6 Visualizing Gravity

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Chapter 6

Visualizing Gravity: the Gravitational FieldMichael Fowler 2/14/06

Introduction

Lets begin with thedefinition of gravitational field:

The gravitational field at any point P in space is defined as the

gravitational force felt by a tiny unit mass placed at P.

So, to visualize the gravitational field, in this room or on a bigger

scale such as the whole Solar System, imagine drawing a vector

representing the gravitational force on a one kilogram mass at many

different points in space, and seeing how the pattern of these vectors

varies from one place to another (in the room, of course, they wontvary much!). We say a tiny unit mass because we dont want the

gravitational field from the test mass itself to disturb the system. This

is clearly not a problem in discussing planetary and solar gravity.

To build an intuition of what various gravitational fields look like,

well examine a sequence of progressively more interesting systems,beginning with a simple point mass and working up to a hollow

spherical shell, this last being what we need to understand the Earthsown gravitational field, both outside and inside the Earth.

Field from a Single Point Mass

This is of course simple: we know this field has strength GM/r2, and

points towards the massthe direction of the attraction. Lets draw it

anyway, or, at least, lets draw in a few vectors showing its strength at

various points:


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This is a rather inadequate representation: theres a lot of blank space,and, besides, the field attracts in three dimensions, there should be

vectors pointing at the mass in the air above (and below) the

paper. But the picture does convey the general idea.

A different way to represent a field is to draw field lines, curvessuch that at every point along the curves length, its direction is the

direction of the field at that point. Of course, for our single mass, the

field lines add little

insight:


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The arrowheads indicate the direction of the force, which points the

same way all along the field line. A shortcoming of the field lines

picture is that although it can give a good general idea of the field,

there is no precise indication of the fieldsstrength at any

point. However, as is evident in the diagram above, there is a clue:

where the lines are closer together, the force is stronger. Obviously,

we could put in a spoke-like field line anywhere, but if we want to

give an indication of field strength, wed have to have additional lines

equally spaced around the mass.

Gravitational Field for Two Masses

The next simplest case is two equal masses. Let us place themsymmetrically above and below thex-axis:

RecallNewtons Universal Law of Gravitation states that any two

masses have a mutual gravitational attraction . A point

mass m = 1 at P will therefore feel gravitational attraction towards

both massesM, and a total gravitational field equal to the vector sum

of these two forces, illustrated by the red arrow in the figure.

The Principle of Superposition

The fact that the total gravitational field is just given by adding the

two vectors together is called the Principle of Superposition. This

may sound really obvious, but in fact it isnt true for every forcefound in physics: the strong forces between elementary particles

dont obey this principle, neither do the strong gravitational fields


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near black holes. But just adding the forces as vectors works fine for

gravity almost everywhere away from black holes, and, as you will

find later, for electric and magnetic fields too. Finally, superposition

works for any number of masses, not just two: the total gravitational

field is the vector sum of the gravitational fields from all the

individual masses. Newton used this to prove that the gravitational

field outside a solid sphere was the same as if all the mass were at the

center by imagining the solid sphere to be composed of many small

massesin effect, doing an integral, as we shall discuss in detail

later. He also invoked superposition in calculating the orbit of the

Moon precisely, taking into account gravity from both the Earth and

the Sun.

Exercise: For the two mass case above, sketch the gravitational field

vector at some other points: look first on the x-axis, then away from

it. What do thefield lines look like for this two mass case? Sketch

them in the neighborhood of the origin.

Field Strength at a Point Equidistant from the Two Masses

It is not difficult to find an exact expression for the gravitational field

strength from the two equal masses at an equidistant point P.

Choose the x,y axes so that the masses lie on they-axis at (0, a) and

(0,-a).

By symmetry, the field at P must point along thex-axis, so all we

have to do is compute the strength of thex-component of the

gravitational force from one mass, and double it.


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If the distance from the point P to one of the masses is s, the

gravitational force towards that mass has strength . This force

has a component along thex-axis equal to , where is theangle between the line from P to the mass and thex-axis, so the total

gravitational force on a small unit mass at P is directed

along thex-axis.

From the diagram, , so the force on a unit mass at P from

the two massesMis

in thex-direction. Note that the force is exactly zero at the origin, and

everywhere else it points towards the origin.

Gravitational Field from a Ring of Mass

Now, as long as we lookonly on thex-axis, this identical formula

works for a ring of mass 2Min they,z plane! Its just a three-dimensional version of the argument above, and can be visualized by

rotating the two-mass diagram above around thex-axis, to give a ring

perpendicular to the paper, or by imagining the ring as made up of

many beads, and taking the beads in pairs opposite each other.

Bottom line: the field from a ring oftotal massM, radius a, at a

point P on the axis of the ring distancex from the center of the ring is

.


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*Field Outside a Massive Spherical Shell

This is an optional section: you can safely skip to the result on the last

line. In fact, you will learn an easy way to derive this result using

Gausss Theorem when you do Electricity and Magnetism. I just put

this section in so you can see that this resultcan be derived by the

straightforward, but quite challenging, method of adding the

individual gravitational attractions from all the bits making up the

spherical shell.

What about the gravitational field from a hollow spherical shell of

matter? Such a shell can be envisioned as a stack of rings.

To find the gravitational field at the point P, we just add the

contributions from all the rings in the stack.

In other words, we divide the spherical shell into narrow zones:imagine chopping an orange into circular slices by parallel cuts,


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perpendicular to the axisbut of course our shell is just the skin of

the orange! One such slice gives a ring of skin, corresponding to the

surface area between two latitudes, the two parallel lines in the

diagram above. Notice from the diagram that this ring of skin willhave radius , therefore circumference and breadth ,

where were taking to be very small. This means that the area of

the ring of skin is

.

So, if the shell has mass per unit area, this ring has

mass , and the gravitational force at P from this ring will

be

.

Now, to find the total gravitational force at P from the entire shell we

have to add the contributions from each of these rings which, taken

together, make up the shell. In other words, we have to integratetheabove expression in

So the gravitational field is:

.

In fact, this is quite a tricky integral: ,x and s are all varying! Itturns out to be is easiest done by switching variables from to s.

Label the distance from P to the center of the sphere by r. Then,

from the diagram, , and a, rare constants,

so ,

and


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Now , and from the diagram ,

so ,

and, writing ,

The derivation was rather lengthy, but the answer is simple:

The gravitational field outside a uniform spherical shell

is GM/r2

towards the center.

And, theres a bonus: for the ring, we only found the fieldalong the

axis, but for the spherical shell, once weve found it in onedirection, the whole problem is solvedfor the spherical shell, the

field must be the same in all directions.

Field Outside a Solid Sphere

Once we know the gravitational field outside a shell of matter is the

same as if all the mass were at a point at the center, its easy to find

the field outside a solid sphere: thats just a nesting set of shells, like

spherical Russian dolls. Adding them up,

The gravitational field outside a uniform sphere is GM/r2

towards

the center.

Theres an added bonus: since we found this result be adding uniform

spherical shells, it is still true if the shells have different densities,

provided the density of each shell is the same in all directions. The

inner shells could be much denser than the outer onesas in fact is

the case for the Earth.


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Field Insidea Spherical Shell

This turns out to be surprisingly simple! We imagine the shell to be

very thin, with a mass density kg per square meter of surface.

Begin by drawing a two-way cone radiating out from the point P, so

that it includes two small areas of the shell on opposite sides: these

two areas will exert gravitational attraction on a mass at P in opposite

directions. It turns out that they exactly cancel.

This is because the ratio of the areasA1 andA2 at

distances r1 and r2 are given by : since the cones have the

same angle, if one cone has twice the height of the other, its base will

have twice the diameter, and thereforefour times the area. Since the

masses of the bits of the shell are proportional to the areas, the ratio of

the masses of the cone bases is also . But the gravitational

attraction at P from these masses goes as , and

that r2 term cancels the one in the areas, so the two opposite areashave equal and opposite gravitational forces at P.

In fact, the gravitational pull from every small part of the shell is

balanced by a part on the opposite sideyou just have to construct a

lot of cones going through P to see this. (There is one slightly tricky

pointthe line from Pto the spheres surface will in general cut thesurface at an angle. However, it will cut the opposite bit of sphere at

the same angle, because any line passing through a sphere hits the two


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surfaces at the same angle, so the effects balance, and the base areas

of the two opposite small cones are still in the ratio of the squares of

the distances r1, r2.)

Field Inside a Sphere: How Does gVary on Going Down aMine?

This is a practical application of the results for shells. On going down

a mine, if we imagine the Earth to be made up of shells, we will be

inside a shell of thickness equal to the depth of the mine, so will

feel nonet gravity from that part of the Earth. However, we will

be closerto the remaining shells, so the force from them will be

intensified.

suppose we descend from the Earths radiusrE to a point

distance rfrom the center of the Earth. What fraction of the Earths

mass is still attracting us towards the center? Lets make life simple

for now and assume the Earths density isuniform, call it kg per

cubic meter.

Then the fraction of the Earths mass that is still attracting us (because

its closer to the center than we areinside the red sphere in the

diagram) is .

The gravitational attraction from this mass at the bottom of the mine,

distance rfrom the center of the Earth, is proportional to

mass/r2

. We have just seen that the mass is itself proportional to r3

,so the actual gravitational force felt must be proportional to .


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That is to say, the gravitational force on going down inside the Earth

is linearly proportional to distance from the center. Since we already

know that the gravitational force on a mass mat the Earths

surface is mg, it follows immediately that in the mine the

gravitational force must be

.

So theres no force at all at the center of the Earthas we would

expect, the masses are attracting equally in all directions.

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22

The Earths Thermal StructureAs the Earths thermal Regime is governed by both theavailable heat and the environment in which it is stored, thischapter begins with a brief review of the Earths heat budget

with respect to the contributions of various external and internalsources and sinks. This is followed by a discussion of thethermal regime of the Earths crust and the associated physicalstorage and transport properties specific heat capacity, thermalconductivity, and diffusivity .Our information on the internalstructure of the Earth and the variation of its physical properties(pressure, temperature, density, and seismic velocities) andchemical composition are derived from seismology, i.e. the

interpretation of travel time curves of earthquakes whichpassed through the Earth. The variation with depth of theobserved seismic velocities and elastic constants combinedwith Maxwells four thermodynamic relations between pressureP, volume V, entropy S (S=Q/T; Q: heat), and temperature Tyield the predominantly radial structure of the Earth. FromMaxwells relation (T/P)S=(V/S)Pone obtains an expression for the adiabatic temperature

gradient in terms of temperature, the volume coefficient of
http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/DiscoveringGravity.htmhttp://galileo.phys.virginia.edu/classes/152.mf1i.spring02/DiscoveringGravity.htmhttp://galileo.phys.virginia.edu/classes/152.mf1i.spring02/GravityIndex.htmhttp://galileo.phys.virginia.edu/classes/152.mf1i.spring02/GravityIndex.htmhttp://galileo.phys.virginia.edu/classes/152.mf1i.spring02/GravPotEnergy.htmhttp://galileo.phys.virginia.edu/classes/152.mf1i.spring02/GravPotEnergy.htmhttp://galileo.phys.virginia.edu/classes/152.mf1i.spring02/GravField.pdfhttp://galileo.phys.virginia.edu/classes/152.mf1i.spring02/GravField.pdfhttp://galileo.phys.virginia.edu/classes/152.mf1i.spring02/GravField.pdfhttp://galileo.phys.virginia.edu/classes/152.mf1i.spring02/GravPotEnergy.htmhttp://galileo.phys.virginia.edu/classes/152.mf1i.spring02/GravityIndex.htmhttp://galileo.phys.virginia.edu/classes/152.mf1i.spring02/DiscoveringGravity.htm


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thermal expansion =(V/T)P/V, and the isobaric specific heatcapacity cP:( )SPT gTz c= ,(1)where g is gravity andsubscripts P and S refer to isobaric and adiabatic conditions,respectively, i.e. constant pressure and constant entropy.Assuming lower mantle values (at about 1500 km depth) ofT=2400 K, g=9.9 m s-2, cP=1200 J kg-1K-1, and=14 K-1, yields an adiabatic temperature gradient of about 0.3 K km-1; the corresponding values for the outer core (at about 3500km depth) of T=4000 K, g=10.1 m s

-2, cP=700 J kg-1K-1, and=14 K-1yield an adiabatic temperature gradient of about 0.8 K km

-1(see e.g. Clauser, 2006).Approximate estimates for theadiabatic temperature inside the Earth can be obtained withtheaid of the dimensionless thermodynamic Grneisenparameter

=KS/ (cP), where KS is the adiabatic incompressibility orbulk modulus and is density:00T d, o r : T T T= =37300 C0 1 0 0 0 2 0 0 0 3 00 0 4 0 0 0 5 0 0 0

T (K)600050004000300020001000Depth (km)100020003000400050006000R ad i us ( k m)0 1 0 0 0 2 0 0 0 3 00 0 4 0 0 0 5 0 0 0T (C)


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L: Lithosphere (0-80 km)A: Asthenosphere (80-220km)TZ: Transition Zone (220-670 km)D'': D'' layer (2741-2891 km)400 km: Phase transition olivine-spinel670 km:Phase transition spinel-perovskiteD''InnerCoreOuterCoreLowerMantleTZAS o l i d u sL63715150289167080T e m p e r a t u r eUpperMantle

8000 C220

Fig. 1. Variation of estimated temperature and melting point inthe Earth with depth (Clauser, 2006, redrawn afterLowrie,1997); Data: Stacey (1992) and Dziewonski & Anderson(1981). Temperature is poorly constrained in thedeepersections, indicated by large error bars (Brown, 1993).From a known temperature T=0

and density=0at a given depth, eq. (2) allows computing the adiabatic

temperature from the density profile in a region where theGrneisen parameter is known (for details: see e.g. Clauser,2006).Fortunately, the Grneisen parameter varies only mildlybetween 0.5 and 1.5 within largeregions of the Earths interior.The currently accepted estimate of the temperature profile ischaracterized by steep gradients in the lithosphere,

asthenosphere and in the lower mantlesD layer (immediatelyabove the core-mantle boundary). Neglecting large lateralvariations in the crust and lithosphere it indicates, on average,temperatures of less than 1000 K in the lithosphere, close to3750 K at the core-mantle boundary, and around 5100 K at thecentre of the Earth (Fig. 1) (Stacey, 1992; Lowrie,1997).However, there are large uncertainties, particularly in themantle and core (Brown, 1993;Beardsmore & Cull, 2001),

indicating ranges for conceivable minimum and maximum


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temperatures of 3000 C4500 C at the core-mantle boundary,4400 C7300 C at the transition between outer and innercore, and a maximum temperature at the centre of the Earth ofless than 8000 C (Fig. 1). From another one of Maxwellsthermodynamic relations,(S/P)T=-(V/T)P, one may derive the fractional variation of the melting pointtemperature Tmpwith depth within the Earth:

GeoidFrom Wikipedia, the free encyclopedia

Geodesy

Fundamentals

Geodesy Geodynamics

GeomaticsCartography

Concepts

Datum Distance Geoid

Figure of the Earth

Geodetic system

Geog. coord. system

Hor. pos. representation

Map projection

Reference ellipsoid

Satellite geodesy

Spatial reference system

Technologies
http://en.wikipedia.org/wiki/Category:Geodesyhttp://en.wikipedia.org/wiki/Category:Geodesyhttp://en.wikipedia.org/wiki/Geodesyhttp://en.wikipedia.org/wiki/Geodesyhttp://en.wikipedia.org/wiki/Geodynamicshttp://en.wikipedia.org/wiki/Geodynamicshttp://en.wikipedia.org/wiki/Geodynamicshttp://en.wikipedia.org/wiki/Geomaticshttp://en.wikipedia.org/wiki/Geomaticshttp://en.wikipedia.org/wiki/Cartographyhttp://en.wikipedia.org/wiki/Cartographyhttp://en.wikipedia.org/wiki/Cartographyhttp://en.wikipedia.org/wiki/Datum_(geodesy)http://en.wikipedia.org/wiki/Datum_(geodesy)http://en.wikipedia.org/wiki/Geographical_distancehttp://en.wikipedia.org/wiki/Geographical_distancehttp://en.wikipedia.org/wiki/Geographical_distancehttp://en.wikipedia.org/wiki/Figure_of_the_Earthhttp://en.wikipedia.org/wiki/Figure_of_the_Earthhttp://en.wikipedia.org/wiki/Geodetic_systemhttp://en.wikipedia.org/wiki/Geodetic_systemhttp://en.wikipedia.org/wiki/Geographic_coordinate_systemhttp://en.wikipedia.org/wiki/Geographic_coordinate_systemhttp://en.wikipedia.org/wiki/Horizontal_position_representationhttp://en.wikipedia.org/wiki/Horizontal_position_representationhttp://en.wikipedia.org/wiki/Map_projectionhttp://en.wikipedia.org/wiki/Map_projectionhttp://en.wikipedia.org/wiki/Reference_ellipsoidhttp://en.wikipedia.org/wiki/Reference_ellipsoidhttp://en.wikipedia.org/wiki/Satellite_geodesyhttp://en.wikipedia.org/wiki/Satellite_geodesyhttp://en.wikipedia.org/wiki/Spatial_reference_systemhttp://en.wikipedia.org/wiki/Spatial_reference_systemhttp://en.wikipedia.org/wiki/Category:Geodesyhttp://en.wikipedia.org/wiki/Spatial_reference_systemhttp://en.wikipedia.org/wiki/Satellite_geodesyhttp://en.wikipedia.org/wiki/Reference_ellipsoidhttp://en.wikipedia.org/wiki/Map_projectionhttp://en.wikipedia.org/wiki/Horizontal_position_representationhttp://en.wikipedia.org/wiki/Geographic_coordinate_systemhttp://en.wikipedia.org/wiki/Geodetic_systemhttp://en.wikipedia.org/wiki/Figure_of_the_Earthhttp://en.wikipedia.org/wiki/Geographical_distancehttp://en.wikipedia.org/wiki/Datum_(geodesy)http://en.wikipedia.org/wiki/Cartographyhttp://en.wikipedia.org/wiki/Geomaticshttp://en.wikipedia.org/wiki/Geodynamicshttp://en.wikipedia.org/wiki/Geodesyhttp://en.wikipedia.org/wiki/Category:Geodesy


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GNSS GPS ...

Standards

ED50 ETRS89 NAD83

NAVD88 SAD69 SRID

UTM WGS84 ...

History

History of geodesy

NAVD29 ...

v

d

e

The geoid is thatequipotential surfacewhich would coincide exactly with the mean

ocean surface of the Earth, if the oceans were in equilibrium, at rest (relative to the

rotating Earth[citation needed]), and extended through the continents (such as with very

narrow canals). According toC.F. Gauss, who first described it, it is the

"mathematical figure of the Earth", a smooth but highly irregular surface that

corresponds not to the actual surface of the Earth's crust, but to a surface which can

only be known through extensivegravitationalmeasurements and calculations.

Despite being an important concept for almost two hundred years in the history

ofgeodesyandgeophysics, it has only been defined to high precision in recent

decades, for instance by works ofPetr Vanekand others. It is often described asthe true physicalfigure of the Earth, in contrast to the idealized geometrical figure of

areference ellipsoid.
http://en.wikipedia.org/wiki/Satellite_navigationhttp://en.wikipedia.org/wiki/Satellite_navigationhttp://en.wikipedia.org/wiki/Global_Positioning_Systemhttp://en.wikipedia.org/wiki/Global_Positioning_Systemhttp://en.wikipedia.org/wiki/Global_Positioning_Systemhttp://en.wikipedia.org/wiki/ED50http://en.wikipedia.org/wiki/ED50http://en.wikipedia.org/wiki/European_Terrestrial_Reference_System_1989http://en.wikipedia.org/wiki/European_Terrestrial_Reference_System_1989http://en.wikipedia.org/wiki/European_Terrestrial_Reference_System_1989http://en.wikipedia.org/wiki/North_American_Datum#North_American_Datum_of_1983http://en.wikipedia.org/wiki/North_American_Datum#North_American_Datum_of_1983http://en.wikipedia.org/wiki/North_American_Datum#North_American_Datum_of_1983http://en.wikipedia.org/wiki/North_American_Vertical_Datum_of_1988http://en.wikipedia.org/wiki/North_American_Vertical_Datum_of_1988http://en.wikipedia.org/wiki/South_American_Datum#SAD69http://en.wikipedia.org/wiki/South_American_Datum#SAD69http://en.wikipedia.org/wiki/South_American_Datum#SAD69http://en.wikipedia.org/wiki/SRIDhttp://en.wikipedia.org/wiki/SRIDhttp://en.wikipedia.org/wiki/SRIDhttp://en.wikipedia.org/wiki/Universal_Transverse_Mercator_coordinate_systemhttp://en.wikipedia.org/wiki/Universal_Transverse_Mercator_coordinate_systemhttp://en.wikipedia.org/wiki/World_Geodetic_Systemhttp://en.wikipedia.org/wiki/World_Geodetic_Systemhttp://en.wikipedia.org/wiki/World_Geodetic_Systemhttp://en.wikipedia.org/wiki/History_of_geodesyhttp://en.wikipedia.org/wiki/History_of_geodesyhttp://en.wikipedia.org/wiki/Sea_Level_Datum_of_1929http://en.wikipedia.org/wiki/Sea_Level_Datum_of_1929http://en.wikipedia.org/wiki/Template:Geodesyhttp://en.wikipedia.org/wiki/Template:Geodesyhttp://en.wikipedia.org/w/index.php?title=Template_talk:Geodesy&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Template_talk:Geodesy&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Template:Geodesy&action=edithttp://en.wikipedia.org/w/index.php?title=Template:Geodesy&action=edithttp://en.wikipedia.org/wiki/Equipotential_surfacehttp://en.wikipedia.org/wiki/Equipotential_surfacehttp://en.wikipedia.org/wiki/Equipotential_surfacehttp://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/Gravityhttp://en.wikipedia.org/wiki/Gravityhttp://en.wikipedia.org/wiki/Gravityhttp://en.wikipedia.org/wiki/Geodesyhttp://en.wikipedia.org/wiki/Geodesyhttp://en.wikipedia.org/wiki/Geodesyhttp://en.wikipedia.org/wiki/Geophysicshttp://en.wikipedia.org/wiki/Geophysicshttp://en.wikipedia.org/wiki/Geophysicshttp://en.wikipedia.org/wiki/Petr_Van%C3%AD%C4%8Dekhttp://en.wikipedia.org/wiki/Petr_Van%C3%AD%C4%8Dekhttp://en.wikipedia.org/wiki/Petr_Van%C3%AD%C4%8Dekhttp://en.wikipedia.org/wiki/Figure_of_the_Earthhttp://en.wikipedia.org/wiki/Figure_of_the_Earthhttp://en.wikipedia.org/wiki/Figure_of_the_Earthhttp://en.wikipedia.org/wiki/Reference_ellipsoidhttp://en.wikipedia.org/wiki/Reference_ellipsoidhttp://en.wikipedia.org/wiki/Reference_ellipsoidhttp://en.wikipedia.org/wiki/Reference_ellipsoidhttp://en.wikipedia.org/wiki/Figure_of_the_Earthhttp://en.wikipedia.org/wiki/Petr_Van%C3%AD%C4%8Dekhttp://en.wikipedia.org/wiki/Geophysicshttp://en.wikipedia.org/wiki/Geodesyhttp://en.wikipedia.org/wiki/Gravityhttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/wiki/Equipotential_surfacehttp://en.wikipedia.org/w/index.php?title=Template:Geodesy&action=edithttp://en.wikipedia.org/w/index.php?title=Template_talk:Geodesy&action=edit&redlink=1http://en.wikipedia.org/wiki/Template:Geodesyhttp://en.wikipedia.org/wiki/Sea_Level_Datum_of_1929http://en.wikipedia.org/wiki/History_of_geodesyhttp://en.wikipedia.org/wiki/World_Geodetic_Systemhttp://en.wikipedia.org/wiki/Universal_Transverse_Mercator_coordinate_systemhttp://en.wikipedia.org/wiki/SRIDhttp://en.wikipedia.org/wiki/South_American_Datum#SAD69http://en.wikipedia.org/wiki/North_American_Vertical_Datum_of_1988http://en.wikipedia.org/wiki/North_American_Datum#North_American_Datum_of_1983http://en.wikipedia.org/wiki/European_Terrestrial_Reference_System_1989http://en.wikipedia.org/wiki/ED50http://en.wikipedia.org/wiki/Global_Positioning_Systemhttp://en.wikipedia.org/wiki/Satellite_navigation


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Map of theundulation of the geoid, in meters (based on theEGM96gravity model and

theWGS84reference ellipsoid).[1]

Description

1. Ocean

2.Reference ellipsoid

3. Localplumb line

4. Continent

5. Geoid

The geoid surface is irregular, unlike thereference ellipsoidwhich is amathematical idealized representation of the physical Earth, butconsiderably smoother than Earth's physical surface. Although thephysical Earth has excursions of +8,000 m (Mount Everest) and 11,000m (Mariana Trench), the geoid's total variation is less than 200 m (106to +85 m)[2]compared to a perfect mathematical ellipsoid.

Sea level, if undisturbed by tides, currents and weather, would assume asurface equal to the geoid. If the continental land masses were criss-

crossed by a series of tunnels or narrow canals, the sea level in thesecanals would also coincide with the geoid. In reality the geoid does not
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have a physical meaning under the continents, butgeodesistsare ableto derive the heights of continental points above this imaginary, yetphysically defined, surface by a technique calledspirit leveling.

Being anequipotential surface, the geoid is by definition a surface to

which the force of gravity is everywhere perpendicular. This means thatwhen travelling by ship, one does not notice the undulations of thegeoid; the local vertical (plumb line) is always perpendicular to the geoidand the local horizontangentialto it. Likewise, spirit levels will always beparallel to the geoid.

Note that aGPSreceiver on a ship may, during the course of a longvoyage, indicate height variations, even though the ship will always be atsea level (tides not considered). This is because GPSsatellites, orbitingabout the center of gravity of the Earth, can only measure heights

relative to a geocentric reference ellipsoid. To obtain one's geoidalheight, a raw GPS reading must be corrected. Conversely, heightdetermined by spirit leveling from a tidal measurement station, as intraditional land surveying, will always be geoidal height. Modern GPSreceivers have a grid implemented inside where they obtain the geoid(e.g. EGM-96) height over the WGS ellipsoid from the current position.Then they are able to correct the height above WGS ellipsoid to theheight above WGS84 geoid. In that case when the height is not zero ona ship it is because of the tides.

[edit]Simplified Example

The gravity field of the earth is neither perfect nor uniform. A flattenedellipsoid is typically used as the idealized earth, but even if the earthwere perfectly spherical, the strength of gravity would not be the sameeverywhere, because density (and therefore mass) varies throughout theplanet. This is due to magma distributions, mountain ranges, deep seatrenches, and so on.

If that perfect sphere were then covered in water, the water would not bethe same height everywhere. Instead, the water level would be higher orlower depending on the particular strength of gravity in that location.

Precise geoid

The 1990s saw important discoveries in theory of geoid computation.The Precise Geoid Solution byVanekand co-workers improved on

theStokesianapproach to geoid computation.[7]

Their solution enables
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millimetre-to-centimetreaccuracyin geoidcomputation, anorder-of-magnitudeimprovement from previous classical solutions.[8][9][10]

[edit]Time-variability

Recent satellite missions, such asGOCEandGRACE, have enabledthe study of time-variable geoid signals. The first products based onGOCE satellite data became available online in June, 2010, through theEuropean Space Agency (ESA)s Earth observation user servicestools.[11][12]ESA launched the satellite in March 2009 on a mission tomap Earth's gravity with unprecedented accuracy and spatial resolution.On 31 March 2011, the new geoid was unveiled at the FourthInternational GOCE User Workshop hosted at the TechnischeUniversitt Mnchen in Munich, Germany.[13]Studies using the time-

variable geoid computed from GRACE data have provided informationon global hydrologic cycles,[14]mass balances oficesheets,[15]andpostglacial rebound.[16]From postglacial reboundmeasurements, time-variable GRACE data can be used to deducetheviscosityofEarth's mantle.[17]

SpheroidFrom Wikipedia, the free encyclopedia

oblate spheroid prolate spheroid

A spheroid, or ellipsoid of revolution is aquadricsurfaceobtained by rotating

anellipseabout one of its principal axes; in other words, anellipsoidwith two

equalsemi-diameters.

If the ellipse is rotated about its major axis, the result is aprolate(elongated)

spheroid, like arugbyball. If the ellipse is rotated about its minor axis, the result is
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anoblate(flattened) spheroid, like alentil. If the generating ellipse is a circle, the

result is asphere.

Because of the combined effects ofgravitationandrotation, theEarth's shape is

roughly that of a sphere slightly flattened in the direction of its axis. For that reason,incartographythe Earth is often approximated by an oblate spheroid instead of a

sphere. The currentWorld Geodetic Systemmodel uses a spheroid whose radius is

6,378.137 km at theequatorand 6,356.752 km at thepoles.

Equation

A spheroid centered at the "y" origin and rotated about the zaxis isdefined by theimplicitequation

where ais the horizontal, transverse radius at the equator, and bisthe vertical, conjugate radius.[1]

[edit]Surface area

A prolate spheroid hassurface area

where is theangular eccentricityof the prolate

spheroid, and e= sin() is its (ordinary)eccentricity.

An oblate spheroid has surface area

where is

theangular eccentricityof the oblate spheroid.[edit]Volume

The volume of a spheroid (of any kind) is .If A=2ais the equatorial diameter, and B=2bis the polar

diameter, the volume is .

[edit]Curvature

If a spheroid is parameterized as
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where is the reduced orparametric latitude, is

thelongitude, and and , then

itsGaussian curvatureis

and itsmean curvatureis

Both of these curvatures are always positive, so that

every point on a spheroid is elliptic.

IsostasyFrom Wikipedia, the free encyclopedia

Isostasy (Greeksos"equal",stsis"standstill") is a term used ingeologyto refer to

the state ofgravitationalequilibrium between

theearth'slithosphereandasthenospheresuch that thetectonic plates"float" at an

elevation which depends on their thickness and density. This concept is invoked to

explain how different topographic heights can exist at the Earth's surface. When a

certain area of lithosphere reaches the state of isostasy, it is said to be in isostatic

equilibrium. Isostasy is not a process that upsets equilibrium, but rather one which

restores it (a negative feedback). It is generally accepted that the earth is a dynamic

system that responds to loads in many different ways. However, isostasy provides

an important 'view' of the processes that are happening in areas that are

experiencing vertical movement. Certain areas (such as theHimalayas) are not in

isostatic equilibrium, which has forced researchers to identify other reasons to

explain their topographic heights (in the case of the Himalayas, which are still rising,

by proposing that their elevation is being "propped-up" by the force of theimpactingIndian plate).
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In the simplest example, isostasy is the principle ofbuoyancywhere an object

immersed in aliquidis buoyed with a force equal to the weight of the displaced

liquid. On a geological scale, isostasy can be observed where the Earth's

stronglithosphereexerts stress on the weakerasthenospherewhich, overgeological

timeflows laterally such that the load of the lithosphere is accommodated by height

adjustments.

The general term 'isostasy' was coined in 1889 by the American geologistClarence

Dutton.

Isostatic models

Three principal models of isostasy are used:

TheAiry-HeiskanenModel

- where different topographic heights are accommodated by

changes incrustalthickness, in which the crust has a constant

density

ThePratt-HayfordModel

- where different topographic heights are accommodated by lateral

changes inrockdensity.

TheVening Meinesz, or Flexural Model

- where thelithosphereacts as anelasticplate and its inherent

rigidity distributes local topographic loads over a broad region by

bending.

Airy and Pratt isostasy are statements of buoyancy, whileflexural isostasy is a statement of buoyancy while deflecting asheet of finite elastic strength.

[edit]Airy
http://en.wikipedia.org/wiki/Buoyancyhttp://en.wikipedia.org/wiki/Buoyancyhttp://en.wikipedia.org/wiki/Buoyancyhttp://en.wikipedia.org/wiki/Liquidhttp://en.wikipedia.org/wiki/Liquidhttp://en.wikipedia.org/wiki/Liquidhttp://en.wikipedia.org/wiki/Lithospherehttp://en.wikipedia.org/wiki/Lithospherehttp://en.wikipedia.org/wiki/Lithospherehttp://en.wikipedia.org/wiki/Asthenospherehttp://en.wikipedia.org/wiki/Asthenospherehttp://en.wikipedia.org/wiki/Asthenospherehttp://en.wikipedia.org/wiki/Geologic_timescalehttp://en.wikipedia.org/wiki/Geologic_timescalehttp://en.wikipedia.org/wiki/Geologic_timescalehttp://en.wikipedia.org/wiki/Geologic_timescalehttp://en.wikipedia.org/wiki/Clarence_Duttonhttp://en.wikipedia.org/wiki/Clarence_Duttonhttp://en.wikipedia.org/wiki/Clarence_Duttonhttp://en.wikipedia.org/wiki/Clarence_Duttonhttp://en.wikipedia.org/wiki/George_Airyhttp://en.wikipedia.org/wiki/George_Airyhttp://en.wikipedia.org/wiki/Veikko_Aleksanteri_Heiskanenhttp://en.wikipedia.org/wiki/Veikko_Aleksanteri_Heiskanenhttp://en.wikipedia.org/wiki/Veikko_Aleksanteri_Heiskanenhttp://en.wikipedia.org/wiki/Crust_(geology)http://en.wikipedia.org/wiki/Crust_(geology)http://en.wikipedia.org/wiki/Crust_(geology)http://en.wikipedia.org/wiki/John_Henry_Pratthttp://en.wikipedia.org/wiki/John_Henry_Pratthttp://en.wikipedia.org/wiki/John_Fillmore_Hayfordhttp://en.wikipedia.org/wiki/John_Fillmore_Hayfordhttp://en.wikipedia.org/wiki/John_Fillmore_Hayfordhttp://en.wikipedia.org/wiki/Rock_(geology)http://en.wikipedia.org/wiki/Rock_(geology)http://en.wikipedia.org/wiki/Densityhttp://en.wikipedia.org/wiki/Densityhttp://en.wikipedia.org/wiki/Densityhttp://en.wikipedia.org/wiki/Vening_Meineszhttp://en.wikipedia.org/wiki/Vening_Meineszhttp://en.wikipedia.org/wiki/Vening_Meineszhttp://en.wikipedia.org/wiki/Lithospherehttp://en.wikipedia.org/wiki/Lithospherehttp://en.wikipedia.org/wiki/Lithospherehttp://en.wikipedia.org/wiki/Elasticity_(physics)http://en.wikipedia.org/wiki/Elasticity_(physics)http://en.wikipedia.org/wiki/Elasticity_(physics)http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=2http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=2http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=2http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=2http://en.wikipedia.org/wiki/Elasticity_(physics)http://en.wikipedia.org/wiki/Lithospherehttp://en.wikipedia.org/wiki/Vening_Meineszhttp://en.wikipedia.org/wiki/Densityhttp://en.wikipedia.org/wiki/Rock_(geology)http://en.wikipedia.org/wiki/John_Fillmore_Hayfordhttp://en.wikipedia.org/wiki/John_Henry_Pratthttp://en.wikipedia.org/wiki/Crust_(geology)http://en.wikipedia.org/wiki/Veikko_Aleksanteri_Heiskanenhttp://en.wikipedia.org/wiki/George_Airyhttp://en.wikipedia.org/wiki/Clarence_Duttonhttp://en.wikipedia.org/wiki/Clarence_Duttonhttp://en.wikipedia.org/wiki/Geologic_timescalehttp://en.wikipedia.org/wiki/Geologic_timescalehttp://en.wikipedia.org/wiki/Asthenospherehttp://en.wikipedia.org/wiki/Lithospherehttp://en.wikipedia.org/wiki/Liquidhttp://en.wikipedia.org/wiki/Buoyancy


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Airy isostasy, in which a constant-density crust floats on a higher-density mantle,

and topography is determined by the thickness of the crust.

The basis of the model is thePascal's law, and particularly itsconsequence that, within a fluid in static equilibrium, thehydrostatic pressure is the same on every point at the sameelevation (surface of hydrostatic compensation). In otherwords: h11 = h22 = h33 = ... hnnFor the simplified picture shown the depth of the mountain beltroots (b1) are:

(h1 + c + b1)c = (cc) + (b1m)

b1(m c) = h1c

where m is the density of the mantle (ca. 3,300kg m-3) and c is the density of the crust (ca.

2,750 kg m-3

). Thus, we may generally consider:b1 5h1In the case of negative topography (i.e., a marine basin), thebalancing of lithospheric columns gives:

cc = (hww) + (b2m) + [(chwb2)c]

b2(m c) = hw(c w)

where m is the density of the mantle (ca. 3,300
http://en.wikipedia.org/wiki/Pascal%27s_lawhttp://en.wikipedia.org/wiki/Pascal%27s_lawhttp://en.wikipedia.org/wiki/Pascal%27s_lawhttp://en.wikipedia.org/wiki/File:Airy_Isostasy.jpghttp://en.wikipedia.org/wiki/File:Airy_Isostasy.jpghttp://en.wikipedia.org/wiki/File:Airy_Isostasy.jpghttp://en.wikipedia.org/wiki/File:Airy_Isostasy.jpghttp://en.wikipedia.org/wiki/File:Airy_Isostasy.jpghttp://en.wikipedia.org/wiki/File:Airy_Isostasy.jpghttp://en.wikipedia.org/wiki/File:Airy_Isostasy.jpghttp://en.wikipedia.org/wiki/File:Airy_Isostasy.jpghttp://en.wikipedia.org/wiki/Pascal%27s_law


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kg m-3), c is the density of the crust (ca. 2,750 kgm

-3) and w is the density of the water (ca. 1,000

kg m-3). Thus, we may generally consider:

b2

3.2

hw

[edit]Pratt

For the simplified model shown the new density is given

by: , where h1 is the height of the mountain and cthe thickness of the crust.

[edit]Vening Meinesz / flexural

This hypothesis was suggested to explain how largetopographic loads such asseamounts(e.g.Hawaiian Islands)could be compensated by regional rather than localdisplacement of the lithosphere. This is the more generalsolution for flexure, as it approaches the locally-compensatedmodels above as the load becomes much larger than a flexuralwavelength or the flexural rigidity of the lithosphere approaches0.

[edit]Isostatic effects of deposition anderosion

When large amounts of sediment are deposited on a particularregion, the immense weight of the new sediment may causethe crust below to sink. Similarly, when large amounts ofmaterial are eroded away from a region, the land may rise tocompensate. Therefore, as a mountain range is eroded down,the (reduced) range rebounds upwards (to a certain extent) tobe eroded further. Some of the rock strata now visible at the

ground surface may have spent much of their history at greatdepths below the surface buried under other strata, to beeventually exposed as those other strata are eroded away andthe lower layers rebound upwards again.

An analogy may be made with aniceberg- it always floats witha certain proportion of its mass below the surface of the water.If more ice is added to the top of the iceberg, the iceberg willsink lower in the water. If a layer of ice is somehow sliced offthe top of the iceberg, the remaining iceberg will rise. Similarly,

the Earth's lithosphere "floats" in the asthenosphere.
http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=3http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=3http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=3http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=4http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=4http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=4http://en.wikipedia.org/wiki/Seamountshttp://en.wikipedia.org/wiki/Seamountshttp://en.wikipedia.org/wiki/Seamountshttp://en.wikipedia.org/wiki/Hawaiian_Islandshttp://en.wikipedia.org/wiki/Hawaiian_Islandshttp://en.wikipedia.org/wiki/Hawaiian_Islandshttp://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=5http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=5http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=5http://en.wikipedia.org/wiki/Iceberghttp://en.wikipedia.org/wiki/Iceberghttp://en.wikipedia.org/wiki/Iceberghttp://en.wikipedia.org/wiki/Iceberghttp://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=5http://en.wikipedia.org/wiki/Hawaiian_Islandshttp://en.wikipedia.org/wiki/Seamountshttp://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=4http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=3


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[edit]Isostatic effects of plate tectonics

When continents collide, the continental crust may thicken attheir edges in the collision. If this happens, much of the

thickened crust may move downwardsrather than up as withthe iceberg analogy. The idea of continental collisions buildingmountains "up" is therefore rather a simplification. Instead, thecrust thickensand the upper part of the thickened crustmaybecome a mountain range.

However, some continental collisions are far more complexthan this, and the region may not be in isostatic equilibrium, sothis subject has to be treated with caution.

[edit]Isostatic effects of ice sheetsMain article:post-glacial rebound

The formation ofice sheetscan cause the Earth's surface tosink. Conversely, isostatic post-glacial rebound is observed inareas once covered by ice sheets that have now melted, suchas around theBaltic SeaandHudson Bay. As the ice retreats,the load on thelithosphereandasthenosphereis reduced andthey reboundback towards their equilibrium levels. In this way,

it is possible to find formersea cliffsand associatedwave-cutplatformshundreds of metres above present-daysea level. Therebound movements are so slow that the uplift caused by theending of the lastglacial periodis still continuing.

In addition to the vertical movement of the land and sea,isostatic adjustment of the Earth also involves horizontalmovements. It can cause changes in thegravitationalfieldandrotation rate of the Earth,polar wander,andearthquakes.

[edit]Eustasy and relative sea level change

Main article:Eustasy

Eustasy is another cause of relative sea level changequitedifferent from isostatic causes. Theterm eustasyor eustaticrefers to changes in the amount ofwater in the oceans, usually due toglobal climate change.When the Earth's climate cools, a greater proportion of water is

stored on land masses in the form of glaciers, snow, etc. Thisresults in falling global sea levels (relative to a stable land
http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=6http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=6http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=6http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=7http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=7http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=7http://en.wikipedia.org/wiki/Post-glacial_reboundhttp://en.wikipedia.org/wiki/Post-glacial_reboundhttp://en.wikipedia.org/wiki/Post-glacial_reboundhttp://en.wikipedia.org/wiki/Ice_sheetshttp://en.wikipedia.org/wiki/Ice_sheetshttp://en.wikipedia.org/wiki/Ice_sheetshttp://en.wikipedia.org/wiki/Baltic_Seahttp://en.wikipedia.org/wiki/Baltic_Seahttp://en.wikipedia.org/wiki/Baltic_Seahttp://en.wikipedia.org/wiki/Hudson_Bayhttp://en.wikipedia.org/wiki/Hudson_Bayhttp://en.wikipedia.org/wiki/Hudson_Bayhttp://en.wikipedia.org/wiki/Lithospherehttp://en.wikipedia.org/wiki/Lithospherehttp://en.wikipedia.org/wiki/Lithospherehttp://en.wikipedia.org/wiki/Asthenospherehttp://en.wikipedia.org/wiki/Asthenospherehttp://en.wikipedia.org/wiki/Asthenospherehttp://en.wikipedia.org/wiki/Sea_cliffhttp://en.wikipedia.org/wiki/Sea_cliffhttp://en.wikipedia.org/wiki/Sea_cliffhttp://en.wikipedia.org/wiki/Wave-cut_platformhttp://en.wikipedia.org/wiki/Wave-cut_platformhttp://en.wikipedia.org/wiki/Wave-cut_platformhttp://en.wikipedia.org/wiki/Wave-cut_platformhttp://en.wikipedia.org/wiki/Sea_levelhttp://en.wikipedia.org/wiki/Sea_levelhttp://en.wikipedia.org/wiki/Sea_levelhttp://en.wikipedia.org/wiki/Glacial_periodhttp://en.wikipedia.org/wiki/Glacial_periodhttp://en.wikipedia.org/wiki/Glacial_periodhttp://en.wikipedia.org/wiki/Earth%27s_gravityhttp://en.wikipedia.org/wiki/Earth%27s_gravityhttp://en.wikipedia.org/wiki/Earth%27s_gravityhttp://en.wikipedia.org/wiki/Earth%27s_gravityhttp://en.wikipedia.org/wiki/Rotation_of_Earthhttp://en.wikipedia.org/wiki/Rotation_of_Earthhttp://en.wikipedia.org/wiki/Rotation_of_Earthhttp://en.wikipedia.org/wiki/Polar_wanderhttp://en.wikipedia.org/wiki/Polar_wanderhttp://en.wikipedia.org/wiki/Polar_wanderhttp://en.wikipedia.org/wiki/Earthquakehttp://en.wikipedia.org/wiki/Earthquakehttp://en.wikipedia.org/wiki/Earthquakehttp://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=8http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=8http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=8http://en.wikipedia.org/wiki/Eustasyhttp://en.wikipedia.org/wiki/Eustasyhttp://en.wikipedia.org/wiki/Eustasyhttp://en.wikipedia.org/wiki/Sea_level_changehttp://en.wikipedia.org/wiki/Sea_level_changehttp://en.wikipedia.org/wiki/Global_climate_changehttp://en.wikipedia.org/wiki/Global_climate_changehttp://en.wikipedia.org/wiki/Global_climate_changehttp://en.wikipedia.org/wiki/Global_climate_changehttp://en.wikipedia.org/wiki/Sea_level_changehttp://en.wikipedia.org/wiki/Eustasyhttp://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=8http://en.wikipedia.org/wiki/Earthquakehttp://en.wikipedia.org/wiki/Polar_wanderhttp://en.wikipedia.org/wiki/Rotation_of_Earthhttp://en.wikipedia.org/wiki/Earth%27s_gravityhttp://en.wikipedia.org/wiki/Earth%27s_gravityhttp://en.wikipedia.org/wiki/Glacial_periodhttp://en.wikipedia.org/wiki/Sea_levelhttp://en.wikipedia.org/wiki/Wave-cut_platformhttp://en.wikipedia.org/wiki/Wave-cut_platformhttp://en.wikipedia.org/wiki/Sea_cliffhttp://en.wikipedia.org/wiki/Asthenospherehttp://en.wikipedia.org/wiki/Lithospherehttp://en.wikipedia.org/wiki/Hudson_Bayhttp://en.wikipedia.org/wiki/Baltic_Seahttp://en.wikipedia.org/wiki/Ice_sheetshttp://en.wikipedia.org/wiki/Post-glacial_reboundhttp://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=7http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=6


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mass). The refilling of ocean basins byglacial meltwaterat theend of ice ages is an example of eustaticsea level rise.

A second significant cause of eustatic sea level rise is thermalexpansion of sea water when the Earth's mean temperature

increases. Current estimates of global eustatic rise from tidegauge records andsatellite altimetryis about +3 mm/a (see2007 IPCC report). Global sea level is also affected by verticalcrustal movements, changes in the rotational rate of the Earth,large scale changes incontinental marginsand changes in thespreading rate of theocean floor.

When the term relativeis used in context with sea levelchange, the implication is that both eustasy and isostasy are atwork, or that the author does not know which cause to invoke.

[edit]Further reading
http://en.wikipedia.org/wiki/Glacial_meltwaterhttp://en.wikipedia.org/wiki/Glacial_meltwaterhttp://en.wikipedia.org/wiki/Glacial_meltwaterhttp://en.wikipedia.org/wiki/Sea_level_risehttp://en.wikipedia.org/wiki/Sea_level_risehttp://en.wikipedia.org/wiki/Sea_level_risehttp://en.wikipedia.org/wiki/Satellite_altimetryhttp://en.wikipedia.org/wiki/Satellite_altimetryhttp://en.wikipedia.org/wiki/Satellite_altimetryhttp://en.wikipedia.org/wiki/Continental_marginhttp://en.wikipedia.org/wiki/Continental_marginhttp://en.wikipedia.org/wiki/Continental_marginhttp://en.wikipedia.org/wiki/Ocean_floorhttp://en.wikipedia.org/wiki/Ocean_floorhttp://en.wikipedia.org/wiki/Ocean_floorhttp://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=9http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=9http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=9http://en.wikipedia.org/w/index.php?title=Isostasy&action=edit&section=9http://en.wikipedia.org/wiki/Ocean_floorhttp://en.wikipedia.org/wiki/Continental_marginhttp://en.wikipedia.org/wiki/Satellite_altimetryhttp://en.wikipedia.org/wiki/Sea_level_risehttp://en.wikipedia.org/wiki/Glacial_meltwater

6 Visualizing Gravity

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