03.potential Fields And Coordinate Systems

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12.540 Principles of Global Positioning Systems Spring 2008

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12.540 Principles of the Global Positioning System Lecture 03 Prof. Thomas Herring

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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 02/12/08

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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 02/12/08

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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 ) 02/12/08

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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 02/12/08

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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 02/12/08

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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 02/12/08

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• 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 02/12/08

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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

http://mathworld.wolfram.com/LegendrePolynomial.html 02/12/08

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Orthogonality 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

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Examples from Matlab • Matlab/Harmonics.m is a small matlab program to plots the associated functions and polynomials • Uses Matlab function: Legendre

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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

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-0.8

-0.6

-0.4

-0.2

0 0.2 Cos(theta)

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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

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-0.8

-0.6

-0.4

-0.2

0 0.2 Cos(theta)

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0.4

0.6

0.8

1

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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

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-0.8

-0.6

-0.4

-0.2

0 0.2 Cos(theta)

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0.4

0.6

0.8

1

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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) 02/12/08

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Surface harmonics

Courtesy of Dr. Svetlana V. Panasyuk, Dimensional Photonics, Inc. Used with permission.

Zonal ---- Terreserals ------------------------Sectorials 02/12/08

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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 02/12/08

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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

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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 02/12/08

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1/r expansion • Pn(cosγ) can be expanded in associated functions as function of θ,λ

x

P

dM d

d' γ

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Spherical harmonics • The Stokes coefficients can be written as volume 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 02/12/08

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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? 02/12/08

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Summary • Examined the spherical harmonic expansion of the Earth’s potential field. • Low order harmonic coefficients set the coordinate. – Degree 1 = 0, Center of mass system; – Degree 2 give moments of inertia and the orientation can be set from the directions of the maximum (and minimum) moments of inertia. (Again these coefficients are computed in one frame and the coefficients tell us how to transform into frame with specific definition.) Not actually done in practice.

• Next we look in more detail into how coordinate systems are actually realized.

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