Editing Geopotential

{{Short description|Energy related to Earth's gravity}}
{{One source|date=July 2014}}

'''Geopotential''' (symbol ''W'') is the [[potential]] of the [[Earth]]'s [[gravity field]]. It has [[SI units]] of [[square metre]] per [[square seconds]] (m<sup>2</sup>/s<sup>2</sup>). For convenience it is often defined as the {{em|negative}} of the [[potential energy]] per unit [[mass]], so that the [[gravity vector]] is obtained as the [[gradient]] of the geopotential, without the negation. In addition to the actual potential (the geopotential), a theoretical '''normal potential''' (symbol ''U'') and their difference, the '''disturbing potential''' ({{math|''T'' {{=}} ''W'' − ''U''}}), can also be defined.

==Concepts==
{{redirect|Geop|the unit prefix|geop-}}

For [[geophysical]] applications, gravity is distinguished from [[gravitation]]. Gravity is defined as the [[resultant force]] of gravitation and the [[centrifugal force (fictitious)|centrifugal force]] caused by the [[Earth's rotation]]. Likewise, the respective [[scalar potential]]s, [[gravitational potential]] and [[centrifugal potential]], can be added to form an [[effective potential]] called the geopotential, <math>W</math>. 
The surfaces of constant geopotential or [[isosurface]]s of the geopotential are called '''''equigeopotential surfaces''''' (sometimes abbreviated as '''''geop'''''),<ref name="Hooijberg 2007 p. 9">{{cite book | last=Hooijberg | first=M. | title=Geometrical Geodesy: Using Information and Computer Technology | publisher=Springer Berlin Heidelberg | year=2007 | isbn=978-3-540-68225-7 | url=https://books.google.com/books?id=NJhdIuyrWmoC&pg=PA9 | access-date=2023-09-11 | page=9}}</ref> also known as ''geopotential level surfaces'', ''equipotential surfaces'', or simply ''level surfaces''.<ref>{{cite web|url=https://glossary.ametsoc.org/wiki/Geopotential_surface |access-date=14 April 2023|website=ametsoc.com|title=Geopotential}}</ref>
 
[[Sea level|Global mean sea surface]] is close to one equigeopotential called the ''[[geoid]]''.<ref>{{cite book |first1=Weikko Aleksanteri |last1=Heiskanen |authorlink1=Veikko Aleksanteri Heiskanen |first2=Helmut |last2=Moritz |title=Physical Geodesy |publisher=[[W.H. Freeman]] |year=1967 |isbn=0-7167-0233-9}}</ref> How the gravitational force and the centrifugal force add up to a force orthogonal to the geoid is illustrated in the figure (not to scale). At latitude 50 deg the off-set between the gravitational force (red line in the figure) and the local vertical (green line in the figure) is in fact 0.098 deg. For a mass point (atmosphere) in motion the centrifugal force no more matches the gravitational and the vector sum is not exactly orthogonal to the Earth surface. This is the cause of the [[coriolis effect]] for atmospheric motion.

[[File:The shape of the rotating Earth.svg|thumb|Balance between gravitational and centrifugal force on the Earth surface]]

The geoid is a gently undulating surface due to the irregular mass distribution inside the Earth; it may be approximated however by an [[ellipsoid of revolution]] called the [[reference ellipsoid]]. The currently most widely used reference ellipsoid, that of the Geodetic Reference System 1980 ([[GRS80]]), approximates the geoid to within a little over ±100 m. One can construct a simple model geopotential <math>U</math> that has as one of its equipotential surfaces this reference ellipsoid, with the same model potential <math>U_0</math> as the true potential <math>W_0</math> of the geoid; this model is called a ''[[normal potential]]''. The difference <math>T=W-U</math> is called the ''disturbing potential''. Many observable quantities of the gravity field, such as gravity anomalies and [[Deflection of the vertical|deflections of the vertical]] ([[plumb-line]]), can be expressed in this disturbing potential.

==Background==
[[Image:NewtonsLawOfUniversalGravitation.svg|thumb|Diagram of two masses attracting one another]]

[[Newton's law of universal gravitation]] states that the gravitational force ''F'' acting between two [[point mass]]es ''m''<sub>1</sub> and ''m''<sub>2</sub> with [[centre of mass]] separation ''r'' is given by
<math display="block">
 \mathbf{F} = - G \frac{m_1 m_2}{r^2}\mathbf{\hat{r}},
</math>
where ''G'' is the [[gravitational constant]], and '''r̂''' is the radial [[unit vector]]. For a non-pointlike object of continuous mass distribution, each mass element ''dm'' can be treated as mass distributed over a small volume, so the [[volume integral]] over the extent of object 2 gives
{{NumBlk|:|<math>
\mathbf{\bar{F}} = - Gm_1 \int\limits_V \frac{\rho_2}{r^2} \mathbf{\hat{r}} \,dx\,dy\,dz
</math>|{{EquationRef|1}}}}
with corresponding [[gravitational potential]]
{{NumBlk|:|<math>
V = - G \int\limits_V \frac{\rho_2}{r} \,dx\,dy\,dz,
</math>|{{EquationRef|2}}}}
where ρ{{sub|2}} = ρ(''x'', ''y'', ''z'') is the [[mass density]] at the [[volume element]] and of the direction from the volume element to point mass 1. <math>u</math>{{clarify|reason=this section has no "u" variable|date=April 2025}} is the gravitational potential energy per unit mass.

[[Earth's gravity]] field can be derived from a gravity [[potential]] (''geopotential'') field as follows:
<math display="block">
 \mathbf{g} = \nabla W = \operatorname{grad}W =
  \frac{\partial W}{\partial X} \mathbf{i} +
  \frac{\partial W}{\partial Y} \mathbf{j} +
  \frac{\partial W}{\partial Z} \mathbf{k},
</math>
which expresses the gravity acceleration vector as the gradient of <math>W</math>, the potential of gravity. The vector triad <math>\{\mathbf{i}, \mathbf{j}, \mathbf{k}\}</math> is the orthonormal set of base vectors in space, pointing along the <math>X, Y, Z</math> coordinate axes.
Here, <math>X</math>, <math>Y</math> and <math>Z</math> are [[geocentric coordinates]].

==Formulation==

Both gravity and its potential contain a contribution from the [[centrifugal force|centrifugal pseudo-force]] due to the Earth's rotation. We can write
<math display="block">
 W = V + \Phi,
</math>
where <math>V</math> is the potential of the ''gravitational field'', <math>W</math> that of the ''gravity field'', and <math>\Phi</math> that of the ''centrifugal field''.

===Centrifugal potential===

The centrifugal [[force per unit mass]]—i.e., acceleration—is given by
<math display="block">
 \mathbf{g}_c = \omega^2 \mathbf{p},
</math>
where
<math display="block">
 \mathbf{p} = X\mathbf{i} + Y\mathbf{j} + 0\cdot\mathbf{k}
</math>
is the vector pointing to the point considered straight from the Earth's rotational axis. 
It can be shown that this pseudo-force field, in a reference frame co-rotating with the Earth, has a potential associated with it in terms of Earth's rotation rate ω:
<math display="block">
 \Phi = \frac{1}{2} \omega^2 (X^2 + Y^2).
</math>
This can be verified by taking the gradient (<math>\nabla</math>) operator of this expression.

The centrifugal potential can also be expressed in terms of [[spherical latitude]] φ and [[geocentric radius]] ''r'':
<math display="block">
 \Phi = 0.5 \, \omega^2 r^2 \sin^2\phi,
</math>
or in terms of perpendicular distance ''ρ'' to the axis or rotation:
<math display="block">
 \Phi = 0.5 \, \omega^2 \rho^2.
</math>

=== Normal potential ===
The Earth is approximately an [[ellipsoid]].
So, it is accurate to approximate the geopotential by a field that has the Earth [[reference ellipsoid]] as one of its equipotential surfaces. 

Like the actual geopotential field ''W'', the normal field ''U'' (not to be confused with the [[potential energy]], also ''U'') is constructed as a two-part sum:
<math display="block">
 U = \Psi + \Phi,
</math>
where <math>\Psi</math> is the ''normal gravitational potential'', and <math>\Phi</math> is the centrifugal potential.

A closed-form exact expression exists in terms of [[ellipsoidal-harmonic coordinates]] (not to be confused with [[geodetic coordinate]]s).<ref name=Torge>Torge, Geodesy. 3rd ed. 2001.</ref>
It can also be expressed as a [[series expansion]] in terms of spherical coordinates; truncating the series results in:<ref name=Torge/>
<math display="block">
 \Psi \approx \frac{GM}{r} \left[1 - \left(\frac{a}{r}\right)^2 J_2 \left(\frac{3}{2} \cos^2 \phi - \frac{1}{2}\right)\right],
</math>
where ''a'' is [[semi-major axis]], and ''J''<sub>2</sub> is the [[second dynamic form factor]].<ref name=Torge>Torge, Geodesy. 3rd ed. 2001.</ref>

The most recent Earth reference ellipsoid is [[GRS80]], or Geodetic Reference System 1980, which the [[Global Positioning System]] uses as its reference. Its geometric parameters are: semi-major axis ''a''&nbsp;= {{val|6378137.0|u=m}}, and flattening ''f''&nbsp;= 1/{{val|298.257222101}}.
If we also require that the enclosed mass ''M'' is equal to the known mass of the Earth (including atmosphere), as involved in the [[standard gravitational parameter]], ''GM'' = {{val|3986005e8|u=m<sup>3</sup>/s<sup>2</sup>}}, we obtain for the ''potential at the reference ellipsoid'':
<math display="block">
 U_0 = 62\,636\,860.850\ \text{m}^2/\text{s}^2.
</math>

Obviously, this value depends on the assumption that the potential goes asymptotically to zero at infinity (<math>R \to \infty</math>), as is common in physics. For practical purposes it makes more sense to choose the zero point of [[normal gravity]] to be that of the [[reference ellipsoid]], and refer the potentials of other points to this.

=== Disturbing potential ===

Once a clean, smooth geopotential field <math>U</math> has been constructed, matching the known GRS80 reference ellipsoid with an equipotential surface (we call such a field a ''normal potential''), it can be subtracted from the true (measured) potential <math>W</math> of the real Earth. The result is defined as ''T'', the '''disturbing potential''':
<math display="block">
 T = W - U.
</math>

The disturbing potential ''T'' is numerically a much smaller than ''U'' or ''W'' and captures the detailed, complex variations of the true gravity field of the actually existing Earth from point to point, as distinguished from the overall global trend captured by the smooth mathematical ellipsoid of the normal potential.

=== Geopotential number<span class="anchor" id="Number"></span> ===

In practical terrestrial work, e.g., [[levelling]], an alternative version of the geopotential is used called '''geopotential number''' <math>C</math>, which are reckoned from the geoid upward:
<math display="block">
 C = -(W - W_0),
</math>
where <math>W_0</math> is the geopotential of the geoid.

==Simple case: nonrotating symmetric sphere==

In the special case of a sphere with a spherically symmetric mass density, ρ = ρ(''s''); i.e., density depends only on the radial distance
<math display="block">
 s = \sqrt{x^2 + y^2 + z^2}.
</math>

These integrals can be evaluated analytically. This is the [[shell theorem]] saying that in this case:
{{NumBlk|:|<math>
 \bar{F} = -\frac{GMm}{R^2}\,\hat{r}
</math>|{{EquationRef|3}}}}
with corresponding [[potential]]
{{NumBlk|:|<math>
 \Psi = -\frac{GM}{r},
</math>|{{EquationRef|4}}}}
where <math>M = \int_V \rho(s) \,dx\,dy\,dz</math> is the total mass of the sphere.

For the purpose of satellite [[orbital mechanics]], the geopotential is typically described by a series expansion into [[spherical harmonics]] (spectral representation). In this context the geopotential is taken as the potential of the gravitational field of the Earth, that is, leaving out the centrifugal potential.

Solving for geopotential in the simple case of a nonrotating sphere, in units of [m<sup>2</sup>/s<sup>2</sup>] or [J/kg]:<ref>{{cite book |last=Holton |first=James R. |authorlink=James R. Holton |title=An Introduction to Dynamic Meteorology |edition=4th |location=Burlington |publisher=[[Elsevier]] |year=2004 |isbn=0-12-354015-1}}</ref>
<math display="block">
 \Psi(h) = \int_0^h g\,dz,
</math>
<math display="block">
 \Psi = \int_0^z \frac{Gm}{(a + z)^2} \,dz.
</math>

Integrate to get
<math display="block">
 \Psi = Gm \left(\frac{1}{a} - \frac{1}{a + z}\right),
</math>
where
: {{math|1=''G'' = {{val|6.673|e=-11|u=Nm<sup>2</sup>/kg<sup>2</sup>}}}} is the gravitational constant,
: {{math|1=''m'' = {{val|5.975|e=24|u=kg}}}} is the mass of the earth,
: {{math|1=''a'' = {{val|6.378|e=6|u=m}}}} is the average radius of the earth,
: {{mvar|z}} is the geometric height in meters.

==See also==
* [[Dynamic height]]
* [[Geoid]]
* [[Geopotential height]]
* [[Geopotential model]]
* [[Normal gravity]]
* [[Physical geodesy]]

==References==
{{Reflist}}

{{Authority control}}

[[Category:Gravimetry]]