Misplaced Pages

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27-603: The IGRF model covers a significant time span, and so is useful for interpreting historical data. (This is unlike the World Magnetic Model , which is intended for navigation in the next few years.) It is updated at 5-year intervals, reflecting the most accurate measurements available at that time. The current 13th edition of the IGRF model (IGRF-13) was released in December 2019 and is valid from 1900 until 2025. For

54-510: A hypergeometric differential equation by a change of variable, and its solutions can be expressed using hypergeometric functions . Since the differential equation is linear, homogeneous (the right hand side =zero) and of second order, it has two linearly independent solutions, which can both be expressed in terms of the hypergeometric function , 2 F 1 {\displaystyle _{2}F_{1}} . With Γ {\displaystyle \Gamma } being

81-492: Is colatitude (the polar angle), a {\displaystyle a} is the Earth's radius, g n m {\displaystyle g_{n}^{m}} and h n m {\displaystyle h_{n}^{m}} are Gauss coefficients, and P n m ( cos ⁡ θ ) {\displaystyle P_{n}^{m}\left(\cos \theta \right)} are

108-634: Is given by L 1 ( G / / K ) ∋ f ↦ f ^ {\displaystyle L^{1}(G//K)\ni f\mapsto {\hat {f}}} where f ^ ( s ) = ∫ 1 ∞ f ( x ) P s ( x ) d x , − 1 ≤ ℜ ( s ) ≤ 0 {\displaystyle {\hat {f}}(s)=\int _{1}^{\infty }f(x)P_{s}(x)dx,\qquad -1\leq \Re (s)\leq 0} Legendre functions P λ of non-integer degree are unbounded at

135-1003: Is necessarily singular when x = ± 1 {\displaystyle x=\pm 1} . The Legendre functions of the second kind can also be defined recursively via Bonnet's recursion formula Q n ( x ) = { 1 2 log ⁡ 1 + x 1 − x n = 0 P 1 ( x ) Q 0 ( x ) − 1 n = 1 2 n − 1 n x Q n − 1 ( x ) − n − 1 n Q n − 2 ( x ) n ≥ 2 . {\displaystyle Q_{n}(x)={\begin{cases}{\frac {1}{2}}\log {\frac {1+x}{1-x}}&n=0\\P_{1}(x)Q_{0}(x)-1&n=1\\{\frac {2n-1}{n}}xQ_{n-1}(x)-{\frac {n-1}{n}}Q_{n-2}(x)&n\geq 2\,.\end{cases}}} The nonpolynomial solution for

162-475: Is omitted in this formulation and is instead accounted for in the Gauss coefficients. The Gauss coefficients are modelled as a piecewise-linear function of time with a 5-year step size. This geophysics -related article is a stub . You can help Misplaced Pages by expanding it . World Magnetic Model The World Magnetic Model ( WMM ) is a large spatial-scale representation of the Earth's magnetic field. It

189-576: The British Geological Survey (BGS). The model, associated software, and documentation are distributed by the NGDC on behalf of National Geospatial-Intelligence Agency (NGA). Updated model coefficients are released at 5-year intervals, with WMM2015 (released Dec 15, 2014) supposed to last until December 31, 2019. However, due to extraordinarily large and erratic movements of the north magnetic pole , an out-of-cycle update (WMM2015v2)

216-482: The Gauss coefficients which define a spherical harmonic expansion of V {\displaystyle V} where r {\displaystyle r} is radial distance from the Earth's center, N {\displaystyle N} is the maximum degree of the expansion, ϕ {\displaystyle \phi } is East longitude, θ {\displaystyle \theta }

243-730: The World Hydrographic Office (WHO) navigation and attitude/heading reference, and the Federal Aviation Administration (FAA). It is also used widely in civilian navigation systems as the magnetic model of the World Geodetic System . For example, WMM is pre-installed in Android and iOS devices to correct for the magnetic declination . The WMM is produced by the U.S. National Geophysical Data Center (NGDC) in collaboration with

270-795: The gamma function , the first solution is P λ μ ( z ) = 1 Γ ( 1 − μ ) [ z + 1 z − 1 ] μ / 2 2 F 1 ( − λ , λ + 1 ; 1 − μ ; 1 − z 2 ) , for    | 1 − z | < 2 , {\displaystyle P_{\lambda }^{\mu }(z)={\frac {1}{\Gamma (1-\mu )}}\left[{\frac {z+1}{z-1}}\right]^{\mu /2}\,_{2}F_{1}\left(-\lambda ,\lambda +1;1-\mu ;{\frac {1-z}{2}}\right),\qquad {\text{for }}\ |1-z|<2,} and

297-1214: The (rising) Pochhammer symbol . The nonpolynomial solution for the special case of integer degree λ = n ∈ N 0 {\displaystyle \lambda =n\in \mathbb {N} _{0}} , and μ = 0 {\displaystyle \mu =0} , is often discussed separately. It is given by Q n ( x ) = n ! 1 ⋅ 3 ⋯ ( 2 n + 1 ) ( x − ( n + 1 ) + ( n + 1 ) ( n + 2 ) 2 ( 2 n + 3 ) x − ( n + 3 ) + ( n + 1 ) ( n + 2 ) ( n + 3 ) ( n + 4 ) 2 ⋅ 4 ( 2 n + 3 ) ( 2 n + 5 ) x − ( n + 5 ) + ⋯ ) {\displaystyle Q_{n}(x)={\frac {n!}{1\cdot 3\cdots (2n+1)}}\left(x^{-(n+1)}+{\frac {(n+1)(n+2)}{2(2n+3)}}x^{-(n+3)}+{\frac {(n+1)(n+2)(n+3)(n+4)}{2\cdot 4(2n+3)(2n+5)}}x^{-(n+5)}+\cdots \right)} This solution

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324-626: The 168 spherical-harmonic "Gauss" coefficients, the model also has an equal number of spherical-harmonic secular variation coefficients predicting the temporal evolution of the field over the upcoming five-year epoch. WMM is the standard geomagnetic model of the United States Department of Defense (DoD), the Ministry of Defence (United Kingdom) , the North Atlantic Treaty Organization (NATO),

351-574: The 3000 km of WMM. At this resolution, it is not only able to model the Earth's magnetic field at the core-mantle boundary ("main field"), but also take into account magnetic anomalies caused by the minerals in the Earth's crust. Associated Legendre function In physical science and mathematics, the Legendre functions P λ , Q λ and associated Legendre functions P λ , Q λ , and Legendre functions of

378-469: The Schmidt quasi-normalized associated Legendre functions of degree n {\displaystyle n} and order m {\displaystyle m} where is the normalization coefficient for the Schmidt quasi-normalized formulation. The ( − 1 ) m {\displaystyle (-1)^{m}} term usually present in the associated Legendre functions

405-1078: The contour winds around the points 1 and z in the positive direction and does not wind around −1 . For real x , we have P s ( x ) = 1 2 π ∫ − π π ( x + x 2 − 1 cos ⁡ θ ) s d θ = 1 π ∫ 0 1 ( x + x 2 − 1 ( 2 t − 1 ) ) s d t t ( 1 − t ) , s ∈ C {\displaystyle P_{s}(x)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(x+{\sqrt {x^{2}-1}}\cos \theta \right)^{s}d\theta ={\frac {1}{\pi }}\int _{0}^{1}\left(x+{\sqrt {x^{2}-1}}(2t-1)\right)^{s}{\frac {dt}{\sqrt {t(1-t)}}},\qquad s\in \mathbb {C} } The real integral representation of P s {\displaystyle P_{s}} are very useful in

432-518: The first and second kind of noninteger degree, with the additional qualifier 'associated' if μ is non-zero. A useful relation between the P and Q solutions is Whipple's formula . For positive integer μ = m ∈ N + {\displaystyle \mu =m\in \mathbb {N} ^{+}} the evaluation of P λ μ {\displaystyle P_{\lambda }^{\mu }} above involves cancellation of singular terms. We can find

459-435: The interval [-1, 1] . In applications in physics, this often provides a selection criterion. Indeed, because Legendre functions Q λ of the second kind are always unbounded, in order to have a bounded solution of Legendre's equation at all, the degree must be integer valued: only for integer degree, Legendre functions of the first kind reduce to Legendre polynomials, which are bounded on [-1, 1] . It can be shown that

486-571: The interval from 1945 to 2015, it is "definitive" (a "DGRF"), meaning that future updates are unlikely to improve the model in any significant way. The IGRF models the geomagnetic field B → ( r , ϕ , θ , t ) {\displaystyle {\vec {B}}(r,\phi ,\theta ,t)} as a gradient of a magnetic scalar potential V ( r , ϕ , θ , t ) {\displaystyle V(r,\phi ,\theta ,t)} The magnetic scalar potential model consists of

513-1018: The limit valid for m ∈ N 0 {\displaystyle m\in \mathbb {N} _{0}} as P λ m ( z ) = lim μ → m P λ μ ( z ) = ( − λ ) m ( λ + 1 ) m m ! [ 1 − z 1 + z ] m / 2 2 F 1 ( − λ , λ + 1 ; 1 + m ; 1 − z 2 ) , {\displaystyle P_{\lambda }^{m}(z)=\lim _{\mu \to m}P_{\lambda }^{\mu }(z)={\frac {(-\lambda )_{m}(\lambda +1)_{m}}{m!}}\left[{\frac {1-z}{1+z}}\right]^{m/2}\,_{2}F_{1}\left(-\lambda ,\lambda +1;1+m;{\frac {1-z}{2}}\right),} with ( λ ) n {\displaystyle (\lambda )_{n}}

540-517: The numbers λ and μ may be complex, and are called the degree and order of the relevant function, respectively. The polynomial solutions when λ is an integer (denoted n ), and μ = 0 are the Legendre polynomials P n ; and when λ is an integer (denoted n ), and μ = m is also an integer with | m | < n are the associated Legendre polynomials. All other cases of λ and μ can be discussed as one, and

567-1219: The second is Q λ μ ( z ) = π   Γ ( λ + μ + 1 ) 2 λ + 1 Γ ( λ + 3 / 2 ) e i μ π ( z 2 − 1 ) μ / 2 z λ + μ + 1 2 F 1 ( λ + μ + 1 2 , λ + μ + 2 2 ; λ + 3 2 ; 1 z 2 ) , for     | z | > 1. {\displaystyle Q_{\lambda }^{\mu }(z)={\frac {{\sqrt {\pi }}\ \Gamma (\lambda +\mu +1)}{2^{\lambda +1}\Gamma (\lambda +3/2)}}{\frac {e^{i\mu \pi }(z^{2}-1)^{\mu /2}}{z^{\lambda +\mu +1}}}\,_{2}F_{1}\left({\frac {\lambda +\mu +1}{2}},{\frac {\lambda +\mu +2}{2}};\lambda +{\frac {3}{2}};{\frac {1}{z^{2}}}\right),\qquad {\text{for}}\ \ |z|>1.} These are generally known as Legendre functions of

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594-889: The second kind , Q n , are all solutions of Legendre's differential equation. The Legendre polynomials and the associated Legendre polynomials are also solutions of the differential equation in special cases, which, by virtue of being polynomials, have a large number of additional properties, mathematical structure, and applications. For these polynomial solutions, see the separate Misplaced Pages articles. The general Legendre equation reads ( 1 − x 2 ) y ″ − 2 x y ′ + [ λ ( λ + 1 ) − μ 2 1 − x 2 ] y = 0 , {\displaystyle \left(1-x^{2}\right)y''-2xy'+\left[\lambda (\lambda +1)-{\frac {\mu ^{2}}{1-x^{2}}}\right]y=0,} where

621-453: The solutions are written P λ , Q λ . If μ = 0 , the superscript is omitted, and one writes just P λ , Q λ . However, the solution Q λ when λ is an integer is often discussed separately as Legendre's function of the second kind, and denoted Q n . This is a second order linear equation with three regular singular points (at 1 , −1 , and ∞ ). Like all such equations, it can be converted into

648-1280: The special case of integer degree λ = n ∈ N 0 {\displaystyle \lambda =n\in \mathbb {N} _{0}} , and μ = m ∈ N 0 {\displaystyle \mu =m\in \mathbb {N} _{0}} is given by Q n m ( x ) = ( − 1 ) m ( 1 − x 2 ) m 2 d m d x m Q n ( x ) . {\displaystyle Q_{n}^{m}(x)=(-1)^{m}(1-x^{2})^{\frac {m}{2}}{\frac {d^{m}}{dx^{m}}}Q_{n}(x)\,.} The Legendre functions can be written as contour integrals. For example, P λ ( z ) = P λ 0 ( z ) = 1 2 π i ∫ 1 , z ( t 2 − 1 ) λ 2 λ ( t − z ) λ + 1 d t {\displaystyle P_{\lambda }(z)=P_{\lambda }^{0}(z)={\frac {1}{2\pi i}}\int _{1,z}{\frac {(t^{2}-1)^{\lambda }}{2^{\lambda }(t-z)^{\lambda +1}}}dt} where

675-568: The study of harmonic analysis on L 1 ( G / / K ) {\displaystyle L^{1}(G//K)} where G / / K {\displaystyle G//K} is the double coset space of S L ( 2 , R ) {\displaystyle SL(2,\mathbb {R} )} (see Zonal spherical function ). Actually the Fourier transform on L 1 ( G / / K ) {\displaystyle L^{1}(G//K)}

702-690: Was developed jointly by the US National Geophysical Data Center and the British Geological Survey . The data and updates are issued by the US National Geospatial-Intelligence Agency and the UK Defence Geographic Centre . The model consists of a degree and order 12 spherical harmonic expansion of the magnetic scalar potential of the geomagnetic main field generated in the Earth's core. Apart from

729-488: Was released in February 2019 (delayed by a few weeks due to the U.S. federal government shutdown ) to accurately model the magnetic field above 55° north latitude until the end of 2019. The next regular update (WMM2020) occurred in December 2019. The Enhanced Magnetic Model (EMM) is a sister product of the NGDC featuring a much higher amount of data to degree and order 790, giving a wavelength of 51 km as opposed to

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