02/13/02 12.540 Lec 03 1 12.540 Principles of the Global Positioning System Lecture 03 Prof. Thomas Herring 02/13/02 12.540 Lec 03 2 Review ? In last lecture we looked at conventional methods of measuring coordinates ? Triangulation, trilateration, and leveling ? Astronomic measurements using external bodies ? Gravity field enters in these determinations 02/13/02 12.540 Lec 03 3 Gravitational potential ? In spherical coordinates: need to solve ? This is Laplace’s equation in spherical coordinates 1 r ? 2 ?r 2 (rV)+ 1 r 2 sinθ ? ?θ (sinθ ?V ?θ )+ 1 r 2 sin 2 θ ? 2 V ?λ 2 = 0 02/13/02 12.540 Lec 03 4 Solution to gravity potential ? The homogeneous form of this equation is a “classic” partial differential equation. ? In spherical coordinates solved by separation of variables, r=radius, λ=longitude and θ=co-latitude V(r,θ,λ)=R(r)g(θ)h(λ) 02/13/02 12.540 Lec 03 5 Solution in spherical coordinates ? The radial dependence of form r n or r -n depending on whether inside or outside body. N is an integer ? Longitude dependence is sin(mλ) and cos(mλ) where m is an integer ? The colatitude dependence is more difficult to solve 02/13/02 12.540 Lec 03 6 Colatitude dependence ? Solution for colatitude function generates Legendre polynomials and associated functions. ? The polynomials occur when m=0 in λ dependence. t=cos(θ) P n (t)= 1 2 n n! d n dt n (t 2 ?1) n 02/13/02 12.540 Lec 03 7 Legendre Functions ? Low order functions. Arbitrary n values are generated by recursive algorithms P o (t)=1 P 1 (t)=t P 2 (t)= 1 2 (3t 2 ?1) P 3 (t)= 1 2 (5t 3 ?3t) P 4 (t)= 1 8 (35t 4 ?30t 2 +3) 02/13/02 12.540 Lec 03 8 Associated Legendre Functions ? The associated functions satisfy the following equation ? The formula for the polynomials, Rodriques’ formula, can be substituted P nm (t)=(?1) m (1?t 2 ) m/2 d m dt m P n (t) 02/13/02 12.540 Lec 03 9 Associated functions ? Pnm(t): n is called degree; m is order ? m<=n. In some areas, m can be negative. In gravity formulations m=>0 P 00 (t)=1 P 10 (t)=t P 11 (t)=?(1?t 2 ) 1/2 P 20 (t)= 1 2 (3t 2 ?1) P 21 (t)=?3t(1?t 2 ) 1/2 P 22 (t)= 3(1?t 2 ) http://mathworld.wolfram.com/LegendrePolynomial.html 02/13/02 12.540 Lec 03 10 Ortogonality conditions ? The Legendre polynomials and functions are orthogonal: P n' (t) ?1 1 ∫ P n (t)dt = 2 2n+1 δ n'n P n'm (t) ?1 1 ∫ P nm (t)dt = 2 2n+1 (n+m)! (n?m)! δ n'n 02/13/02 12.540 Lec 03 11 Examples from Matlab ? Matlab/Harmonics.m is a small matlab program to plots the associated functions and polynomials ? Uses Matlab function: Legendre 02/13/02 12.540 Lec 03 12 Polynomials -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Polynomials: Degrees 2-5; Order 0 Cos(theta) Legendre Function TextEnd P2 b, P3 g, P4 r, P5 m 02/13/02 12.540 Lec 03 13 “Sectoral Harmonics” m=n -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1000 -800 -600 -400 -200 0 200 Sectoral harmonics: Degrees 2-5, order m=n Cos(theta) Legendre Function TextEnd P2 b, P3 g, P4 r, P5 m 02/13/02 12.540 Lec 03 14 Normalized -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1.5 -1 -0.5 0 0.5 1 1.5 Normalized Sectoral harmonics: Degrees 2-5, order m=n Cos(theta) Legendre Function TextEnd P2 b, P3 g, P4 r, P5 m 2 2m+1 (n+m)! (n?m)! 02/13/02 12.540 Lec 03 15 Surface harmonics ? To represent field on surface of sphere; surface harmonics are often used ? Be cautious of normalization. This is only one of many normalizations ? Complex notation simple way of writing cos(mλ) and sin(mλ) Y nm (θ,λ)= 2m+1 4π (n?m)! (n+m)! P nm (θ)e imλ 02/13/02 12.540 Lec 03 16 Surface harmonics Zonal ---- Terreserals ------------------------Sectorials 02/13/02 12.540 Lec 03 17 Gravitational potential ? The gravitational potential is given by: ?Where ρ is density, ? G is Gravitational constant 6.6732x10 -11 m 3 kg -1 s -2 (N m 2 kg -2 ) ? r is distance ? The gradient of the potential is the gravitational acceleration V = Gρ r dV ∫∫∫ 02/13/02 12.540 Lec 03 18 Spherical Harmonic Expansion ? The Gravitational potential can be written as a series expansion ? Cnm and Snm are called Stokes coefficients V =? GM r a r ? ? ? ? ? ? n=0 ∞ ∑ n P nm (cosθ) C nm cos(mλ)+S nm sin(mλ) [] m=0 n ∑ 02/13/02 12.540 Lec 03 19 Stokes coefficients ? The Cnm and Snm for the Earth’s potential field can be obtained in a variety of ways. ? One fundamental way is that 1/r expands as: 1 r = d' n d n+1 n=0 ∞ ∑ P n (cosγ) 02/13/02 12.540 Lec 03 20 1/r expansion ?Pn(cosγ) can be expanded in associated functions as function of θ,λ P γ d d' dM x 02/13/02 12.540 Lec 03 21 Spherical harmonics ? The Stokes coefficents can be written as volumn integrals ?C 00 = 1 if mass is correct ?C 10 , C 11 , S 11 = 0 if origin at center of mass ?C 21 and S 21 = 0 if Z-axis along maximum moment of inertia 02/13/02 12.540 Lec 03 22 Global coordinate systems ? If the gravity field is expanded in spherical harmonics then the coordinate system can be realized by adopting a frame in which certain Stokes coefficients are zero. ? What about before gravity field was well known?