Gravitational Potential - Principles of the Global Positioning System

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 2 Gravitational potential • In spherical coordinates: need to solve 1 ∂2 1 ∂ ∂V 1 ∂ 2V (rV ) + 2 (sin θ ) + 2 2 =0 2 2 r ∂r r sin θ ∂θ ∂θ r sin θ ∂λ • This is Laplace’s equation in spherical coordinates 3 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 ) 4 Solution in spherical coordinates • The radial dependence of form rn 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 5 Colatitude dependence • Solution for colatitude function generates Legendre polynomials and associated functions. • The polynomials occur when m=0 in λ dependence. t=cos(θ) 1 dn 2 n Pn (t) = n (t −1) 2 n! dt n 6 Legendre Functions Po (t) = 1 P1 (t) = t 1 2 P2 (t) = (3t −1) 2 1 3 P3 (t) = (5t − 3t) 2 1 P4 (t) = (35t 4 − 30t 2+3) 8 • Low order functions. Arbitrary n values are generated by recursive algorithms 7 Associated Legendre Functions • The associated functions satisfy the following equation m d Pnm (t) = (−1)m (1− t 2 )m /2 m Pn (t) dt • The formula for the polynomials, Rodriques’ formula, can be substituted 8 Associated functions P00 (t) = 1 P10 (t) = t P11 (t) = −(1− t 2 )1/2 1 2 P20 (t) = (3t −1) 2 P21 (t) = −3t(1− t 2 )1/2 P22 (t) = 3(1− t 2 ) • Pnm(t): n is called degree; m is order • m<=n. In some areas, m can be negative. In gravity formulations m=>0 9 Ortogonality conditions • The Legendre polynomials and functions are orthogonal: 1 ∫ −1 Pn ' (t)Pn (t)dt = 2 δn 'n 2n +1 2 (n + m)! ∫ Pn'm (t)Pnm (t)dt = 2n +1 (n − m)!δn'n −1 1 10 Examples from Matlab • Matlab/Harmonics.m is a small matlab program to plots the associated functions and polynomials • Uses Matlab function: Legendre 11 Polynomials Polynomials: Degrees 2-5; Order 0 1 0.8 0.6 Legendre Function 0.4 0.2 0 -0.2 TextEnd -0.4 -0.6 -0.8 P2 b, P3 g, P4 r, P5 m -1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 Cos(theta) 0.4 0.6 0.8 1 12 “Sectoral Harmonics” m=n Sectoral harmonics: Degrees 2-5, order m=n 200 0 Legendre Function -200 -400 TextEnd -600 -800 P2 b, P3 g, P4 r, P5 m -1000 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 Cos(theta) 0.4 0.6 0.8 1 13 Normalized Normalized Sectoral harmonics: Degrees 2-5, order m=n 1.5 1 Legendre Function 0.5 2 (n + m)! 2m +1 (n − m)! 0 TextEnd -0.5 -1 P2 b, P3 g, P4 r, P5 m -1.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 Cos(theta) 0.4 0.6 0.8 1 14 Surface harmonics • To represent field on surface of sphere; surface harmonics are often used 2m +1 (n − m)! Ynm (θ , λ ) = Pnm (θ )eimλ 4 π (n + m)! • Be cautious of normalization. This is only one of many normalizations • Complex notation simple way of writing cos(mλ) and sin(mλ) 15 Surface harmonics Zonal ---- Terreserals ------------------------Sectorials 16 Gravitational potential • The gravitational potential is given by: V= ∫∫∫ Gρ dV r • Where ρ is density, • G is Gravitational constant 6.6732x10-11 m3kg-1s-2 (N m2kg-2) • r is distance • The gradient of the potential is the gravitational acceleration 17 Spherical Harmonic Expansion • The Gravitational potential can be written as a series expansion ∞ n n GM ⎛ a ⎞ V =− ⎜ ⎟ ∑ r n=0 ⎝ r ⎠ ∑P nm (cosθ )[Cnm cos(mλ ) + Snm sin(mλ )] m=0 • Cnm and Snm are called Stokes coefficients 18 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: ∞ d'n 1 = ∑ n+1 Pn (cos γ ) r n=0 d 19 1/r expansion • Pn(cosγ) can be expanded in associated functions as function of θ,λ x P dM d d' γ 20 Spherical harmonics • The Stokes coefficents can be written as volumn integrals • C00 = 1 if mass is correct • C10, C11, S11 = 0 if origin at center of mass • C21 and S21 = 0 if Z-axis along maximum moment of inertia 21 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? 22