effect and wave-induced Stokes drift. This force acts on water independently of the wind stress.

This force is named after Gaspard-Gustave Coriolis and George Gabriel Stokes, two nineteenth-century scientists. Important initial studies into the effects of the Earth's rotation on the wave motion – and the resulting forcing effects on the mean ocean circulation – were done by Ursell & Deacon (1950), Hasselmann (1970) and Pollard (1970).

The Coriolis–Stokes forcing on the mean circulation in an Eulerian reference frame was first given by Hasselmann (1970):

${\displaystyle \rho {\boldsymbol {f}}\times {\boldsymbol {u}}_{S},}$

to be added to the common Coriolis forcing ${\displaystyle \rho {\boldsymbol {f}}\times {\boldsymbol {u}}.}$ Here ${\displaystyle {\boldsymbol {u}}}$ is the mean flow velocity in an Eulerian reference frame and ${\displaystyle {\boldsymbol {u}}_{S}}$ is the Stokes drift velocity – provided both are horizontal velocities (perpendicular to ${\displaystyle {\hat {\boldsymbol {z}}}}$). Further ${\displaystyle \rho }$ is the fluid density, ${\displaystyle \times }$ is the cross product operator, ${\displaystyle {\boldsymbol {f}}=f{\hat {\boldsymbol {z}}}}$ where ${\displaystyle f=2\Omega \sin \phi }$ is the Coriolis parameter (with ${\displaystyle \Omega }$ the Earth's rotation angular speed and ${\displaystyle \sin \phi }$ the sine of the latitude) and ${\displaystyle {\hat {\boldsymbol {z}}}}$ is the unit vector in the vertical upward direction (opposing the Earth's gravity).

Since the Stokes drift velocity ${\displaystyle {\boldsymbol {u}}_{S}}$ is in the wave propagation direction, and ${\displaystyle {\boldsymbol {f}}}$ is in the vertical direction, the Coriolis–Stokes forcing is perpendicular to the wave propagation direction (i.e. in the direction parallel to the wave crests). In deep water the Stokes drift velocity is ${\displaystyle {\boldsymbol {u}}_{S}={\boldsymbol {c}}\,(ka)^{2}\exp(2kz)}$ with ${\displaystyle {\boldsymbol {c}}}$ the wave's phase velocity, ${\displaystyle k}$ the wavenumber, ${\displaystyle a}$ the wave amplitude and ${\displaystyle z}$ the vertical coordinate (positive in the upward direction opposing the gravitational acceleration).

Equatorial bulge

An equatorial bulge is a difference between the equatorial and polar diameters of a planet, due to the centrifugal force exerted by the rotation about the body's axis. A rotating body tends to form an oblate spheroid rather than a sphere.

Gal (unit)

The gal (symbol: Gal), sometimes called galileo after Galileo Galilei, is a unit of acceleration used extensively in the science of gravimetry. The gal is defined as 1 centimeter per second squared (1 cm/s2). The milligal (mGal) and microgal (µGal) refer respectively to one thousandth and one millionth of a gal.

The gal is not part of the International System of Units (known by its French-language initials "SI"). In 1978 the CIPM decided that it was permissible to use the gal "with the SI until the CIPM considers that [its] use is no longer necessary". However, use of the gal is deprecated by ISO 80000-3:2006.

The gal is a derived unit, defined in terms of the centimeter–gram–second (CGS) base unit of length, the centimeter, and the second, which is the base unit of time in both the CGS and the modern SI system. In SI base units, 1 Gal is equal to 0.01 m/s2.

The acceleration due to Earth’s gravity (see standard gravity) at its surface is 976 to 983 Gal, the variation being due mainly to differences in latitude and elevation. Mountains and masses of lesser density within the Earth's crust typically cause variations in gravitational acceleration of tens to hundreds of milligals (mGal). The gravity gradient (variation with height) above Earth's surface is about 3.1 µGal per centimeter of height (3.1×10−6 s−2), resulting in a maximal difference of about 2 Gal (0.02 m/s2) from the top of Mount Everest to sea level.Unless it is being used at the beginning of a sentence or in paragraph or section titles, the unit name gal is properly spelled with a lowercase g. As with the torr and its symbol, the unit name (gal) and its symbol (Gal) are spelled identically except that the latter is capitalized.

Galilei number

In fluid dynamics, the Galilei number (Ga), sometimes also referred to as Galileo number (see discussion), is a dimensionless number named after Italian scientist Galileo Galilei (1564-1642).

It may be regarded as proportional to gravity forces divided by viscous forces. The Galilei number is used in viscous flow and thermal expansion calculations, for example to describe fluid film flow over walls. These flows apply to condensers or chemical columns.

${\displaystyle \mathrm {Ga} =\mathrm {Re} ^{2}\,\mathrm {Ri} ={\frac {g\,L^{3}}{\nu ^{2}}}}$
Gravimeter

A gravimeter is an instrument used to measure gravitational acceleration. Every mass has an associated gravitational potential. The gradient of this potential is an acceleration. A gravimeter measures this gravitational acceleration.

The first gravimeters were vertical accelerometers, specialized for measuring the constant downward acceleration of gravity on the earth's surface. The earth's vertical gravity varies from place to place over the surface of the Earth by about +/- 0.5%. It varies by about +/- 1000 nm/s^2 ("nanometers per second squared") at any location because of the changing positions of the sun and moon relative to the earth.

The change from calling a device an "accelerometer" to calling it a "gravimeter" occurs at approximately the point where it has to make corrections for earth tides.

Though similar in design to other accelerometers, gravimeters are typically designed to be much more sensitive. Their first uses were to measure the changes in gravity from the varying densities and distribution of masses inside the earth, from temporal "tidal" variations in the shape and distribution of mass in the oceans, atmosphere and earth.

Gravimeters can detect vibrations and gravity changes from human activities. Depending on the interests of the researcher or operator, this might be counteracted by integral vibration isolation and signal processing.

The resolution of the gravimeters can be increased by averaging samples over longer periods. Fundamental characteristic of gravimeters are the accuracy of a single measurement (a single "sample"), and the sampling rate (samples per second).

${\displaystyle {\text{Resolution}}={{\text{SingleMeasurementResolution}} \over {\sqrt {\text{NumberOfSamples}}}}}$

for example:

${\displaystyle {\text{Resolution per minute}}={{\text{Resolution per second}} \over {\sqrt {60}}}}$

Gravimeters display their measurements in units of gals (cm/s2), nanometers per second squared, and parts per million, parts per billion, or parts per trillion of the average vertical acceleration with respect to the earth. Some newer units are pm/s2 (picometers per second squared), fm/s2 (femto), am/s2 (atto) for very sensitive instruments.

Gravimeters are used for petroleum and mineral prospecting, seismology, geodesy, geophysical surveys and other geophysical research, and for metrology. Their fundamental purpose is to map the gravity field in space and time.

Most current work is earth-based, with a few satellites around earth, but gravimeters are also applicable to the moon, sun, planets, asteroids, stars, galaxies and other bodies. Gravitational wave experiments monitor the changes with time in the gravitational potential itself, rather than the gradient of the potential which the gravimeter is tracking. This distinction is somewhat arbitrary. The subsystems of the gravitational radiation experiments are very sensitive to changes in the gradient of the potential. The local gravity signals on earth that interfere with gravitational wave experiments are disparagingly referred to as "Newtonian noise", since Newtonian gravity calculations are sufficient to characterize many of the local (earth-based) signals.

The term "absolute gravimeter" has most often been used to label gravimeters which report the local vertical acceleration due to the earth. "Relative gravimeter" usually refer to differential comparisons of gravity from one place to another. They are designed to subtract the average vertical gravity automatically. They can be calibrated at a location where the gravity is known accurately, and then transported to the location where the gravity is to be measured. Or they can calibrated in absolute units at their operating location.

There are many methods for displaying acceleration fields, also called "gravity fields". This includes traditional 2D maps, but increasingly 3D video. Since gravity and acceleration are the same, "acceleration field" might be preferable, since "gravity" is an oft misused prefix.

Gravitation of the Moon

The acceleration due to gravity on the surface of the Moon is about 1.625 m/s2, about 16.6% that on Earth's surface or 0.166 ɡ. Over the entire surface, the variation in gravitational acceleration is about 0.0253 m/s2 (1.6% of the acceleration due to gravity). Because weight is directly dependent upon gravitational acceleration, things on the Moon will weigh only 16.6 % (≈ 1/6) of what they weigh on the Earth.

Gravity gradiometry

Gravity gradiometry is the study and measurement of variations in the acceleration due to gravity. The gravity gradient is the spatial rate of change of gravitational acceleration.

Gravity gradiometry is used by oil and mineral prospectors to measure the density of the subsurface, effectively by measuring the rate of change of gravitational acceleration (or jerk) due to underlying rock properties. From this information it is possible to build a picture of subsurface anomalies which can then be used to more accurately target oil, gas and mineral deposits. It is also used to image water column density, when locating submerged objects, or determining water depth (bathymetry). Physical scientists use gravimeters to determine the exact size and shape of the earth and they contribute to the gravity compensations applied to inertial navigation systems.

Gravity of Earth

The gravity of Earth, denoted by g, is the net acceleration that is imparted to objects due to the combined effect of gravitation (from distribution of mass within Earth) and the centrifugal force (from the Earth's rotation).In SI units this acceleration is measured in metres per second squared (in symbols, m/s2 or m·s−2) or equivalently in newtons per kilogram (N/kg or N·kg−1). Near Earth's surface, gravitational acceleration is approximately 9.8 m/s2, which means that, ignoring the effects of air resistance, the speed of an object falling freely will increase by about 9.8 metres per second every second. This quantity is sometimes referred to informally as little g (in contrast, the gravitational constant G is referred to as big G).

The precise strength of Earth's gravity varies depending on location. The nominal "average" value at Earth's surface, known as standard gravity is, by definition, 9.80665 m/s2. This quantity is denoted variously as gn, ge (though this sometimes means the normal equatorial value on Earth, 9.78033 m/s2), g0, gee, or simply g (which is also used for the variable local value).

The weight of an object on Earth's surface is the downwards force on that object, given by Newton's second law of motion, or F = ma (force = mass × acceleration). Gravitational acceleration contributes to the total gravity acceleration, but other factors, such as the rotation of Earth, also contribute, and, therefore, affect the weight of the object.

Gravity does not normally include the gravitational pull of the Moon and Sun, which are accounted for in terms of tidal effects.

It is a vector (physics) quantity, whose direction coincides with a plumb bob.

Kilopondmetre

The Kilopondmetre is an obsolete unit of torque and energy in the gravitational metric system. It is abbreviated kp·m or m·kp, older publications often use m­kg and kg­m as well.

Torque is a product of the length of a lever and the force applied to the lever. One kilopond is the force applied to one kilogram due to gravitational acceleration; this force is exactly 9.80665 N.

This means 1 kp·m = 9.80665 kg·m2/s2 = 9.80665 N·m.

Mathematical coincidence

A mathematical coincidence is said to occur when two expressions with no direct relationship show a near-equality which has no apparent theoretical explanation.

For example, there is a near-equality close to the round number 1000 between powers of 2 and powers of 10:

Some mathematical coincidences are used in engineering when one expression is taken as an approximation of another.

Microsome

In cell biology, microsomes are heterogenous vesicle-like artifacts (~20-200 nm diameter) re-formed from pieces of the endoplasmic reticulum (ER) when eukaryotic cells are broken-up in the laboratory; microsomes are not present in healthy, living cells.Microsomes can be concentrated and separated from other cellular debris by differential centrifugation. Unbroken cells, nuclei, and mitochondria sediment out at 10,000 g, whereas soluble enzymes and fragmented ER, which contains cytochrome P450 (CYP), remain in solution (g is the Earth's gravitational acceleration). At 100,000 g, achieved by faster centrifuge rotation, ER sediments out of solution as a pellet but the soluble enzymes remain in the supernatant. In this way, cytochrome P450 in microsomes is concentrated and isolated. Microsomes have a reddish-brown color, due to the presence of the heme. Because of the need for a multi-part protein-system, microsomes are necessary to analyze the metabolic activity of CYPs. These CYPs are highly abundant in livers of rats, mice and humans, but present in all other organs and organisms as well.

To get microsomes containing a specific CYP or for high amounts of active enzyme, microsomes are prepared from Sf9 insect cells or in yeast via heterologous expression. Alternatively expression in Escherichia coli of whole or truncated proteins can also be performed. Therefore, microsomes are a valuable tool for investigating the metabolism of compounds (enzyme inhibition, clearance and metabolite identification) and for examining drug-drug interactions by in vitro-research. Researchers often select microsome lots based on the enzyme activity level of specific CYPs. Some lots are available to study specific populations (example: lung microsomes from smokers or non-smokers) or divided into classifications to meet target CYP activity levels for inhibition and metabolism studies.

Researchers use microsomes to mimic the activity of the endoplasmic reticulum in a test tube and conduct experiments that require protein synthesis on a membrane; they provide a way for scientists to figure out how proteins are being made on the ER in a cell by reconstituting the process in a test tube.

Mu2 Gruis

Mu2 Gruis, Latinized from μ2 Gruis, is a yellow-hued star or star system in the southern constellation of Grus. It is a suspected astrometric binary, showing a variation in proper motion due to gravitational acceleration. Mu2 Gruis is visible to the naked eye with an apparent visual magnitude of 5.10. The distance to this system, as determined using an annual parallax shift of 12.29 mas as seen from the Earth, is around 270 light years. The primary component is an evolved giant star with a stellar classification of G8 III. It is a periodic variable star, showing a change in brightness with an amplitude of 0.004 magnitude at the rate of 7.50983 times per day.

Overburden pressure

Overburden pressure, also called lithostatic pressure, confining pressure or vertical stress, is the pressure or stress imposed on a layer of soil or rock by the weight of overlying material.

The Oxford Dictionary of Earth Sciences describes 'confining pressure' as "the combined hydrostatic stress and lithostatic stress; i.e. the total weight of the interstitial pore water and rock above a specified depth." Confining pressure might influence ductile behavior of rocks as well. Ductile behavior is enhanced where high confining pressures are combined with high temperatures and low rates of strain, conditions characteristic of deeper crustal levels.

The overburden pressure at a depth z is given by

${\displaystyle p(z)=p_{0}+g\int _{0}^{z}\rho (z)\,dz}$

where ρ(z) is the density of the overlying rock at depth z and g is the acceleration due to gravity. p0 is the datum pressure, the pressure at the surface.

In deriving the above equation it is assumed that gravitational acceleration g is a constant over z, since it is placed outside the integral. In reality, g is a (non-constant) function of z and should appear inside the integral. But since g varies little over depths which are a very small fraction of the Earth's radius, it is placed outside the integral in practice for most near-surface applications which require an assessment of lithostatic pressure. In deep-earth geophysics/geodynamics, gravitational acceleration varies significantly over depth and g may not be assumed to be constant.

This should be compared with the equivalent concept of hydrostatic pressure in hydrodynamics.

Slab pull

Slab pull is that part of the motion of a tectonic plate that is caused by its subduction. Plate motion is partly driven by the weight of cold, dense plates sinking into the mantle at oceanic trenches. This force and slab suction account for almost all of the force driving plate tectonics. The ridge push at rifts contributes only 5 to 10%.

Carlson et al. (1983) in Lallemandet al. (2005) defines the slab pull force as:

${\displaystyle F_{sp}=K\times \Delta \rho \times L\times {\sqrt {A}}}$

Where:

K is 4.2g (gravitational acceleration = 9.81 m/s2) according to McNutt (1984);
Δρ = 80 kg/m3 is the mean density difference between the slab and the surrounding asthenosphere;
L is the slab length calculated only for the part above 670 km (the upper/lower mantle boundary);
A is the slab age in Ma at the trench.

The slab pull force manifests itself between two extreme forms:

Between these two examples there is the evolution of the Farallon plate: from the huge slab width with the Nevada, the Sevier and Laramide orogenies; the Mid-Tertiary ignimbrite flare-up and later left as Juan de Fuca and Cocos plates, the Basin and Range Province under extension, with slab break off, smaller slab width, more edges and mantle return flow.

Some early models of plate tectonics envisioned the plates riding on top of convection cells like conveyor belts. However, most scientists working today believe that the asthenosphere does not directly cause motion by the friction of such basal forces. The North American Plate is nowhere being subducted, yet it is in motion. Likewise the African, Eurasian and Antarctic Plates. The subducting slabs around the Pacific Ring of Fire cool down the Earth and its Core-mantle boundary, around the African Plate the upwelling mantle plumes from the Core-mantle boundary produce rifting. The overall driving force for plate motion and its energy source remain subjects of ongoing research.

Standard gravity

The standard acceleration due to gravity (or standard acceleration of free fall), sometimes abbreviated as standard gravity, usually denoted by ɡ0 or ɡn, is the nominal gravitational acceleration of an object in a vacuum near the surface of the Earth. It is defined by standard as 9.80665 m/s2 (about 32.17405 ft/s2). This value was established by the 3rd CGPM (1901, CR 70) and used to define the standard weight of an object as the product of its mass and this nominal acceleration. The acceleration of a body near the surface of the Earth is due to the combined effects of gravity and centrifugal acceleration from the rotation of the Earth (but which is small enough to be neglected for most purposes); the total (the apparent gravity) is about 0.5% greater at the poles than at the Equator.Although the symbol ɡ is sometimes used for standard gravity, ɡ (without a suffix) can also mean the local acceleration due to local gravity and centrifugal acceleration, which varies depending on one's position on Earth (see Earth's gravity). The symbol ɡ should not be confused with G, the gravitational constant, or g, the symbol for gram. The ɡ is also used as a unit for any form of acceleration, with the value defined as above; see g-force.

The value of ɡ0 defined above is a nominal midrange value on Earth, originally based on the acceleration of a body in free fall at sea level at a geodetic latitude of 45°. Although the actual acceleration of free fall on Earth varies according to location, the above standard figure is always used for metrological purposes. In particular, it gives the conversion factor between newton and kilogram-force, two units of force.

Surface gravity

The surface gravity, g, of an astronomical or other object is the gravitational acceleration experienced at its surface at the equator, including the effects of rotation. The surface gravity may be thought of as the acceleration due to gravity experienced by a hypothetical test particle which is very close to the object's surface and which, in order not to disturb the system, has negligible mass.

Surface gravity is measured in units of acceleration, which, in the SI system, are meters per second squared. It may also be expressed as a multiple of the Earth's standard surface gravity, g = 9.80665 m/s². In astrophysics, the surface gravity may be expressed as log g, which is obtained by first expressing the gravity in cgs units, where the unit of acceleration is centimeters per second squared, and then taking the base-10 logarithm. Therefore, the surface gravity of Earth could be expressed in cgs units as 980.665 cm/s², with a base-10 logarithm (log g) of 2.992.

The surface gravity of a white dwarf is very high, and of a neutron star even higher. The neutron star's compactness gives it a surface gravity of up to 7×1012 m/s² with typical values of order 1012 m/s² (that is more than 1011 times that of Earth). One measure of such immense gravity is that neutron stars have an escape velocity of around 100,000 km/s, about a third of the speed of light. For black holes, the surface gravity must be calculated relativistically.

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