Geopotential
Template:Short description Template:One source
Geopotential (symbol W) is the potential of the Earth's gravity field. It has SI units of square metre per square seconds (m2/s2). For convenience it is often defined as the Template:Em 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 (Template:Math), can also be defined.
ConceptsEdit
For geophysical applications, gravity is distinguished from gravitation. Gravity is defined as the resultant force of gravitation and the centrifugal force caused by the Earth's rotation. Likewise, the respective scalar potentials, 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 isosurfaces of the geopotential are called equigeopotential surfaces (sometimes abbreviated as geop),<ref name="Hooijberg 2007 p. 9">Template:Cite book</ref> also known as geopotential level surfaces, equipotential surfaces, or simply level surfaces.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Global mean sea surface is close to one equigeopotential called the geoid.<ref>Template:Cite book</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.
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 deflections of the vertical (plumb-line), can be expressed in this disturbing potential.
BackgroundEdit
Newton's law of universal gravitation states that the gravitational force F acting between two point masses m1 and m2 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 Template:NumBlk = - Gm_1 \int\limits_V \frac{\rho_2}{r^2} \mathbf{\hat{r}} \,dx\,dy\,dz </math>|Template:EquationRef}} with corresponding gravitational potential Template:NumBlk where ρ2 = 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>Template:Clarify 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.
FormulationEdit
Both gravity and its potential contain a contribution from the 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 potentialEdit
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 potentialEdit
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 coordinates).<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 J2 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 = Template:Val, and flattening f = 1/Template:Val. 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 = Template:Val, 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 potentialEdit
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 numberEdit
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 sphereEdit
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: Template:NumBlk with corresponding potential Template:NumBlk 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 [m2/s2] or [J/kg]:<ref>Template:Cite book</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
- Template:Math is the gravitational constant,
- Template:Math is the mass of the earth,
- Template:Math is the average radius of the earth,
- Template:Mvar is the geometric height in meters.