Geopotential model

In geophysics, a geopotential model is the theoretical analysis of measuring and calculating the effects of Earth's gravitational field. F ¯ = − G ∫ V ρ r 2 r ^ d x d y d z {displaystyle mathbf {ar {F}} =-Gint limits _{V}{frac { ho }{r^{2}}}mathbf {hat {r}} ,dx,dy,dz}     (1) u   =   − ∫ V ρ G r d x d y d z {displaystyle u = -int limits _{V} ho {frac {G}{r}},dx,dy,dz}     (2) F ¯ = − G M R 2   r ^ {displaystyle {ar {F}}=-{frac {GM}{R^{2}}} {hat {r}}}     (3) u = − G M r {displaystyle u=-{frac {GM}{r}}}     (4) ∂ F x ∂ x + ∂ F y ∂ y + ∂ F z ∂ z = 0 {displaystyle {frac {partial F_{x}}{partial x}}+{frac {partial F_{y}}{partial y}}+{frac {partial F_{z}}{partial z}}=0}     (5) ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 + ∂ 2 u ∂ z 2 = 0 {displaystyle {frac {partial ^{2}u}{partial x^{2}}}+{frac {partial ^{2}u}{partial y^{2}}}+{frac {partial ^{2}u}{partial z^{2}}}=0}     (6) g ( x , y , z ) = 1 r n + 1 P n m ( sin ⁡ θ ) cos ⁡ m φ , 0 ≤ m ≤ n , n = 0 , 1 , 2 , … h ( x , y , z ) = 1 r n + 1 P n m ( sin ⁡ θ ) sin ⁡ m φ , 1 ≤ m ≤ n , n = 1 , 2 , … {displaystyle {egin{aligned}g(x,y,z)&={frac {1}{r^{n+1}}}P_{n}^{m}(sin heta )cos mvarphi ,,&quad 0leq mleq n,,&quad n=0,1,2,dots \h(x,y,z)&={frac {1}{r^{n+1}}}P_{n}^{m}(sin heta )sin mvarphi ,,&quad 1leq mleq n,,&quad n=1,2,dots end{aligned}}}     (7) x = r cos ⁡ θ cos ⁡ φ y = r cos ⁡ θ sin ⁡ φ z = r sin ⁡ θ , {displaystyle {egin{aligned}&x=rcos heta cos varphi \&y=rcos heta sin varphi \&z=rsin heta ,,end{aligned}}}     (8) u = − μ r + ∑ n = 2 N z J n P n 0 ( sin ⁡ θ ) r n + 1 + ∑ n = 2 N t ∑ m = 1 n P n m ( sin ⁡ θ ) ( C n m cos ⁡ m φ + S n m sin ⁡ m φ ) r n + 1 {displaystyle u=-{frac {mu }{r}}+sum _{n=2}^{N_{z}}{frac {J_{n}P_{n}^{0}(sin heta )}{r^{n+1}}}+sum _{n=2}^{N_{t}}sum _{m=1}^{n}{frac {P_{n}^{m}(sin heta )(C_{n}^{m}cos mvarphi +S_{n}^{m}sin mvarphi )}{r^{n+1}}}}     (9) u = − μ r ( 1 + ∑ n = 2 N z J n ~ P n 0 ( sin ⁡ θ ) ( r R ) n + ∑ n = 2 N t ∑ m = 1 n P n m ( sin ⁡ θ ) ( C n m ~ cos ⁡ m φ + S n m ~ sin ⁡ m φ ) ( r R ) n ) {displaystyle u=-{frac {mu }{r}}left(1+sum _{n=2}^{N_{z}}{frac {{ ilde {J_{n}}}P_{n}^{0}(sin heta )}{{({frac {r}{R}})}^{n}}}+sum _{n=2}^{N_{t}}sum _{m=1}^{n}{frac {P_{n}^{m}(sin heta )({ ilde {C_{n}^{m}}}cos mvarphi +{ ilde {S_{n}^{m}}}sin mvarphi )}{{({frac {r}{R}})}^{n}}} ight)}     (10) φ ^ = − sin ⁡ φ x ^ + cos ⁡ φ y ^ θ ^ = − sin ⁡ θ   ( cos ⁡ φ   x ^ + sin ⁡ φ y ^ ) + cos ⁡ θ z ^ r ^ = cos ⁡ θ   ( cos ⁡ φ   x ^   +   sin ⁡ φ   y ^ ) +   sin ⁡ θ   z ^ {displaystyle {egin{aligned}&{hat {varphi }}=-sin varphi {hat {x}}+cos varphi {hat {y}}\&{hat { heta }}=-sin heta (cos varphi {hat {x}}+sin varphi {hat {y}})+cos heta {hat {z}}\&{hat {r}}=cos heta (cos varphi {hat {x}} + sin varphi {hat {y}})+ sin heta {hat {z}}end{aligned}}}     (11) F θ = − 1 r   ∂ u ∂ θ = − J 2   1 r 4 3   cos ⁡ θ   sin ⁡ θ F r = − ∂ u ∂ r = J 2   1 r 4 3 2   ( 3 sin 2 ⁡ θ   −   1 ) {displaystyle {egin{aligned}&F_{ heta }=-{frac {1}{r}} {frac {partial u}{partial heta }}=-J_{2} {frac {1}{r^{4}}}3 cos heta sin heta \&F_{r}=-{frac {partial u}{partial r}}=J_{2} {frac {1}{r^{4}}}{frac {3}{2}} left(3sin ^{2} heta - 1 ight)end{aligned}}}     (12) F x = − ∂ u ∂ x = J 2   x r 7 ( 6 z 2 − 3 2 ( x 2 + y 2 ) ) F y = − ∂ u ∂ y = J 2   y r 7 ( 6 z 2 − 3 2 ( x 2 + y 2 ) ) F z = − ∂ u ∂ z = J 2   z r 7 ( 3 z 2 − 9 2 ( x 2 + y 2 ) ) {displaystyle {egin{aligned}&F_{x}=-{frac {partial u}{partial x}}=J_{2} {frac {x}{r^{7}}}left(6z^{2}-{frac {3}{2}}(x^{2}+y^{2}) ight)\&F_{y}=-{frac {partial u}{partial y}}=J_{2} {frac {y}{r^{7}}}left(6z^{2}-{frac {3}{2}}(x^{2}+y^{2}) ight)\&F_{z}=-{frac {partial u}{partial z}}=J_{2} {frac {z}{r^{7}}}left(3z^{2}-{frac {9}{2}}(x^{2}+y^{2}) ight)end{aligned}}}     (13) F θ = − 1 r ∂ u ∂ θ = − J 3 1 r 5 3 2 cos ⁡ θ ( 5 sin 2 ⁡ θ − 1 ) F r = − ∂ u ∂ r = J 3 1 r 5 2 sin ⁡ θ ( 5 sin 2 ⁡ θ − 3 ) {displaystyle {egin{aligned}&F_{ heta }=-{frac {1}{r}}{frac {partial u}{partial heta }}=-J_{3}{frac {1}{r^{5}}}{frac {3}{2}}cos heta left(5sin ^{2} heta -1 ight)\&F_{r}=-{frac {partial u}{partial r}}=J_{3}{frac {1}{r^{5}}}2sin heta left(5sin ^{2} heta -3 ight)end{aligned}}}     (14) F x = − ∂ u ∂ x = J 3 x z r 9 ( 10 z 2 − 15 2 ( x 2 + y 2 ) ) F y = − ∂ u ∂ y = J 3 y z r 9 ( 10 z 2 − 15 2 ( x 2 + y 2 ) ) F z = − ∂ u ∂ z = J 3 1 r 9 ( 4 z 2   ( z 2 − 3 ( x 2 + y 2 ) ) + 3 2 ( x 2 + y 2 ) 2 ) {displaystyle {egin{aligned}&F_{x}=-{frac {partial u}{partial x}}=J_{3}{frac {xz}{r^{9}}}left(10z^{2}-{frac {15}{2}}(x^{2}+y^{2}) ight)\&F_{y}=-{frac {partial u}{partial y}}=J_{3}{frac {yz}{r^{9}}}left(10z^{2}-{frac {15}{2}}(x^{2}+y^{2}) ight)\&F_{z}=-{frac {partial u}{partial z}}=J_{3}{frac {1}{r^{9}}}left(4z^{2} left(z^{2}-3(x^{2}+y^{2}) ight)+{frac {3}{2}}(x^{2}+y^{2})^{2} ight)end{aligned}}}     (15) ϕ   =   R ( r )   Θ ( θ )   Φ ( φ ) {displaystyle phi = R(r) Theta ( heta ) Phi (varphi )}     (16) ∂ 2 f ∂ x 2   +   ∂ 2 f ∂ y 2   +   ∂ 2 f ∂ z 2   =   1 r 2 ∂ ∂ r ( r 2 ∂ f ∂ r ) + 1 r 2 cos ⁡ θ ∂ ∂ θ ( cos ⁡ θ ∂ f ∂ θ ) + 1 r 2 cos 2 ⁡ θ ∂ 2 f ∂ φ 2 {displaystyle {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} ight)+{1 over r^{2}cos heta }{partial over partial heta }left(cos heta {partial f over partial heta } ight)+{1 over r^{2}cos ^{2} heta }{partial ^{2}f over partial varphi ^{2}}}     (17) r 2 ϕ ( ∂ 2 ϕ ∂ x 2 + ∂ 2 ϕ ∂ y 2 + ∂ 2 ϕ ∂ z 2 )   = 1 R d d r ( r 2 d R d r ) + 1 Θ cos ⁡ θ d d θ ( cos ⁡ θ d Θ d θ ) + 1 Φ cos 2 ⁡ θ d 2 Φ d φ 2 {displaystyle {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}}} ight) ={frac {1}{R}}{frac {d}{dr}}left(r^{2}{frac {dR}{dr}} ight)+{frac {1}{Theta cos heta }}{frac {d}{d heta }}left(cos heta {frac {dTheta }{d heta }} ight)+{frac {1}{Phi cos ^{2} heta }}{frac {d^{2}Phi }{dvarphi ^{2}}}}     (18) 1 R d d r ( r 2 d R d r )   =   λ {displaystyle {frac {1}{R}}{frac {d}{dr}}left(r^{2}{frac {dR}{dr}} ight) = lambda }     (19) 1 Θ cos ⁡ θ d d θ ( cos ⁡ θ d Θ d θ ) + 1 Φ cos 2 ⁡ θ d 2 Φ d φ 2   =   − λ {displaystyle {frac {1}{Theta cos heta }}{frac {d}{d heta }}left(cos heta {frac {dTheta }{d heta }} ight)+{frac {1}{Phi cos ^{2} heta }}{frac {d^{2}Phi }{dvarphi ^{2}}} = -lambda }     (20) 1 Φ d 2 Φ d φ 2   =   − m 2 {displaystyle {frac {1}{Phi }}{frac {d^{2}Phi }{dvarphi ^{2}}} = -m^{2}}     (21) 1 Θ   cos ⁡ θ   d d θ ( cos ⁡ θ d Θ d θ )   + λ   cos 2 ⁡ θ =   m 2 {displaystyle {frac {1}{Theta }} cos heta {frac {d}{d heta }}left(cos heta {frac {dTheta }{d heta }} ight) +lambda cos ^{2} heta = m^{2}}     (22) Φ ( φ )   = a   cos ⁡ m φ   +   b   sin ⁡ m φ {displaystyle Phi (varphi ) =a cos mvarphi + b sin mvarphi }     (23) d d x ( ( 1 − x 2 ) d Θ d x ) + ( λ − m 2 1 − x 2 ) Θ = 0 {displaystyle {frac {d}{dx}}left((1-x^{2}){frac {dTheta }{dx}} ight)+left(lambda -{frac {m^{2}}{1-x^{2}}} ight)Theta =0}     (24) d d x ( ( 1 − x 2 )   d P n d x )   +   n ( n + 1 )   P n   =   0 {displaystyle {frac {d}{dx}}left((1-x^{2}) {frac {dP_{n}}{dx}} ight) + n(n+1) P_{n} = 0}     (25) d d x ( ( 1 − x 2 )   d P n m d x )   +   ( n ( n + 1 ) − m 2 1 − x 2 )   P n m   =   0 {displaystyle {frac {d}{dx}}left((1-x^{2}) {frac {dP_{n}^{m}}{dx}} ight) + left(n(n+1)-{frac {m^{2}}{1-x^{2}}} ight) P_{n}^{m} = 0}     (26) ϕ = 1 r n + 1   P n m ( sin ⁡ θ )   ( a   cos ⁡ m φ   +   b   sin ⁡ m φ ) {displaystyle phi ={frac {1}{r^{n+1}}} P_{n}^{m}(sin heta ) (a cos mvarphi + b sin mvarphi )}     (27) P 0 ( x ) = 1 P n ( x ) = 1 2 n n !   d n ( x 2 − 1 ) n d x n n ≥ 1 {displaystyle {egin{aligned}&P_{0}(x)=1\&P_{n}(x)={frac {1}{2^{n}n!}} {frac {d^{n}(x^{2}-1)^{n}}{dx^{n}}}quad ngeq 1\end{aligned}}}     (28) P 0 ( x ) = 1 P 1 ( x ) = x P 2 ( x ) = 1 2 ( 3 x 2 − 1 ) P 3 ( x ) = 1 2 ( 5 x 3 − 3 x ) P 4 ( x ) = 1 8 ( 35 x 4 − 30 x 2 + 3 ) P 5 ( x ) = 1 8 ( 63 x 5 − 70 x 3 + 15 x ) {displaystyle {egin{aligned}&P_{0}(x)=1\&P_{1}(x)=x\&P_{2}(x)={frac {1}{2}}left(3x^{2}-1 ight)\&P_{3}(x)={frac {1}{2}}left(5x^{3}-3x ight)\&P_{4}(x)={frac {1}{8}}left(35x^{4}-30x^{2}+3 ight)\&P_{5}(x)={frac {1}{8}}left(63x^{5}-70x^{3}+15x ight)\end{aligned}}}     (29) P n m ( x )   = ( 1 − x 2 ) m 2   d m P n d x m 1 ≤ m ≤ n {displaystyle P_{n}^{m}(x) =(1-x^{2})^{frac {m}{2}} {frac {d^{m}P_{n}}{dx^{m}}}quad 1leq mleq n}     (30) In geophysics, a geopotential model is the theoretical analysis of measuring and calculating the effects of Earth's gravitational field. 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 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: with corresponding gravitational potential 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. 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 These integrals can be evaluated analytically. This is the shell theorem saying that in this case: with corresponding potential where M = ∫Vρ(s)dxdydz is the total mass of the sphere.