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| In [[geophysics]], a '''geopotential model''' is the theoretical analysis of measuring and calculating the effects of the [[Earth]]'s [[gravitational field]].
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| ==Newton's law==
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| [[Image:NewtonsLawOfUniversalGravitation.svg|150px|"150px"|right|thumb|Diagram of two masses attracting one another]]
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| [[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
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| :<math>\mathbf{F} = - G \frac{m_1 m_2}{r^2}\mathbf{\hat{r}}</math>
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| where ''G'' is the [[gravitational constant]] and '''r̂''' is the radial [[unit vector]]. For an object of continuous mass distribution, each mass element ''dm'' can be treated as a point mass, so the [[volume integral]] over the extent of the object gives:
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| {{NumBlk|:|<math>
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| \mathbf{\bar{F}} = - G \int\limits_V \frac{\rho }{r^2}\mathbf{\hat{r}}\,dx\,dy\,dz
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| </math>|{{EquationRef|1}}}} | |
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| with corresponding [[gravitational potential]]
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| {{NumBlk|:|<math>
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| u\ =\ -\int\limits_V \rho \frac{G}{r}\, dx\,dy\,dz
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| </math>|{{EquationRef|2}}}}
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| where ρ = ρ(''x, y, z'') is the [[mass density]] at the [[volume element]] and of the direction from the volume element to the point mass.
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| ===The case of a homogeneous sphere===
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| In the special case of a sphere with a spherically symmetric mass density then ρ = ρ(''s''), i.e. density depends only on the radial distance
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| :<math>s = \sqrt{x^2+y^2+z^2} \,.</math>
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| These integrals can be evaluated analytically. This is the [[shell theorem]] saying that in this case:
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| {{NumBlk|:|<math>
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| \bar{F} = -\frac{GM}{R^2}\ \hat{r}
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| </math>|{{EquationRef|3}}}} | |
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| with corresponding [[potential]]
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| {{NumBlk|:|<math>
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| u = -\frac{GM}{r}
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| </math>|{{EquationRef|4}}}}
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| where ''M'' = ∫<sub>V</sub>ρ(''s'')''dxdydz'' is the total mass of the sphere.
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| ==The deviations of the gravitational field of the Earth from that of homogeneous sphere==
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| In reality the shape of the Earth is not exactly spherical, mainly because of its rotation around the polar axis that makes its shape slightly oblate. If this shape would have been perfectly known together with the exact mass density ρ = ρ(''x, y, z'') the integrals ({{EquationNote|1}}) and ({{EquationNote|2}}) could have been evaluated with numerical methods to find a more accurate model for the gravitational field of the Earth. But the situation is in fact the opposite, by observing the orbits of spacecraft (and the Moon) the gravitational field of the Earth can be determined quite accurately and the best estimate of the mass of the Earth is obtained by dividing the product ''GM'' as determined from the analysis of spacecraft orbit with a value for ''G'' determined to a lower relative accuracy using other physical methods.
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| From the defining equations ({{EquationNote|1}}) and ({{EquationNote|2}}) it is clear (taking the partial derivatives of the integrand) that outside the body in empty space the following differential equations are valid for the field caused by the body:
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| {{NumBlk|:|<math>
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| \frac{\partial F_x }{\partial x} + \frac{\partial F_y }{\partial y} + \frac{\partial F_z }{\partial z} = 0 </math>|{{EquationRef|5}}}}
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| {{NumBlk|:|<math>
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| \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = 0 </math>|{{EquationRef|6}}}}
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| Functions of the form <math>\phi\ =\ R(r)\ \Theta(\theta)\ \Phi(\varphi)</math>
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| where (''r'', θ, φ) are the [[spherical coordinates]] which satisfy the partial differential equation ({{EquationNote|6}}) (the [[Laplace's equation|Laplace equation]]) are called [[spherical harmonic function]].
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| They take the forms:
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| {{NumBlk|:|<math>
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| \begin{align}
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| g(x,y,z) & =\frac{1}{r^{n+1}} P^m_n(\sin \theta) \cos m\varphi \,,&\quad 0 \le m \le n \,,&\quad n=0,1,2,\dots \\
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| h(x,y,z) & =\frac{1}{r^{n+1}} P^m_n(\sin \theta) \sin m\varphi \,,&\quad 1 \le m \le n \,,&\quad n=1,2,\dots
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| \end{align}
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| </math>|{{EquationRef|7}}}} | |
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| where [[spherical coordinates]] (''r'', θ, φ) are used, given here in terms of cartesian (''x, y, z'') for reference:
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| {{NumBlk|:|<math>
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| \begin{align}
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| &x = r \cos \theta \cos \varphi \\
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| &y = r \cos \theta \sin \varphi \\
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| &z = r \sin \theta\,,
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| \end{align}
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| </math>|{{EquationRef|8}}}}
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| also ''P''<sup>0</sup><sub>''n''</sub> are the [[Legendre polynomials]] and ''P<sup>m</sup><sub>n</sub>'' for 1 ≤ ''m'' ≤ ''n'' are the [[Associated Legendre polynomials|associated Legendre functions]].
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| The first spherical harmonics with ''n'' = 0,1,2,3 are presented in the table below.
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| :{| class="wikitable"
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| |-
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| ! ''n'' !! Spherical harmonics
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| |-
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| | 1
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| || <math>\frac{1}{r}</math>
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| |-
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| |rowspan="3"| 2
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| ||<math>\frac{1}{r^2} P^0_1(\sin\theta) = \frac{1}{r^2} \sin\theta</math>
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| |-
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| |<math>\frac{1}{r^2} P^1_1(\sin\theta) \cos\varphi= \frac{1}{r^2} \cos\theta \cos\varphi</math>
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| |-
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| |<math>\frac{1}{r^2} P^1_1(\sin\theta) \sin\varphi= \frac{1}{r^2} \cos\theta \sin\varphi</math>
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| |-
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| |rowspan="5"| 3
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| ||<math>\frac{1}{r^3} P^0_2(\sin\theta) = \frac{1}{r^3} \frac{1}{2} (3\sin^2\theta -1)</math>
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| |-
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| |<math>\frac{1}{r^3} P^1_2(\sin\theta) \cos\varphi = \frac{1}{r^3} 3 \sin\theta \cos\theta\ \cos\varphi</math>
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| |-
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| |<math>\frac{1}{r^3} P^1_2(\sin\theta) \sin\varphi = \frac{1}{r^3} 3 \sin\theta \cos\theta \sin\varphi</math>
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| |-
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| |<math>\frac{1}{r^3} P^2_2(\sin\theta) \cos2\varphi = \frac{1}{r^3} 3 \cos^2 \theta\ \cos2\varphi</math>
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| |-
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| |<math>\frac{1}{r^3} P^2_2(\sin\theta) \sin2\varphi = \frac{1}{r^3} 3 \cos^2 \theta \sin 2\varphi</math>
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| |-
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| |rowspan="7"| 4
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| ||<math>\frac{1}{r^4} P^0_3(\sin\theta) = \frac{1}{r^4} \frac{1}{2} \sin\theta\ (5\sin^2\theta -3)</math>
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| |-
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| |<math>\frac{1}{r^4} P^1_3(\sin\theta) \cos\varphi = \frac{1}{r^4} \frac{3}{2}\ (5\ \sin^2\theta - 1) \cos\theta \cos\varphi
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| </math>
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| |-
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| |<math>\frac{1}{r^4} P^1_3(\sin\theta) \sin\varphi = \frac{1}{r^4} \frac{3}{2}\ (5\ \sin^2\theta - 1) \cos\theta \sin\varphi
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| </math>
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| |-
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| |<math>\frac{1}{r^4} P^2_3(\sin\theta) \cos 2\varphi = \frac{1}{r^4} 15 \sin\theta \cos^2 \theta \cos 2\varphi</math>
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| |-
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| |<math>\frac{1}{r^4} P^2_3(\sin\theta) \sin 2\varphi = \frac{1}{r^4} 15 \sin\theta \cos^2 \theta \sin 2\varphi</math>
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| |-
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| |<math>\frac{1}{r^4} P^3_3(\sin\theta) \cos 3\varphi = \frac{1}{r^4} 15 \cos^3 \theta \cos 3\varphi</math>
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| |-
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| |<math>\frac{1}{r^4} P^3_3(\sin\theta) \sin 3\varphi = \frac{1}{r^4} 15 \cos^3 \theta \sin 3\varphi</math>
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| |-
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| |}
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| The model for the Earth gravitational field is that its potential is a sum
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| {{NumBlk|:|<math>
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| u = -\frac{\mu }{r} + \sum_{n=2}^{N_z} \frac{J_n P^0_n(\sin\theta) }{r^{n+1}} + \sum_{n=2}^{N_t} \sum_{m=1}^n \frac{ P^m_n(\sin\theta) (C_n^m \cos m\varphi + S_n^m \sin m\varphi)}{r^{n+1}}</math>|{{EquationRef|9}}}}
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| where <math>\mu=GM</math> and the coordinates ({{EquationNote|8}}) are relative the standard geodetic reference system extended into space with origin in the center of the [[reference ellipsoid]] and with ''z''-axis in the direction of the polar axis.
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| The '''zonal terms''' refer to terms of the form:
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| :<math>\frac{P^0_n(\sin\theta) }{r^{n+1}} \quad n=0,1,2,\dots</math>
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| and the '''tesseral terms''' terms refer to terms of the form:
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| :<math>\frac{ P^m_n(\sin\theta) \cos m\varphi}{r^{n+1}}\,, \quad 1 \le m \le n \quad n=1,2,\dots</math>
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| :<math>\frac{ P^m_n(\sin\theta) \sin m\varphi}{r^{n+1}}</math>
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| The zonal and tesseral terms for ''n'' = 1 are left out in ({{EquationNote|9}}).
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| The different coefficients ''J<sub>n</sub>'', ''C<sub>n</sub><sup>m</sup>'', ''S<sub>n</sub><sup>m</sup>'', are then given the values for which the best possible agreement between the computed and the observed spacecraft orbits is obtained. | |
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| As ''P''<sup>0</sup><sub>n</sub>(''x'') = −''P''<sup>0</sup><sub>n</sub>(−''x'') non-zero coefficients ''J<sub>n</sub>'' for '''odd ''n''''' correspond to a lack of symmetry "north/south" relative the equatorial plane for the shape/mass-distribution of the Earth. Non-zero coefficients ''C<sub>n</sub><sup>m</sup>'', ''S<sub>n</sub><sup>m</sup>'' correspond to a lack of rotational symmetry around the polar axis for the shape/mass-distribution of the Earth, i.e. to a "tri-axiality" of the Earth
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| For large values of ''n'' the coefficients above (that are divided by ''r''<sup>(''n'' + 1)</sup> in ({{EquationNote|9}})) take very large values when for example kilometers and seconds are used as units. In the literature it is common to introduce some arbitrary "reference radius" ''R'' close to the radius of the Earth and to work with the dimensionless coefficients
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| :<math>\tilde{J_n} = -\frac{J_n}{\mu\ R^n}</math>
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| :<math>\tilde{C_{n}^m} = -\frac{C_{n}^m}{\mu\ R^n}</math>
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| :<math>\tilde{S_{n}^m} = -\frac{S_{n}^m}{\mu\ R^n}</math>
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| and to write the potential as
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| {{NumBlk|:|<math>
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| u = -\frac{\mu }{r} \left(1 + \sum_{n=2}^{N_z} \frac{\tilde{J_n} P^0_n(\sin\theta) }{{(\frac{r}{R})}^n} + \sum_{n=2}^{N_t} \sum_{m=1}^n \frac{ P^m_n(\sin\theta) (\tilde{C_{n}^m} \cos m\varphi + \tilde{S_{n}^m} \sin m\varphi)}{{(\frac{r}{R})}^n}\right)</math>|{{EquationRef|10}}}}
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| The dominating term (after the term −μ/''r'') in ({{EquationNote|9}}) is the "''J''<sub>2</sub> term":
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| :<math>u = \frac{J_2\ P^0_2(\sin\theta)}{r^3} = J_2 \frac{1}{r^3} \frac{1}{2} (3\sin^2\theta -1) = J_2 \frac{1}{r^5} \frac{1}{2} (3 z^2 -r^2)</math>
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| Relative the coordinate system
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| {{NumBlk|:|<math>
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| \begin{align}
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| &\hat{\varphi}=-\sin \varphi \hat{x} + \cos \varphi \hat{y} \\
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| &\hat{\theta}=-\sin \theta\ (\cos \varphi\ \hat{x} + \sin \varphi \hat{y})+ \cos\theta \hat{z} \\
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| &\hat{r}= \cos \theta\ (\cos \varphi\ \hat{x}\ +\ \sin \varphi\ \hat{y})+\ \sin\theta\ \hat{z}
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| \end{align}
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| </math>|{{EquationRef|11}}}}
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| [[File:Spherical coordinates unit vectors.svg|thumb|right|Figure 1: The unit vectors <math>\hat{\varphi}\ ,\ \hat{\theta}\ ,\ \hat{r}</math>]]
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| illustrated in figure 1 the components of the force caused by the "''J''<sub>2</sub> term" are
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| {{NumBlk|:|<math>
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| \begin{align}
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| &F_\theta = -\frac{1}{r}\ \frac{\partial u }{\partial \theta} = -J_2\ \frac{1}{r^4} 3\ \cos\theta\ \sin\theta \\
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| &F_r = -\frac{\partial u }{\partial r} = J_2\ \frac{1}{r^4} \frac{3}{2}\ \left(3\sin^2\theta\ -\ 1\right)
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| \end{align}
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| </math>|{{EquationRef|12}}}}
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| In the rectangular coordinate system (''x, y, z'') with unit vectors (''x̂ ŷ ẑ'') the force components are:
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| {{NumBlk|:|<math>
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| \begin{align}
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| &F_x = -\frac{\partial u }{\partial x} = J_2\ \frac{x}{r^7} \left(6z^2 - \frac{3}{2} (x^2 + y^2)\right) \\
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| &F_y = -\frac{\partial u }{\partial y} = J_2\ \frac{y}{r^7} \left(6z^2 - \frac{3}{2} (x^2 + y^2)\right) \\
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| &F_z = -\frac{\partial u }{\partial z} = J_2\ \frac{z}{r^7} \left(3z^2 - \frac{9}{2} (x^2 + y^2)\right)
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| \end{align}
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| </math>|{{EquationRef|13}}}}
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| The components of the force corresponding to the "''J''<sub>3</sub> term"
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| :<math>u = \frac{J_3 P^0_3(\sin\theta) }{r^4} = J_3 \frac{1}{r^4} \frac{1}{2} \sin\theta (5\sin^2\theta -3) = J_3 \frac{1}{r^7} \frac{1}{2} z (5 z^2 - 3 r^2)</math>
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| are | |
| {{NumBlk|:|<math>
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| \begin{align}
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| &F_\theta = -\frac{1}{r} \frac{\partial u }{\partial \theta} = -J_3 \frac{1}{r^5} \frac{3}{2} \cos\theta \left(5 \sin^2\theta -1\right) \\
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| &F_r = -\frac{\partial u }{\partial r} = J_3 \frac{1}{r^5} 2 \sin\theta \left(5\sin^2\theta - 3\right)
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| \end{align}
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| </math>|{{EquationRef|14}}}}
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| and
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| {{NumBlk|:|<math>
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| \begin{align}
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| &F_x = -\frac{\partial u }{\partial x} = J_3 \frac{x z}{r^9} \left(10 z^2 - \frac{15}{2} (x^2 + y^2)\right) \\
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| &F_y = -\frac{\partial u }{\partial y} = J_3 \frac{y z}{r^9} \left(10 z^2 - \frac{15}{2} (x^2 + y^2)\right) \\
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| &F_z = -\frac{\partial u }{\partial z} = J_3 \frac{1}{r^9} \left(4 z^2\ \left( z^2 - 3 (x^2 + y^2)\right) + \frac{3}{2} (x^2 + y^2)^2\right)
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| \end{align}
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| </math>|{{EquationRef|15}}}}
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| The exact numerical values for the coefficients deviate (somewhat) between different Earth models but for the lowest coefficients they all agree almost exactly.
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| For JGM-3 the values are:
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| :μ = 398600.440 km<sup>3</sup>⋅s<sup>−2</sup>
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| :''J''<sub>2</sub> = 1.7555 × 10<sup>10</sup> km<sup>5</sup>⋅s<sup>−2</sup>
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| :''J''<sub>3</sub> = −2.619 × 10<sup>11</sup> km<sup>6</sup>⋅s<sup>−2</sup>
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| With a "reference radius" ''R'' of 6378.1363 km corresponding dimensionless parameters are
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| :<math>\tilde{J_2} = -1.0826 \times 10^{-3}</math>
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| :<math>\tilde{J_3} = 2.532 \times 10^{-6}</math>
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| For example, at a radius of 6600 km (about 200 km over the Earth's surface) ''J''<sub>3</sub>/(''J''<sub>2</sub>''r'') is about 0.002, i.e
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| the correction to the "''J''<sub>2</sub> force" from the "''J''<sub>3</sub> term" is in the order of 2 promille. The negative value of ''J''<sub>3</sub> implies that for a mass point in the equatorial plane of the Earth the gravitational force is tilted slightly towards south due to the lack of symmetry for the mass distribution of the Earth "north/south".
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| ==Recursive algorithms used for the numerical propagation of spacecraft orbits==
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| Spacecraft orbits are computed by the [[Numerical methods for ordinary differential equations|numerical integration]] of the [[equations of motion|equation of motion]]. For this the gravitational force, i.e. the [[gradient]] of the potential, must be computed. Efficient [[Recursion (mathematics)|recursive algorithms]] have been designed to compute the gravitational force for any <math>N_z</math> and <math>N_t</math> and such algorithms are used in standard orbit propagation software
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| ==Available models==
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| The earliest Earth models in general use by [[NASA]] and [[ESRO]]/[[ESA]] were the "Goddard Earth Models" developed by [[Goddard Space Flight Center]] denoted "GEM-1", "GEM-2", "GEM-3", and so on. Later the "Joint Earth Gravity Models" denoted "JGM-1", "JGM-2", "JGM-3" developed by [[Goddard Space Flight Center]] in cooperation with universities and private companies became available. The newer models generally provided higher order terms than their precursors. The [[EGM96]] uses ''N<sub>z</sub>'' = ''N<sub>t</sub>'' = 360 resulting in 130317 coefficients.
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| For a normal Earth satellite for which an orbit determination/prediction accuracy of a few meters is sufficient the "JGM-3" truncated to ''N<sub>z</sub>'' = ''N<sub>t</sub>'' = 36 (1365 coefficients) is usually sufficient. Inaccuracies from the modeling of the air-drag and to a lesser extent the solar radiation pressure will exceed the inaccuracies caused by the gravitation modeling errors.
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| ==Spherical harmonics==
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| {{Main|Spherical harmonics}}
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| The following is a compact account of the spherical harmonics used to model the gravitational field of the Earth. The spherical harmonics are derived from the approach of looking for harmonic functions of the form
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| {{NumBlk|:|<math>
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| \phi\ =\ R(r)\ \Theta(\theta)\ \Phi(\varphi)
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| </math>|{{EquationRef|16}}}} | |
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| where (''r'', θ, φ) are the [[spherical coordinates]] defined by the equations ({{EquationNote|8}}). By straightforward calculations one gets that for any function ''f''
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| {{NumBlk|:|<math>
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| \frac{\partial^2 f}{\partial x^2}\ +\ \frac{\partial^2 f}{\partial y^2}\ +\ \frac{\partial^2 f}{\partial z^2}\ =\ {1 \over r^2}{\partial \over \partial r}\left(r^2 {\partial f \over \partial r}\right)
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| + {1 \over r^2\cos\theta}{\partial \over \partial \theta}\left(\cos\theta {\partial f \over \partial \theta}\right)
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| + {1 \over r^2\cos^2\theta}{\partial^2 f \over \partial \varphi^2} </math>|{{EquationRef|17}}}}
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| Introducing the expression ({{EquationNote|16}}) in ({{EquationNote|17}}) one gets that
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| {{NumBlk|:|<math>
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| \frac{r^2}{\phi}\left(\frac{\partial^2 \phi}{\partial x^2}+\frac{\partial^2 \phi}{\partial y^2}+\frac{\partial^2 \phi}{\partial z^2}\right)\ =
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| \frac{1}{R}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)+\frac{1}{\Theta\cos\theta}\frac{d}{d\theta}\left(\cos\theta \frac{d\Theta}{d\theta}\right)+\frac{1}{\Phi\cos^2\theta}\frac{d^2\Phi}{d\varphi^2}
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| </math>|{{EquationRef|18}}}}
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| As the term
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| :<math>\frac{1}{R}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)</math>
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| only depends on the variable <math>r</math> and the sum
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| :<math>\frac{1}{\Theta\cos\theta}\frac{d}{d\theta}\left(\cos\theta \frac{d\Theta}{d\theta}\right) + \frac{1}{\Phi\cos^2\theta}\frac{d^2\Phi}{d\varphi^2} </math>
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| only depends on the variables θ and φ. One gets that φ is harmonic if and only if
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| {{NumBlk|:|<math>
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| \frac{1}{R}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)\ =\ \lambda
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| </math>|{{EquationRef|19}}}} | |
| and
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| {{NumBlk|:|<math>
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| \frac{1}{\Theta\cos\theta}\frac{d}{d\theta}\left(\cos\theta \frac{d\Theta}{d\theta}\right) + \frac{1}{\Phi\cos^2\theta}\frac{d^2\Phi}{d\varphi^2}\ =\ -\lambda
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| </math>|{{EquationRef|20}}}}
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| for some constant <math>\lambda</math>
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| From ({{EquationNote|20}}) then follows that
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| :<math>\frac{1}{\Theta}\ \cos\theta\ \frac{d}{d\theta}\left(\cos\theta \frac{d\Theta}{d\theta}\right)\ + \lambda\ \cos^2\theta\ +\ \frac{1}{\Phi}\frac{d^2\Phi}{d\varphi^2}\ =\ 0</math>
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| The first two terms only depend on the variable <math>\theta</math> and the third only on the variable <math>\varphi</math>.
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| From the definition of φ as a spherical coordinate it is clear that Φ(φ) must be periodic with the period 2π and one must therefore have that
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| {{NumBlk|:|<math>
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| \frac{1}{\Phi}\frac{d^2\Phi}{d\varphi^2}\ =\ -m^2
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| </math>|{{EquationRef|21}}}}
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| and
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| {{NumBlk|:|<math>
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| \frac{1}{\Theta}\ \cos\theta\ \frac{d}{d\theta}\left(\cos\theta \frac{d\Theta}{d\theta}\right)\ + \lambda\ \cos^2\theta=\ m^2
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| </math>|{{EquationRef|22}}}}
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| for some integer ''m'' as the family of solutions to ({{EquationNote|21}}) then are
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| {{NumBlk|:|<math>
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| \Phi(\varphi)\ =a\ \cos m\varphi\ +\ b\ \sin m\varphi
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| </math>|{{EquationRef|23}}}}
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| With the variable substitution
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| :<math>x=\sin \theta</math>
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| equation ({{EquationNote|22}}) takes the form
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| {{NumBlk|:|<math>
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| \frac{d}{dx}\left((1-x^2) \frac{d\Theta}{dx}\right)+\left(\lambda -\frac{m^2}{1-x^2} \right)\Theta=0
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| </math>|{{EquationRef|24}}}}
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| From ({{EquationNote|19}}) follows that in order to have a solution <math>\phi</math> with
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| :<math>R(r) = \frac{1}{r^{n+1}}</math>
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| one must have that
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| :<math>\lambda = n (n+1)</math>
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| If ''P<sub>n</sub>''(''x'') is a solution to the differential equation
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| {{NumBlk|:|<math>
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| \frac{d}{dx}\left((1-x^2)\ \frac{dP_n}{dx}\right)\ +\ n(n+1)\ P_n\ =\ 0
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| </math>|{{EquationRef|25}}}} | |
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| one therefore has that the potential corresponding to ''m'' = 0
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| :<math>\phi = \frac{1}{r^{n+1}}\ P_n(\sin\theta)</math>
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| '''which is rotational symmetric around the z-axis''' is an harmonic function
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| If <math>P_{n}^{m}(x)</math> is a solution to the differential equation
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| {{NumBlk|:|<math>
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| \frac{d}{dx}\left((1-x^2)\ \frac{dP_{n}^{m}}{dx}\right)\ +\ \left(n(n+1) -\frac{m^2}{1-x^2} \right)\ P_{n}^{m}\ =\ 0
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| </math>|{{EquationRef|26}}}}
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| with ''m'' ≥ 1 one has the potential
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| {{NumBlk|:|<math>
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| \phi = \frac{1}{r^{n+1}}\ P_{n}^{m}(\sin\theta)\ (a\ \cos m\varphi\ +\ b\ \sin m\varphi)</math>|{{EquationRef|27}}}}
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| where ''a'' and ''b'' are arbitrary constants is a harmonic function that depends on φ and therefore is '''not''' rotational symmetric around the z-axis
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| The differential equation ({{EquationNote|25}}) is the Legendre differential equation for which the [[Legendre polynomials]] defined
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| {{NumBlk|:|<math>
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| \begin{align}
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| & P_{0}(x) = 1 \\
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| & P_{n}(x)=\frac{1}{2^n n!}\ \frac{d^n(x^2-1)^n}{dx^n} \quad n \ge 1 \\
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| \end{align}
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| </math>|{{EquationRef|28}}}}
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| are the solutions.
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| The arbitrary factor 1/(2<sup>n</sup>''n''!) is selected to make ''P<sub>n</sub>''(−1)=−1 and ''P<sub>n</sub>''(1) = 1 for odd ''n'' and ''P<sub>n</sub>''(−1) = ''P<sub>n</sub>''(1) = 1 for even ''n''.
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| The first six Legendre polynomials are:
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| {{NumBlk|:|<math>
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| \begin{align}
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| & P_{0}(x) = 1 \\
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| & P_{1}(x) = x \\
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| & P_{2}(x)=\frac{1}{2} \left(3x^2-1\right) \\
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| & P_{3}(x)=\frac{1}{2} \left(5x^3-3x\right) \\
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| & P_{4}(x)=\frac{1}{8} \left(35x^4-30x^2+3\right) \\
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| & P_{5}(x)=\frac{1}{8} \left(63x^5-70x^3+15x\right) \\
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| \end{align}
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| </math>|{{EquationRef|29}}}} | |
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| The solutions to differential equation ({{EquationNote|26}}) are the associated [[Legendre polynomials|Legendre functions]]
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| {{NumBlk|:|<math>
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| P_{n}^{m}(x)\ = (1-x^2)^{\frac{m}{2}}\ \frac{d^n P_n}{dx^n} \quad 1 \le m \le n
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| </math>|{{EquationRef|30}}}}
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| One therefore has that
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| :<math>
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| P_{n}^{m}(\sin\theta)=\cos^m \theta\ \frac{d^n P_n}{dx^n} (\sin\theta)
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| </math>
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| ==References==
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| * El'Yasberg "Theory of flight of artificial earth satellites", '''Israel program for Scientific Translations (1967)'''
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| * Lerch, F.J., Wagner, C.A., Smith, D.E., Sandson, M.L., Brownd, J.E., Richardson, J.A.,"Gravitational Field Models for the Earth (GEM1&2)", Report X55372146, Goddard Space Flight Center, Greenbelt/Maryland, 1972
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| * Lerch, F.J., Wagner, C.A., Putney, M.L., Sandson, M.L., Brownd, J.E., Richardson, J.A., Taylor, W.A., "Gravitational Field Models GEM3 and 4" ,Report X59272476, Goddard Space Flight Center, Greenbelt/Maryland, 1972
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| * Lerch, F.J., Wagner, C.A., Richardson, J.A., Brownd, J.E., "Goddard Earth Models (5 and 6)", Report X92174145, Goddard Space Flight Center, Greenbelt/Maryland, 1974
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| * Lerch, F.J., Wagner, C.A., Klosko, S.M., Belott, R.P., Laubscher, R.E., Raylor, W.A., "Gravity Model Improvement Using Geos3 Altimetry (GEM10A and 10B)", 1978 Spring Annual Meeting of the American Geophysical Union, Miami, 1978
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| * Lerch, F.J., Klosko, S.M., Laubscher, R.E., Wagner, C.A., "Gravity Model Improvement Using Geos3 (GEM9 and 10)", Journal of Geophysical Research, Vol. 84, B8, p. 3897-3916, 1979
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| * Lerch,F.J., Putney, B.H., Wagner, C.A., Klosko, S.M. ,"Goddard earth models for oceanographic applications (GEM 10B and 10C)", Marine-Geodesy, 5(2), p. 145-187, 1981
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| * Lerch, F.J., Klosko, S.M., Patel, G.B., "A Refined Gravity Model from Lageos (GEML2)", 'NASA Technical Memorandum 84986, Goddard Space Flight Center, Greenbelt/Maryland, 1983
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| * Lerch, F.J., Nerem, R.S., Putney, B.H., Felsentreger, T.L., Sanchez, B.V., Klosko, S.M., Patel, G.B., Williamson, R.G., Chinn, D.S., Chan, J.C., Rachlin, K.E., Chandler, N.L., McCarthy, J.J., Marshall, J.A., Luthcke, S.B., Pavlis, D.W., Robbins, J.W., Kapoor, S., Pavlis, E.C., " Geopotential Models of the Earth from Satellite Tracking, Altimeter and Surface Gravity Observations: GEMT3 and GEMT3S", NASA Technical Memorandum 104555, Goddard Space Flight Center , Greenbelt/Maryland, 1992
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| * Lerch, F.J., Nerem, R.S., Putney, B.H., Felsentreger, T.L., Sanchez, B.V., Marshall, J.A., Klosko, S.M., Patel, G.B., Williamson, R.G., Chinn, D.S., Chan, J.C., Rachlin, K.E., Chandler, N.L., McCarthy, J.J., Luthcke, S.B., Pavlis, N.K., Pavlis, D.E., Robbins, J.W., Kapoor, S., Pavlis, E.C., "A Geopotential Model from Satellite Tracking, Altimeter and Surface Gravity Data: GEMT3", Journal of Geophysical Research, Vol. 99, No. B2, p. 2815-2839, 1994
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| * Nerem, R.S., Lerch, F.J., Marshall, J.A., Pavlis, E.C., Putney, B.H., Tapley, B.D., Eanses, R.J., Ries, J.C., Schutz, B.E., Shum, C.K., Watkins, M.M., Klosko, S.M., Chan, J.C., Luthcke, S.B., Patel, G.B., Pavlis, N.K., Williamson, R.G., Rapp, R.H., Biancale, R., Nouel, F., "Gravity Model Developments for Topex/Poseidon: Joint Gravity Models 1 and 2", Journal of Geophysical Research, Vol. 99, No. C12, p. 24421-24447, 1994a
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| == External links ==
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| * http://cddis.nasa.gov/lw13/docs/papers/sci_lemoine_1m.pdf
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| * http://geodesy.geology.ohio-state.edu/course/refpapers/Tapley_JGR_JGM3_96.pdf
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| [[Category:Spaceflight concepts]]
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| [[Category:Gravitation]]
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| [[Category:Earth orbits]]
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