《全球定位系统原理》（英文版）Lecture 3 Gravitational potential

12.540 Principles of the Global Positioning System Lecture 03 Prof. Thomas Herring 02/1302 12.540Lec03

02/13/02 12.540 Lec 03 1 12.540 Principles of the Global Positioning System Lecture 03 Prof. Thomas Herring

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 02/1302 12.540Lec03

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

Gravitational potential In spherical coordinates: need to solve 1 6 1 OV (r)+ (sine)+-22 ra r- since r- sin 0 an This is laplace s equation in spherical coordinates 02/1302 12.540Lec03

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 sin θ ∂ ∂θ(sin θ ∂V ∂θ ) + 1 r 2 sin 2 θ ∂ 2 V ∂λ 2 = 0

Solution to gravity potential The homogeneous form of this equation is a"" partial differential equation In spherical coordinates solved by separation oT variables, r=radius 入= longitude andθ=co- latitude V(r,,)=R()g(6)h() 02/1302 12.540Lec03

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)

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(mn) and cos(mn)where m is an integer The colatitude dependence is more difficult to solve 02/1302 12.540Lec03

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(m λ) and cos(m λ) where m is an integer • The colatitude dependence is more difficult to solve

Colatitude dependence Solution for colatitude function generates Legendre polynomials and associated functions The polynomials occur when m=0 in n dependence. t=cos(0) Pn()= 2 nl dt 02/1302 12.540Lec03

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 ) Pn ( t ) = 1 2 n n! d n d t n ( t 2 −1) n

Legendre Functions Low order functions P()=t Arbitrary n P2()=(312-1) 2 values are P3()=(5t3-3) generated by recursive P24(t)=(35t-3012+3 algorithms 02/1302 12.540Lec03

02/13/02 12.540 Lec 03 7 Legendre Functions • Low order functions. Arbitrary n values are generated by recursive algorithms Po ( t ) = 1 P1( t ) = t P2 ( t ) = 1 2 (3 t 2 −1) P3 ( t ) = 1 2 (5 t 3 − 3 t ) P4 ( t ) = 1 8 (3 5 t 4 − 30 t 2 +3)

Associated Legendre Functions The associated functions satisfy the following equation Pn(t)=(-1y(1-2)m2 d 2P() The formula for the polynomials Rodriques formula, can be substituted 02/1302 12.540Lec03

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 Pnm ( t ) = ( −1) m (1 − t 2 ) m/2 d m d t m Pn ( t )

Associated functions P0(t)=1 Pnm(t): n is called P0()=t degree; m is order P1()=-(1-t2)2 m0 http://mathworld.wolframcom/legendrepoLynomial.html 02/1302 12.540Lec03

02/13/02 12.540 Lec 03 9 Associated functions • Pnm(t): n is called degree; m is order • m0 P00 ( t ) = 1 P10 ( t ) = t P11 ( t ) = −(1 − t 2 )1/ 2 P20 ( t ) = 1 2 (3 t 2 −1) P21 ( t ) = − 3 t(1 − t 2 )1/ 2 P22 ( t ) = 3( 1 − t 2 ) http://mathworld.wolfram.com/LegendrePolynomial.html

Ortogonality conditions The Legendre polynomials and functions are orthogonal P()P(t)dt=2 2n+1 Pim(t)pm(t)dt 2(n+m)! 2n+1(n-m) nn 02/1302 12.540Lec03

02/13/02 12.540 Lec 03 10 Ortogonality conditions • The Legendre polynomials and functions are orthogonal: Pn' ( t ) −1 1 ∫ Pn ( t )dt = 2 2 n + 1 δ n'n Pn'm ( t ) −1 1 ∫ Pnm ( t )dt = 2 2 n + 1 ( n + m)! ( n − m)! δ n'n