03.potential fields and coordinate systems

8/14/2019 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 theGlobal Positioning System

Lecture 03Prof. Thomas Herring


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Review

In last lecture we looked at conventionalmethods of measuring coordinates Triangulation, trilateration, and leveling Astronomic measurements using

external bodies Gravity field enters in these

determinations


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Gravitational potential

In spherical coordinates: need to solve

This is Laplaces equation in sphericalcoordinates

1

r

2

r 2(rV ) + 1

r 2 sin

(sin

V

) + 1

r 2 sin 2

2V

2=0


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Solution to gravity potential

The homogeneous form of this equationis a classic partial differential equation. In spherical coordinates solved by

separation of variables, r=radius,=longitude and =co-latitude

V (r , , ) = R(r )g( )h( )


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Solution in sphericalcoordinates

The radial dependence of form rn

or r-n

depending on whether inside or outsidebody. N is an integer

Longitude dependence is sin(m ) andcos(m ) where m is an integer

The colatitude dependence is moredifficult to solve


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Colatitude dependence

Solution for colatitude functiongenerates Legendre polynomials andassociated functions.

The polynomials occur when m=0 in dependence. t=cos( )

Pn (t ) =1

2 n n!d n

dt n(t 2 1)n


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Legendre Functions

Low orderfunctions.Arbitrary nvalues aregenerated by

recursivealgorithms

Po (t ) =1P1(t ) = t

P2 (t ) = 12 (3t 2 1)

P3 (t ) =1

2

(5 t 3 3t )

P4 (t ) =18

(35 t 4 30 t 2+3)


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Associated LegendreFunctions

The associated functions satisfy thefollowing equation

The formula for the polynomials,Rodriques formula, can be substituted

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

dt mPn (t )


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Orthogonality conditions

The Legendre polynomials andfunctions are orthogonal:

Pn '(t )1

1

Pn (t )dt = 22n +1 n 'n

Pn 'm (t )1

1

Pnm (t )dt =

2

2n +1(n +m)!(n m)!

n 'n


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Examples from Matlab

Matlab/Harmonics.m is a small matlabprogram to plots the associatedfunctions and polynomials

Uses Matlab function: Legendre
http://../Matlab/Harmonics.mhttp://../Matlab/Harmonics.m


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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

1Polynomials: Degrees 2-5; Order 0

Cos(theta)

L e g e n

d r e

F u n c

t i o n

TextEnd

P2 b, P3 g, P4 r, P5 mP2 b, P3 g, P4 r, P5 m


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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

200Sectoral harmonics: Degrees 2-5, order m=n

Cos(theta)

L e g e n

d r e

F u n c

t i o n

TextEnd

P2 b, P3 g, P4 r, P5 mP2 b, P3 g, P4 r, P5 m


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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

1.5Normalized Sectoral harmonics: Degrees 2-5, order m=n

Cos(theta)

L e g e n

d r e

F u n c

t i o n

TextEnd

P2 b, P3 g, P4 r, P5 m

2

2m +1(n +m)!(n m)!


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Surface harmonics To represent field on surface of sphere;

surface harmonics are often used

Be cautious of normalization. This is only oneof many normalizations

Complex notation simple way of writingcos(m ) and sin(m )

Y nm ( , ) =2m +1

4 (n m)!(n +m)!Pnm ( )e

im


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Surfaceharmonics

Zonal ---- Terreserals ------------------------Sectorials

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


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Gravitational potential The gravitational potential is given by:

Where is density, G is Gravitational constant 6.6732x10 -11

m3kg -1s -2 (N m 2kg -2) r is distance The gradient of the potential is the

gravitational acceleration

V = G r

dV


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Spherical HarmonicExpansion

The Gravitational potential can be written asa series expansion

Cnm and Snm are called Stokes coefficients

V = GM

r ar

n=0

n

Pnm (cos ) C nm cos( m ) +Snm sin( m )[ ]m=0

n


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Stokes coefficients

The Cnm and Snm for the Earthspotential field can be obtained in avariety of ways.

One fundamental way is that 1/rexpands as:

1r

= d 'n

d n+1n=0

Pn (cos )


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1/r expansion Pn(cos ) can be expanded in

associated functions as function of ,

P

dd'

dMx


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Global coordinate systems

If the gravity field is expanded inspherical harmonics then the coordinatesystem can be realized by adopting aframe in which certain Stokescoefficients are zero.

What about before gravity field was wellknown?


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Summary Examined the spherical harmonic expansion of theEarths 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 computedin one frame and the coefficients tell us how to transform intoframe with specific definition.) Not actually done in practice.

Next we look in more detail into how coordinatesystems are actually realized.

03.potential fields and coordinate systems

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