1 12.540 Principles of the Global Positioning System Lecture 03 Prof. Thomas Herring Room 54-820A; 253-5941 [email protected]http://geoweb.mit.edu/~tah/12.540 02/13/13 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
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12.540 Principles of the Global Positioning System Lecture 03geoweb.mit.edu/~tah/12.540/12.540_Lec03.pdf · 02/13/13! 12.540 Lec 03! 27! Summary" • Examined the spherical harmonic
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12.540 Principles of the Global Positioning System
Gravitational potential"• The gravitational potential is given by:"
• 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"
€
V =GρrdV∫∫∫
02/13/13 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 = −GMr
ar# $ % & ' (
n=0
∞
∑n
Pnm (cosθ) Cnm cos(mλ)+ Snm sin(mλ)[ ]m=0
n
∑
10
02/13/13 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:"
• Where d’ is the distance to dM and d is the distance to the external point, γ is the angle between the two vectors (figure next slide)"€
1r
=" d n
dn+1n=0
∞∑ Pn (cosγ)
02/13/13 12.540 Lec 03 20
1/r expansion"• Pn(cosγ) can be expanded in
associated functions as function of θ,λ"
P
γ
dd'
dM
x
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Computing Stoke coefficients"• Substituting the expression for 1/r and
converting γ to co-latitude and longitude dependence yields:"
12.540 Lec 03 21 €
Pn (γ) =4π2n +1
Ynm*
m=0
n∑ ( % θ , % λ )Ynm (θ,λ)
V =GdM
r∫∫∫ = 4π dM
2n +1% d n
dn+1 Ynm*
m=0
n∑ ( % θ , % λ )Ynm (θ,λ)
n=0
∞∑∫∫∫
The integral and summation can be reversed yielding integrals for the Cnm and Snm Stokes coefficients.
02/13/13
Low degree Stokes coefficients"
• By substituting into the previous equation we obtain:"
!!!!
€
C10 =GM " z!dM C11 =GM∫∫∫ " x!dM∫∫∫
S11 =GM " y!dM∫∫∫
!!!!
€
C20 =GM
22z2∫∫∫ − x2 − y2dM
C21 =GM xzdM S21 =GM yzdM∫∫∫∫∫∫
C22 =GM4
x2 − y2dM S22 =GM2
xydM∫∫∫∫∫∫
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02/13/13
Moments of Inertia"• Equation for moments of inertia are:"
• The diagonal elements in increasing magnitude are often labeled A B and C with A and B very close in value (sometimes simply A and C are used)"
!!!!
€
I =
y2 + z2dM∫∫∫ xydM∫∫∫ xzdM∫∫∫
xydM∫∫∫ z2 + x2dM∫∫∫ yzdM∫∫∫
xzdM∫∫∫ yzdM∫∫∫ x2 + y2dM∫∫∫
#
$
% % % %
&
'
( ( ( (
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02/13/13
Relationship between moments of inertia and Stokes coefficients"
• With a little bit of algebra it is easy to show that:"
€
C20 =GM(A + B2
−C)
C22 =14GM(B − A)
S22 =12GMI12
C21 S21 are related to I13 and I23
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02/13/13 12.540 Lec 03 25
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"
02/13/13 12.540 Lec 03 26
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?"
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02/13/13 12.540 Lec 03 27
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."