5 earth gravitation
TRANSCRIPT
Earth Gravitation
SOLO HERMELIN
Updated: 17.01.13
16.12.14
1
http://www.solohermelin.com
Table of Content
SOLO
2
Earth Gravitation
Introduction to Gravitation
Gravitation of a Point Mass M (Netwon Gravitation Law)
Gravitation of a Distribution of Mass Defined by the Density ( )Srρ
Gravitation of a Uniform Distribution of Mass in a Spherical Volume
( )( ).ConstrS =ρ
Physical Meaning of the Low Degree and Order SHCs
Reference Earth Model
Clairaut's theoremMac Cullagh’s Approximation
World Geodetic System (WGS 84)
Reference Ellipsoid
Air Vehicle in Ellipsoidal Earth Atmosphere
References
Gravity
Nicolaus Copernicus (1473 – 1543)“De Revolutionibus Orbium Coelestium” (On the Revolutions of the Heavenly Spheres”, Nuremberg 1543)
SOLO
Gravity
Kepler’s Laws of Planetary Motion
From 1601 to 1606 Kepler tried to fit various Geometric Curves to Tycho Brache’s Data on Mars Orbit. The result is“Kepler’s Law of Planetary Motion”
Tycho Brahe(1546-1601)
Johaness Kepler (1571-1630)
SOLO
KEPLER’S LAWS OF PLANETARY MOTION 1609-1619
•FIRST LAW THE ORBIT OF EACH PLANET IS AN ELLIPSE, WITH THE SUN AT A FOCUS.
b
a
•SECOND LAW THE LINE JOINING THE PLANET TO THE SUN SWEEPS OUT EQUAL AREAS IN EQUAL TIME.
a dAh
dt2=
b
•THIRD LAW THE SQUARE OF THE PERIOD OF A PLANET IS PROPORTIONAL TO THE CUBE OF ITS MEAN DISTANCE FROM THE SUN.
a 2/322a
GMGM
abTP
ππ ==
b
GravitySOLO
GALILEO GALILEI (1564-1642( “DISCOURSES AND MATHEMATICAL DEMONSTRATIONS CONCERNING TWO NEW SCIENCES ” 1636
THE FIRST TWO CHAPTERS DEAL WITH STRENGTH OF MATERIALS . THIS IS ESSENTIALLY THE FIRST NEW SCIENCE.
THE THIRD AND FOURTH SUBJECT IS :
• UNIFORM MOTION WITH CONSTANT ACCELERATION AND MOTION OF PROJECTILES
•THE LAW OF INERTIA
•THE COMPOSITION OF MOTIONS ACCORDING TO VECTOR ADDITION (“GALILEAN TRANSFORMATION”) AND
•THE STUDY OF UNIFORMLY ACCELERATED MOTION
GravitySOLO
NEWTON’S LAWS OF MOTION
“THE MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY”1687
• FIRST LAW EVERY BODY CONTINUES IN ITS STATE OF REST OR OF UNIFORM MOTION IN A STRAIGHT LINE UNLESS IT IS COMPELLED TO CHANGE THAT STATE BY FORCES IMPRESSED UPON IT.
• SECOND LAW THE RATE OF CHANGE OF MOMENTUM IS PROPORTIONAL TO THE FORCE IMPRESSED AND IS IN THE SAME DIRECTION AS THAT FORCE.
• THIRD LAW TO EVERY ACTION THERE IS ALWAYS OPPOSED AN EQUAL REACTION.
GravitySOLO
NEWTON’S LAW OF UNIVERSAL GRAVITATION
ANY TWO BODY ATTRACT ONE ONOTHER WITH A FORCE PROPORTIONAL TO THE PRODUCT OF THE MASSES AND INVERSLY PROPORTIONAL TO THE SQUARE OF THE DISTANCE BETWEEN THEM.
THE UNIVERSAL GRAVITATIONAL CONSTANT INSTANTANEUS PROPAGATION OF THE FORCE ALONG THE DIRECTION BETWEEN THE MASES (“ACTION AT A DISTANCE”).
M m
GravitySOLO
THE LAGEOS SATELLITE MONITORSITS POSITION RELATIVE TO THE EARTHUSING REFLECTED LIGHT.
THE TORSION BALANCE EXPERIMENTS:* HENRY CAVENDISH 1797 * RESEARCH GROUP OF UNIVERSITY OF WASHINGTON, SEATTLE (EOTWASH)
CAVENDISH EXPERIMENTS HAVE NOT BEEN ABLE TO TEST THE GRAVITATIONAL FORCE AT SEPARATIONSSMALLER THAN A MILLIMETER.
IF THERE ARE n EXTRA DIMENSIONS (TO THE 3 SPACE + 1 TIME) CURLED UP WITH DIAMETERS R, AT SCALES
SMALLER THAN R THE GENERALIZED NEWTON POTENTIAL WILL BE: Rrr
MGrV
nn <<=+1)(
Gravity
Henry Cavendish(1731 – 1810)
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SOLO
10
SOLOEarth Gravitation
Gravitation of a Point Mass M (Netwon Gravitation Law)
SFSF
SF
SF
SFFM rrr
rr
rr
rr
MmGgmF
−=
−−
−−== :2
According to Newton the Gravitation Force of Point Mass M on the Point mass mF is given by
MF
The acceleration of Mass m due to Gravity is
( ) ( ) rrfr
r
r
MGrg
=
−=
2
( ) ( ) ( )r
r
r
rfr
r
rfrf
∂∂=∇
∂∂=∇
Any function of the form has the property that( ) rrf
Because of this 1.we can write2.we have
( ) ( )rUrg −∇=
( )( )
( )( )
( )
( )( )
( )
( )( )
( )
( )( ) ( )[ ]1
1111
rUrUmrUdmrdrUmrdrgmrdrF F
rB
rA
F
rB
rA
rB
rA
rB
rA
−=−=⋅∇−=⋅=⋅ ∫∫∫∫
The Work necessary to move the Mass mF in the Gravity Field of Point Mass M,from point to point , is not a function of the trajectory chosen but onthe values of the scalar function U (called Potential) at those two points.
( )rA ( )1rB
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11
Earth Gravitation
Gravitation of a Point Mass M (Netwon Gravitation Law)
( ) ( ) ( )( )
( )
( )
( )
( )
( )
r
MG
r
MGrd
r
MGrd
r
r
r
MGrdrgrUrU
f
rB
rA
rB
rA
rB
rA
f
fff
−=−=⋅
−=⋅=− ∫∫∫ 22
The Gravitation Potential U of a Point Mass M is given by
The Gravitation Equi-Potential Surface of a Point Mass M is Spherical Surface centered at M location.
Gravitation of a Distribution of Mass Defined by the Density ( )Srρ
The Gravitation Potential of the Distribution is obtaining byIntegrating the Gravitation Potentials of the Point Masses dmover the Volume V
( ) ( ) ( )∫∫ −
−=−
−=V
SS
S
M S
S Vdrr
rG
rr
rmdGrU
ρ
If we take the reference rf → ∞ we get 0. ==fr
MGConst
( )r
MGrU
−=
( ) .Constr
MGrU +−=
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SOLO
12
Earth Gravitation
( ) S
SSS
rrMd
r
r
r
rr
G
rr
MdGrUd >
+
−
−=−
−=2
cos21
1
γ
( ) ( ) SSSSSSSS
onDistributiDensity
SSS ddrdrrVdrMd ϕθθθϕρθϕρ sin,,,, 2==
( ) ( )S
r SS
SSSSSSSS rr
r
r
r
r
ddrdrr
r
GrU
S S S
>
+
−
−= ∫ ∫ ∫ϕ θ γ
ϕθθθϕρ2
2
cos21
sin,,
( ) Sn
n
n
SSS rrPr
r
r
r
r
r >
=
+
− ∑
∞
=
−
0
2/12
coscos21 γγ
Pn (u) are Lagrange Polynomials, of order n, given explicitly by Rodrigues’ Formula:
( ) ( )n
nn
kn ud
ud
nuP
1
!2
1 2 −=
Gravitation of a Distribution of Mass Defined by the Density ( )Srρ
( ) ( ) ( ) ( ) 2/13&,1 2210 −=== uuPuuPuPExamples:
Using Spherical Coordinates (rS, φS, θS)
The Potential Differential of any distribution of mass is given by:
SOLO
13
Earth Gravitation
( ) ( ) ( ) S
r Md
SSSSSSSSn
n
n
S rrddrdrrPr
r
r
GrU
S S S
>
−= ∫ ∫ ∫ ∑
∞
=ϕ θ
ϕθθθϕργ
sin,,cos 2
0
From Spherical Trigonometry:
( )SSS ϕϕθθθθγ −+= cossinsincoscoscos
According to Addition Theorem for Spherical Harmonics:
( ) ( ) ( ) ( )( ) ( )[ ] ( ) ( )∑
=
−+−+=
n
mS
mn
mnSSnnn PPm
mn
mnPPP
1
coscoscos!
!2coscoscos θθϕϕθθγ
( ) ( ) ( )uPud
duuP nm
mmm
n
2/21: −=
Where Pnm(u) is the Associated Lagrange Function of the First Kind of Degree n
and Order m, given by:
Gravitation of a Distribution of Mass Defined by the Density ( )Srρ
( ) ( )uPuP nn =0 ( ) ( ) ( ) ( )222
2
121
2 13,13 ttPtttP −=−=They satisfy:
The Potential of any distribution of mass is given by:
See Presentation on“Legendre Functions”
SOLO
14
Earth Gravitation
The Potential of any distribution of mass is given by:
( ) ( ) ( )
( ) ( ) Sn r Md
SSSSSSSSn
n
S
k
S
r Md
SSSSSSSSk
n
n
S
rrddrdrrPa
r
r
a
r
G
rrddrdrrPr
r
r
GrU
S S S
S S S
>
−=
>
−=
∑ ∫ ∫ ∫
∫ ∫ ∫ ∑∞
=
∞
=
0
2
2
0
sin,,cos
sin,,cos
ϕ θ
ϕ θ
ϕθθθϕργ
ϕθθθϕργ
According to Addition Theorem for Spherical Harmonics:
( ) ( ) ( ) ( )( ) ( )[ ] ( ) ( )∑
=
−+−+=
n
mS
mn
mnSSnnn PPm
mn
mnPPP
1
coscoscos!
!2coscoscos θθϕϕθθγ
Using those results we obtain:
( ) 1cos0 =γP
( ) ( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫
∑ ∑ ∫ ∫ ∫
∑ ∫ ∫ ∫∫ ∫ ∫
∞
=
∞
= =
∞
=
+−−
+−−
−
−=
1
2
1 0
2
1
22
1
0
0
sin,,sincossincos!
!2
sin,,coscoscoscos!
!2
sin,,coscossin,,cos
n r
SSSSSSSSSSn
n
Sn
n
n
n
m r
SSSSSSSSSSm
n
n
Smn
n
n r
SSSSSSSSSn
n
Sn
n
M
r Md
SSSSSSSSS
S S S
S S S
S S SS S S
ddrdrrmPa
rmP
r
a
mn
mn
r
G
ddrdrrmPa
rmP
r
a
mn
mn
r
G
ddrdrrPa
rP
r
a
r
GddrdrrP
a
r
r
GrU
ϕ θ
ϕ θ
ϕ θϕ θ
ϕθθθϕρϕθϕθ
ϕθθθϕρϕθϕθ
ϕθθθϕρθθϕθθθϕργ
Gravitation of a Distribution of Mass Defined by the Density ( )Srρ
a – any reference length
SOLO
15
Earth Gravitation
( ) ( ) ( ) ( ) ( )( )
+
+
+−= ∑∑∑
∞
= =
∞
= 1 110 sincoscoscos1
n
n
mnmnm
mn
n
nnn
n
mSmCPr
aCP
r
a
r
MGrU ϕϕθθ
( ) ( )∫ ∫∫
==
S S Sr Md
SSSSSSSSSn
n
Snn ddrdrrP
a
r
MCC
ϕ θ
ϕθθθϕρθ
sin,,cos1
: 20
( )( ) ( ) ( )
( ) ( )∫ ∫ ∫
+−=
S S Sr Md
SSSSSSSSS
SS
mn
n
S
nm
nm ddrdrrm
mP
a
r
mn
mn
S
C
ϕ θ
ϕθθθϕρϕϕ
θ
sin,,sin
coscos
!
!2: 2
Gravitation of a Distribution of Mass Defined by the Density ( )Srρ
The Potential of any distribution of mass is given by:
SOLO
16
Earth Gravitation
(1) A tremendous simplification results if the mass distribution is symmetric about the z axis, i.e. ρ is a function only of rS and θS. Since
( ) ( ) ,2,10cossin2
0
2
0
=== ∫∫ jdjdj SSSS
ππ
ϕϕϕϕ
the coefficients Cmn and Smn vanish identically.
(2) In addition to axial symmetry, the origin of coordinates coincides with the center of mass, the constant C1 is identically zero.
( ) ( ) 0cos1
sin,cos1
: 2
cos
11 =
=
= ∫∫ ∫ ∫
m
SS
r md
SSSSSSSSS md
a
r
MddrdrrP
a
r
MC
S S S S
θϕθθθρθϕ θ θ
C1 is proportional to the first moment of the mass M with respect to the xy plane.
Gravitation of a Distribution of Mass Defined by the Density ( )Srρ
In this case
( ) ( ) 01
:cos1 nnn
nn
n
CJJPr
a
r
MGrU −=
−−= ∑
∞
=
θ
SOLO
17
Earth Gravitation
(3) Finally, if the mass is distributed in homogeneous concentric layers, i.e. ρ is a function only of rS , then Cn vanishes identically for all n
( ) ( ) 0sincos:2
0
0
00
22
=
= ∫∫∫
+ ππ
ϕθθθρ SSSSk
R
SS
n
Sn ddPrdr
a
r
M
aC
Gravitation of a Distribution of Mass Defined by the Density ( )Srρ
( )r
MGrU =In this case
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SOLO
18
Earth Gravitation
Gravitation of a Uniform Distribution of Mass in a Spherical Volume( )( ).ConstrS =ρ
For a Static Spherical Volume with an Uniform DistributedMass the Gravitation Potential at a distance r > R is given by
( )r
MG
r
RGVd
rrGrU S
V
SSS
−=−=−
−= ∫3/41 3πρρ
The Gravitation Potential for a Static Spherical Volume with an Uniform Distributed Mass is equivalent to a Point MassGravitation Potential concentrated at the Center of the Sphere. The Equi-Potential Surfaces will be Concentric Spherical Surfaces outside the Spherical Mass.
SOLO
19
Earth Gravitation
The Potential of the Reference Ellipsoid is given by:
where:UG – Gravitational Potential (m2/s2)GM – Earth’s Gravitational Constantr - Distance from the Earth’s Center of MassrS - Distance from the Earth’s Center of Mass to d m
Gravitation of a Uniform Distribution of Mass in a Ellipsoid Volume( )( ).Constr =ρ
( )Srr
mdGrUd
−=
Typically the Potential is expanded in a series. This can be done in two ways, which lead to the same result:1.By expanding the term and integrate the result term by term.2. By writing the Potential as solution of Lalace’s Equation using Spherical Harmonics.
Srr −
1
SOLO
20
Earth Gravitation
The Potential Equation of Gravity Field
where:UG – Gravitational Potential (m2/s2)GM – Earth’s Gravitational Constantr - Distance from the Earth’s Center of MassrS - Distance from the Earth’s Center of Mass to dm
( ) ( )
( ) ( )
( )rG
VdrrrG
Vdrr
rGrU
S
S
V
SSS
V
SS
SS
ρπ
δπρ
ρ
4
4
122
−=
−−=
−
∇−=∇
∫∫∫
∫∫∫
( ) ( )∫∫∫∫ −−=
−−=
SV S
SS
M S rr
VdrG
rr
mdGrU
ρStart with:
Inside the mass M we have:
( ) ( )rGrU ρπ42 −=∇Poisson Equation:
( ) 02 =∇ rULaplace Equation:
Outside the mass M we have:
Siméon Denis Poisson
1781-1840
Pierre-Simon Laplace(1749-1827)
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SOLO
21
Earth Gravitation
Physical Meaning of the Low Degree and Order SHCs
Gravitation of a Distribution of Mass Defined by the Density ( )Srρ
As pointed out in the previous slides the first term of the Spherical Harmonic Coefficients (SHC) of Gravity Potential is equal to the Potential for Spherical Mass. The remaining terms then represent the Gravitational Potential due to Non-spherical, Non-uniform Mass.
( ) ( ) ( ) ( ) ( )( )
+
+
+−= ∑∑∑
∞
= =
∞
= 1 110 sincoscoscos1
n
n
mnmnm
mn
n
nnn
n
mSmCPr
aCP
r
a
r
MGrU ϕϕθθ
We have (see Figure)
=
=
S
SS
SS
S
s
s
s
S r
z
y
x
r
θϕθϕθ
cos
sinsin
cossin
( ) 0&1
cos1
: 10
cos
1
1
110 ===
== ∫∫ S
a
zMdz
MaMdP
a
r
MCC G
M
S
M
SS
S
θ
θ
( )
=
=
=
=
∫∫ ∫ ay
axMd
y
x
MaMd
a
r
MMdP
a
r
MS
C
G
G
M S
S
M M S
SS
S
S
SS
S
/
/2
2
sin
cossin
2
sin
coscos
21
11
1
11
11
ϕϕ
θϕϕ
θ
( ) ( ) 2
121
1 1 ttP −=where we used Legendre Polynomials. GGG zyx
,, are the coordinates of the Mass Center of Gravity
SOLO
22
Earth Gravitation
Physical Meaning of the Low Degree and Order SHCs
Gravitation of a Distribution of Mass Defined by the Density ( )Srρ
( ) ( ) ( ) ( ) ( )( )
+
+
+−= ∑∑∑
∞
= =
∞
= 1 110 sincoscoscos1
n
n
mnmnm
mn
n
nnn
n
mSmCPr
aCP
r
a
r
MGrU ϕϕθθ
=
=
S
SS
SS
S
s
s
s
S r
z
y
x
r
θϕθϕθ
cos
sinsin
cossin
+−−=
−=
2
1
2
312220
SSSS
SSSSSS
yyxxzzzzrr
III
MaII
MaC
( )∫
=
M
SS MdPa
r
MS
C
0
1cos
2 02
2
20
20 θ
( ) ( )
+−++=
+−=−−−=−==2
31
2
1
2
3
2
1cos3coscos
2222
2
222
22
22222
2
2
22
2022
2SS
SSSSS
SSSSSSS
SS
SS yx
zyxa
yxz
aa
zyxz
a
rP
a
rP
a
r θθθ
( ) ∫∫∫
=
=
=
=
M zy
zx
SS
SS
M S
SSSS
M S
SS
S
MaI
MaIMd
zy
zx
MaMdr
MaMdP
a
r
MS
C
SS
SS
2
2
2
2
2
12
2
21
21
/
/1
sin
coscossin3
3
1
sin
coscos
!3
12
ϕϕ
θθϕϕ
θ
( )( ) ( )
( )∫∫∫
−=
−=
−⋅
=
=
M yx
xxyy
SS
SS
M SS
SSSS
M S
SS
S
MaI
MaIIMd
yx
yx
MaMdr
MaMdP
a
r
MS
C
SS
SSSS
2
222
2
2222
2
22
2
22
22
2/
4/
24
1
cossin2
sincossin3
34
1
2sin
2coscos
!4
!12
ϕϕϕϕ
θϕϕ
θ
( ) ( ) ( ) ( )222
2
121
2 13,13 ttPtttP −=−=where we used
( ) ( ) ( ) ( ) ( )SSSSSSSSSSSSSS zzyyxx
M
SSSrr
M
SSzz
M
SSyy
M
SSxx IIIMdzyxIMdyxIMdzxIMdzyI ++=++=+=+=+= ∫∫∫∫ 2
1::,:,: 222222222
Define Moments of Inertia
SOLO
23
Earth Gravitation
Physical Meaning of the Low Degree and Order SHCs
Gravitation of a Distribution of Mass Defined by the Density ( )Srρ
SOLO
24
Earth Gravitation
Physical Meaning of the Low Degree and Order SHCs
Gravitation of a Distribution of Mass Defined by the Density ( )Srρ
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SOLO
25
Earth Gravitation
Earth is a Rotating Non-Spherical Body, with a slightly non-uniform mass distribution. The force acting on an external mass is due to Gravitation and Centrifugal Accelerations. To find a Model for the Gravitation Acceleration the Earth is approximated by an Ellipsoid
The flattering of the Earth was already discovered by the end of the 18th century.It was noticed that the distance between a degree of Latitude as measured, for instance with a sextant, differs from that expected from a Sphere:RE (θ1 – θ2) ≠ RE dθ, with RE the radius of the Earth, θ1 and θ2 two different Latitudes
f - the flattening of a meridian section of the Earth, defined as:
a
baf
−=:
R = 6,371.000 kma = 6,378.136 kmb = 6,356.751 km
Reference Earth Model
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SOLO
26
Reference Ellipsoid
Meridian Ellipse Equation: 12
2
2
2
=+b
z
a
p
Slope of the Normal to Ellipse:
2
2
tanbp
az
zd
pd =−=φ
The Slope of the Geocentric Line to the same point
p
zO =φtan
OO RzRp φφ sincos ==
Deviation Angle between Geographic and GeodeticAt Ellipsoid Surface
Ob
a φφ tantan2
2
=
= −
Ob
a φφ tantan2
21 ( ) ( ) φφφφ tan1tan1tantan 22
2
2
fea
bO −=−==
a
baf
−=:
22
222 2: ff
a
bae −=−=
Earth Gravitation
SOLO
27
Reference Ellipsoid
Meridian Ellipse Equation:
Oφφδ −=
12
2
2
2
=+b
z
a
p
Slope of the Normal to Ellipse:
2
2
tanbp
az
zd
pd =−=φ
The Slope of the Geocentric Line to the same point
p
zO =φtan
−=
−+
−
=+
−=
+−= 1
11
1
1tantan1
tantantan
2
2
2
2
2
2
2
2
2
22
22
2
2
b
a
a
zp
ap
pa
ba
pz
bpazpz
bpaz
O
O
φφφφδ
OO RzRp φφ sincos ==
( )
( ) ( )OO
f
O fba
R
b
ba
a
baR
ba
ba φφφδ 2sin2sin2
tan2sin2
tan
1
2
11
1222
221 ≈
+
−=
−=
≈≈<<
−−
Deviation Angle between Geographic and GeodeticAt Ellipsoid Surface
Earth Gravitation
SOLO
28
Reference Ellipsoid
For a point at a Height h near the Ellipsoid the value of δ must be corrected:
u−= 1δδ
From the Law of Sine we have:
Deviation Angle between Geographic and GeodeticAt Altitude h from Ellipsoid Surface
( ) R
h
hR
huu ≈+
≈=− 11 sin
sin
sin
sin
δδπ
Since u and δ1 are small: 1δR
hu ≈
The corrected value of δ is:
( )OfR
h
R
hu φδδδ 2sin11 11
−=
−=−=
Therefore:
( )OO fR
h φλδφφ 2sin1
−+=+=
Earth Gravitation
SOLO
29
World Geodetic System (WGS 84)
where λ – Longitude e – Eccentricity = 0.08181919
Reference Earth Model
In Earth Center Earth Fixed Coordinate –ECEF-System (E)the Vehicle Position is given by:
( )( )( )( )
+++
=
=
φφλφλ
sin
cossin
coscos
HR
HR
HR
z
y
x
P
M
N
N
E
E
E
E
( )NhH
e
aRN
+=−
= 2/12 sin1 φ
Another variable, used frequently, is the radius of the Ellipsoid referred as the Meridian Radius
( )( ) 2/32
2
sin1
1
φe
eaRM
−−=
Earth Gravitation
SOLO
30
Reference Ellipsoid
b
a
Oφ
?O – Geographic Latitude? – Geodetic Latitude
Equator
North Pole
Tangentto Ellipsoid
pNR
zP
a
r
φ
β
Meridian Ellipse
Ellipsoid Equation:
( ) 2222
2
2
22
11 aebb
z
a
yx −==++
Define: 22: yxp +=
12
2
2
2
=+b
z
a
p
Differentiate: 022
=+b
dzz
a
dpp
z
p
a
b
dp
dz2
2
cot =−=φFrom the Figure:
φtan2
2
pa
bz =
( )φφ 2224
222
4
2
2
2
tantan1 baa
pp
a
b
a
p +=+=
φφφ
φφφ
2222
2
2222
2
sincos
sin,
sincos
cos
ba
bz
ba
ap
+=
+=
Earth Gravitation
SOLO
31
Reference Ellipsoid
( )φφ
φφφ
φφ
φφφ
22
2
2222
2
222222
2
sin1
cos1
sincos
sin
sin1
cos
sincos
cos
e
ea
ba
bz
e
a
ba
ap
−−=
+=
−=
+=
( ) 222 1 aeb −=
Earth Gravitation
SOLO
32
Reference Ellipsoid
( ) 222 1 aeb −=
12
2
2
2
=+b
z
a
p
0
0
sin
cos
φφ
rz
rp
==
1sincos
20
2
20
22 =
+
bar
φφ
( )2/1
20
2
20
2
0
sincos−
+=
bar
φφφ
( )
( ) ( )022/1
02
2/1
02
2
2/1
02
2
222/1
02
2
2
02
0
sin1sin21
sin1sin1sincos
φφ
φφφφφ
fafa
b
ba
b
a
a
baa
b
baa
b
aar
f
−≈+≈
+−+=
−+=
+=
−
−
≈
−−
Earth Gravitation
SOLO
33
Reference Ellipsoid
The Meridians and Parallels are the Lines of Curvature of the Ellipsoid. The principal Radii of Curvature are therefore in the Plane of Meridian (Meridian Radius of Curvature RM) and in the Plane of Prime Vertical, perpendicular to Meridian Plane (Radius of Curvature in the Prime Vertical RN)
Radiuses of Curvature of the Ellipsoid
Meridian Ellipse Equation: ( ) 2222
2
2
2
11 aebb
z
a
p −==+
From this Equation, at any point (x,y) on the Ellipse, we have:
φtan
12
2
−=−=az
bp
pd
zd
32
4
32
2222
2
2
2
2
22
2
22
2
2
2 111
za
b
za
pbza
a
b
z
p
a
b
z
p
za
b
pd
zd
z
p
za
b
pd
zd −=+−=
+−=
−−=
From the Ellipse Equation:
( )φ
φφ2
22
2
2
2222
2
2
2
2
2
2
2
2
cos
sin1
1
1tan1111
e
a
p
ee
a
p
b
a
p
z
a
p −=
−−+=
+=
( )( )
( ) 2/122
2
2
2
2/122 sin1
sin1tan
sin1
cos
φφφ
φφ
e
eap
a
bz
e
ap
−−==→
−=
From the Figure on right: ( ) 2/122 sin1cos φφ e
apRN
−==
Earth Gravitation
SOLO
34
Reference Ellipsoid
Let develop the RN and RM (continue):
we have at any point (p,z) on the Ellipse:
φtan
12
2
−=−=az
bp
pd
zd ( )3
22
32
4
2
2 11
z
ea
za
b
pd
zd −−=−=
The Radius of Curvature of the Ellipse at the point (p,z) is:
( )( )
( )( )
( ) 2/322
2
2/322
3323
22
2/3
2
2
2
2/32
sin1
1
sin1
sin1
1
tan1
11
:φφ
φφe
ea
e
ea
ea
pdzd
pdzd
RM−
−=−
−−
+
=
+
=
( )( )
( ) 2/122
2
2/122 sin1
sin1
sin1
cos
φφ
φφ
e
eaz
e
ap
−−=
−=
( )( ) 2/322
2
sin1
1:
φe
eaRM
−−=
Radiuses of Curvature of the Ellipsoid
Earth Gravitation
SOLO
35
Reference Ellipsoid
( )( ) 2/322
2
sin1
1:
φe
eaRM
−−=
( ) 2/122 sin1cos φφ e
apRN
−==
a
baf
−=: 22
222 2: ff
a
bae −=−=Using
( )( )[ ] ( ) ( ) [ ] ++−≈
+−++−≈
−−−= φφ
φ2222
2/322
2
sin321sin22
3121
sin21
1: ffaffffa
ff
faRM
( )[ ] ( ) [ ]φφφ
2222/122
sin31sin22
31
sin21faffa
ff
aRN +≈
+−+≈
−−=
[ ]φ2sin321 ffaRM +−≈
[ ]φ2sin31 faRN +≈
We used and we neglect f2 terms( )( )
+−++=− !2
11
1
1 nnxn
x n
Radiuses of Curvature of the Ellipsoid
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Earth Gravitation
SOLO
36
Reference Earth Model
Clairaut's theorem
Clairaut's theorem, published in 1743 by Alexis Claude Clairaut in his “Théorie de la figure de la terre, tirée des principes de l'hydrostatique”, synthesized physical and geodetic evidence that the Earth is an oblate rotational ellipsoid. It is a general mathematical law applying to spheroids of revolution. It was initially used to relate the gravity at any point on the Earth's surface to the position of that point, allowing the ellipticity of the Earth to be calculated from measurements of gravity at different latitudes.
Clairaut's formula for the acceleration due to gravity g on the surface of a spheroid at latitude , was:ϕ
where G is the value of the acceleration of gravity at the equator, m the ratio of the centrifugal force to gravity at the equator, and f the flattening of a meridian section of the earth, defined as:
a
baf
−=:
Alexis Claude Clairaut )1713 – 1765(
−+= φ2sin2
51 fmGg
Earth Gravitation
Return to Table of Content
SOLO
37
Reference Earth Model
Mac Cullagh’s Approximation (1845)
James MacCullagh (1809 – 1847)
Mac Cullagh’s used the Approximation:
( )( )
( )
( ) ( )
−
−
+
+−≈ ∫∫
≈
MP
SS
P
ra
M
M P
SS
P
Mda
r
Mr
aMd
a
r
Mr
aM
r
GrU
S
CM
S
θθθθ
θθθθ
cos
22
cos
22
cos
1
cos
1
22
0
11
2
1cos31
2
1cos3cos
1cos1
∫∫∫
+−++=
+−=−=M
SSSSS
M
SSS
M
SS dMyx
zyxMa
dMyx
zMa
dMa
r
MC
23
1
2
1
2
1cos31:
2222
2
222
2
2
2
2
20
θ
222220 2
1
2
31J
Ma
ACIII
MaII
MaC SSSS
SSSSSS
yyxxzzzzrr −=−−=
+−−=
−=
( ) ( ) ( ) ( ) ( )SSSSSSSSSSSSSS zzyyxx
M
SSSrr
M
SSzz
M
SSyy
M
SSxx IIIMdzyxIMdyxIMdzxIMdzyI ++=++=+=+=+= ∫∫∫∫ 2
1::,:,: 222222222
Define Moments of Inertia
2:,: SSSS
SS
yyxxzz
IIAIC
+==
( )( )
Ma
ACJJ
r
aMGM
r
GrU
P
22
2
23
2
:2
1cos3
2
−=−+−≈
θ
θ
Earth Gravitation
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SOLO
38
Reference Earth Model
The definition of geodetic latitude (φ) and longitude (λ) on an ellipsoid. The normal to the surface does not pass through the centre
Reference Ellipsoid
Geodetic latitude: the angle between the normal and the equatorial plane. The standard notation in English publications is ϕ
Geocentric latitude: the equatorial plane and the radius from the centre to a point on the surface. The relation between the geocentric latitude (ψ) and the geodetic latitude ( ) is ϕderived in the above references as
The definition of geodetic (or geographic) and geocentric latitudes
( ) ( )[ ]φφψ tan1tan 21 e−= −
To use previous results, where we used spherical coordinates, rS, φS, θS we will use
θπψϕλ −==2
&
Earth Gravitation
SOLO
39
Earth Gravitation
Centrifugal Potential
Since Earth is a Rotating Body, a Centrifugal Acceleration is exerted on a Mass m, at a (x,y,z) Position, given by
( ) Cyx
yxVyx
C Uyxyx
yxVa ∇=+Ω=
+
+Ω=
+Ω=
1111 2
22
22
where:Ω = 7292115.1467x10-11 rad/s – Earth Angular Velocity
The Centrifugal Earth Potential UC at (x,y,z) Position is given by
( ) ( )( ) .
.1111..
2
2
Constydyxdx
ConstydxdyxConstrdUConstUdU yxyxCCC
++Ω=
++⋅+Ω=+⋅∇=+=
∫∫∫∫∫
Centrifugal Potential
Earth Gravitation Model
( )222
2
1yxUC +Ω=
By choosing Const. = 0, we obtain
SOLO
40
Earth Gravitation
Potential of the Rotating Reference Ellipsoid
The Potential of the Rotating Reference Ellipsoid is the sum of the Gravitational and Centrifugal Potentials:
CG UUW +=
( ) .2
1 222 ConstyxUC ++Ω=
The Gravitational Potential of the Reference Ellipsoid (assuming Uniform Density Distribution) is:
( ) ( )
+
+−= ∑∑
= =
max
2 0
sincossin1n
n
n
mnmnmnm
n
G mSmCPr
a
r
MGU λλψ
where:UG – Gravitational Potential (m2/s2)GM – Earth’s Gravitational Constantr - Distance from the Earth’s Center of Massa - Semi-Major Axis of the WG 84 Ellipsoidn,m – Degree and Order respectivelyψ– Geocentric Latitudeλ – Geocentric Longitude = Geodetic Longitude
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SOLO
41
Earth Gravitation
The WGS 84 Gravity Model is defined in terms of normalized coefficients:
( ) ( ) ( ) ( ) ( )( )
+
+
+−= ∑∑∑
∞
= =
∞
= 1 110 sincossinsin1
n
n
mnmnm
mn
n
nnn
n
mSmCPr
aCP
r
a
r
MGrU λλψψ
( )∫
==
M
Sn
n
Snn MdP
a
r
MCC ψsin
1:0
( )( ) ( ) ( )
( )∫
+−=
M S
SS
mn
n
S
nm
nm Mdm
mP
a
r
mn
mn
S
C
λλ
ψsin
cossin
!
!2:
( ) ( ) ( ) ( ) ( )( )
+
+
+−= ∑∑∑
∞
= =
∞
= 1 110 sincossinsin1
n
n
mnmnm
mn
n
nnn
n
mSmCPr
aCP
r
a
r
MGrU λλψψ
( ) ( ) ( )( ) ( )
≠=
=
+−+=
02
01sin
!
!12sin
m
mkP
mn
mnknP m
nm
n ψψ
( )( )( )
≠=
=
−+
+=
02
01
!
!
12
1
m
mk
S
C
mn
mn
knS
C
nm
nm
nm
nm
Cnm and Snm are called Spherical Harmonic Coefficients (SHC).
The Gravitation Potential of the Earth Model is given by:Earth Gravitation Model
World Geodetic System (WGS 84)
SOLO
42
World Geodetic System (WGS 84)
Reference Earth Model
Spherical Harmonics
Visual representations of the first few spherical harmonics. Red portions represent regions where the function is positive, and green portions represent regions where the function is negative.
Earth Gravitation
SOLO
43
World Geodetic System (WGS 84)
Reference Earth Model
Carlo Somigliana (1860 –1955)
The Theoretical Gravity on the surface of the Ellipsoidis given by the Somigliana Formula (1929)
84
22
2
2222
22
sin1
sin1
sincos
sincos
WGS
epe
e
k
ba
ba
φφγ
φφ
φγφγγ
−+=
+
+=
where
1: −=e
p
a
bk
γγ
2
22
:a
bae
−= - Ellipsoid Eccentricity
a - Ellipsoid Semi-major Axis = 6378137.0 m
b - Ellipsoid Semi-minor Axis = 6356752.314 m
γp – Gravity at the Poles = 983.21849378 cm/s2
γe – Gravity at the Equator = 978.03267714 cm/s2
– Geodetic Latitudeϕ
The Theory of the Equipotential Ellipsoid was first given byP. Pizzetti (1894)
Earth Gravitation
SOLO
44
World Geodetic System (WGS 84)
Reference Earth Model
The coordinate origin of WGS 84 is meant to be located at the Earth's center of mass; the error is believed to be less than 2 cm. The WGS 84 meridian of zero longitude is the IERS Reference Meridian. 5.31 arc seconds or 102.5 meters (336.3 ft) east of the Greenwich meridian at the latitude of the Royal Observatory.The WGS 84 datum surface is an oblate spheroid (ellipsoid) with major (transverse) radius a = 6378137 m at the equator and flattening f = 1/298.257223563. The polar semi-minor (conjugate) radius b then equals a times (1−f), or b = 6356752.3142 m. Presently WGS 84 uses the EGM96 (Earth Gravitational Model 1996) Geoid, revised in 2004. This Geoid defines the nominal sea level surface by means of a spherical harmonics series of degree 360 (which provides about 100 km horizontal resolution).[7] The deviations of the EGM96 Geoid from the WGS 84 Reference Ellipsoid range from about −105 m to about +85 m.[8] EGM96 differs from the original WGS 84 Geoid, referred to as EGM84.
Earth Gravitation
SOLO
45
World Geodetic System (WGS 84)
Reference Earth Model
Earth Gravitation
SOLO
46
World Geodetic System (WGS 84)Reference Earth Model
Geoid product, the 15-minute, worldwide Geoid Height for EGM96 The difference between the Geoid and the Reference Ellipsoid exhibit the following statistics: Mean = - 0.57 m, Standard Deviation = 30.56 mMinimum = -106.99 m, Maximum = 85.39 m
Earth Gravitation
SOLO
47
World Geodetic System (WGS – 84)
Reference Earth Model
Parameters Notation Value
Ellipsoid Semi-major Axis a 6.378.137 m
Ellipsoid Flattening (Ellipticity) f 1/298.257223563(0.00335281066474)
Second Degree Zonal Harmonic Coefficient of the Geopotential C2,0 -484.16685x10-6
Angular Velocity of the Earth Ω 7.292115x10-5 rad/s
The Earth’s Gravitational Constant (Mass of Earth includes Atmosphere)
GM 3.986005x1014 m3/s2
Mass of Earth (Includes Atmosphere) M 5.9733328x1024 kg
Theoretical (Normal) Gravity at the Equator (on the Ellipsoid) γe 9.7803267714 m/s2
Theoretical (Normal) Gravity at the Poles (on the Ellipsoid) γp 9.8321863685 m/s2
Mean Value of Theoretical (Normal) Gravity γ 9.7976446561 m/s2
Geodetic and Geophysical Parameters of the WGS-84 Ellipsoid
Earth Gravitation
SOLO
48
World Geodetic System (WGS 84)Reference Earth Model
a
baf
−=:f - Ellipsoid Flattening (Ellipticity)
a - Ellipsoid Semi-major Axis
b - Ellipsoid Semi-minor Axis
e - Ellipsoid Eccentricity 22
222 2: ff
a
bae −=−=
( ) 211 eafab −=−=
Reference Ellipsoid
Earth Gravitation
SOLO
49
World Geodetic System (WGS 84)
where λ – Longitude e – Eccentricity = 0.08181919
Reference Earth Model
In Earth Center Earth Fixed Coordinate –ECEF-System (E)the Vehicle Position is given by:
( )( )( )( )
+++
=
=
φφλφλ
sin
cossin
coscos
HR
HR
HR
z
y
x
P
M
N
N
E
E
E
E
( )NhH
e
aRN
+=−
= 2/12 sin1 φ
Another variable, used frequently, is the radius of the Ellipsoid referred as the Meridian Radius
( )( ) 2/32
2
sin1
1
φe
eaRM
−−=
Radius of Curvature in the Prime Vertical
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Earth Gravitation
50
SOLO Air Vehicle in Ellipsoidal Earth AtmosphereSIMULATION EQUATIONS
51
SOLO
7. Forces Acting on the Vehicle (continue – 4)
Gravitation Acceleration
( ) ( )
−
−
−==
zgygxg
gg100
0
0
0
010
0
0
0
001
χχχχ
γγ
γγ
σσσσ cs
sc
cs
sc
cs
scC EWE
W
( ) gg
−=
γσγσ
γ
cc
cs
sW
2sec/174.322sec/81.90
2
0
00gg ftmg
HR
R==
+=
The derivation of Gravitation Acceleration assumes an Ellipsoidal Symmetrical Earth.The Gravitational Potential U (R, ) is given byϕ
( ) ( )
( ) ( )φ
φµφ
,
sin1,2
RUg
PR
aJ
RRU
EE
n n
n
n
∇=
−⋅−= ∑∞
=
μ – The Earth Gravitational Constanta – Mean Equatorial Radius of the EarthR=[xE
2+yE2+zE
2]]/2 is the magnitude of the Geocentric Position Vector
– Geocentric Latitude (sin =zϕ ϕ E/R)Jn – Coefficients of Zonal Harmonics of the Earth Potential FunctionPn (sin ) – Associated Legendre Polynomialsϕ
Air Vehicle in Ellipsoidal Earth Atmosphere
52
SOLO
7. Forces Acting on the Vehicle (continue – 5)
Gravitation Acceleration
Retaining only the first three terms of theGravitational Potential U (R, ) we obtain:ϕ
R
z
R
z
R
z
R
aJ
R
z
R
aJ
Rg
R
y
R
z
R
z
R
aJ
R
z
R
aJ
Rg
R
x
R
z
R
z
R
aJ
R
z
R
aJ
Rg
EEEEz
EEEEy
EEEEx
E
E
E
⋅
+⋅−⋅
⋅−
−⋅
⋅−⋅−=
⋅
+⋅−⋅
⋅−
−⋅
⋅−⋅−=
⋅
+⋅−⋅
⋅−
−⋅
⋅−⋅−=
342638
515
2
31
342638
515
2
31
342638
515
2
31
2
2
4
44
42
22
22
2
2
4
44
42
22
22
2
2
4
44
42
22
22
µ
µ
µ
φ
φλ
φλ
sin
cossin
coscos
=
⋅=
⋅=
R
zR
yR
x
E
E
E
( ) 2/1222EEE zyxR ++=
Air Vehicle in Ellipsoidal Earth Atmosphere
53
SOLO
23. Local Level Local North (LLLN) Computations for an Ellipsoidal Earth Model
( )( )( )
( )( ) 2
2210
20
20
20
5
21
20
60
sin
sin1
sin321
sin1
sec/10292116557.7
sec/051646.0
sec/780333.9
26.298/.1
10378135.6
Ae
e
p
m
e
HR
RLatggg
LatfRR
LatffRR
LatfRR
rad
mg
mg
f
mR
++=
+=
+−=
−=
⋅=Ω
=
==
⋅=
−
LatHR
V
HR
V
HR
V
Ap
EastDown
Am
NorthEast
Ap
EastNorth
tan+
−=
+−=
+=
ρ
ρ
ρ
Lat
Lat
Down
East
North
sin
0
cos
Ω−=Ω=Ω
Ω=Ω
DownDownDown
EastEast
NorthNorthNorth
Ω+==
Ω+=
ρφρφρφ
East
North
Lat
LatLong
ρ
ρ
−=
=
•
•
cos
( )
( ) ∫
∫•
•
+=
+=
t
t
dtLatLattLat
dtLongLongtLong
0
0
0
0
SIMULATION EQUATIONS
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Air Vehicle in Ellipsoidal Earth Atmosphere
SOLO
54
References
Return to Table of Content
R.,H.,Battin, “Astronautical Guidance”, McGraw-Hill, 1964
George M. Siouris, “Aerospace Avionics System”, Academic Press, 1993, Appendix B
S. Hermelin, “Legendre Functions”
Broxmeyer, C,. “Inertial Navigation Systems”, McGraw-Hill, 1964
Averil B. Chatfield, “Fundamentals of High Accuracy Inertial Navigation”, Progress in Astronautics and Aeronautics 174, 1997
X. Li, H-J. Götze, “Tutorial: Ellipsoid, Geoid, Gravity, Geodesy, and Geophysics”, Geophysics, Vol. 66, No. 6, Nov-Dec. 2001
http://en.wikipedia.org/wiki/
Department of Defense, World Geodetic System 84, NIMA (National Imagery and Mapping Agency) TR8350.2, Third Edition
http://earth-info.nga.mil/GandG/images/ww15mgh2.gif
http://earth-info.nga.mil/GandG/publications/tr8350.2/wgs84fin.pdf
Earth Gravitation
55
SOLO
TechnionIsraeli Institute of Technology
1964 – 1968 BSc EE1968 – 1971 MSc EE
Israeli Air Force1970 – 1974
RAFAELIsraeli Armament Development Authority
1974 – 2013
Stanford University1983 – 1986 PhD AA
56
Vector AnalysisSOLO
Vector Operations in Various Coordinate Systems
Laplacian 2∇=∇⋅∇• Cartesian:
2
2
2
2
2
22
z
U
y
U
x
UU
∂∂+
∂∂+
∂∂=∇
• Cylindrical:
2
2
2
2
22
2
2
2
2
2
22 1111
z
UUUU
z
UUUU
∂∂+
∂∂+
∂∂+
∂∂=
∂∂+
∂∂+
∂∂
∂∂=∇
θρρρρθρρρ
ρρ
• Spherical:
( )φθθ
θθθ
φθ2
2
2222
22
sin
1sin
sin
11,,
∂∂+
∂∂
∂∂+
∂∂
∂∂=∇ U
r
U
rr
Ur
rrrU
57
SOLO
Laplace Differential Equation in Spherical Coordinates
0sin
1sin
sin
112
2
2222
22 =
∂∂+
∂∂
∂∂+
∂∂
∂∂=∇
φθθθ
θθU
r
U
rr
Ur
rrU
Let solve this equation by the method of Separation of Variables, by assuming a solution of the form :
( ) ( ) ( )φθφθ ,,, SrRrU =
Spherical Coordinates:
θϕθϕθ
cos
sinsin
cossin
rz
ry
rx
===
In Spherical Coordinates the Laplace equation becomes:
Substituting in the Laplace Equation and dividing by U gives:
0sinsinsin
112
2
222
2=
∂∂+
∂∂
∂∂+
φθ
θθ
θθ
SS
Srrd
Rdr
rd
d
Rr
The first term is a function of r only, and the second of angular coordinates. For the sum to be zero each must be a constant, therefore:
λφθ
θθ
θθ
λ
−=
∂∂+
∂∂
∂∂
=
2
2
2
2
sinsinsin
1
1
SS
S
rd
Rdr
rd
d
R
58
SOLOLaplace Differential Equation in Spherical Coordinates
λ=
rd
Rdr
rd
d
R21
( ) ( )ϕθ ,SrR=Φ
We get:
022
22 =−+ R
rd
Rdr
rd
Rdr λor:
Assume a solution of the form: R = C rα, where α is a constant to be defined and C is determined by the boundary conditions. Substituting in the Differential Equation gives
( )[ ] 021 =−−− αλααα rC
or:( ) 01 =−+ λαα
Let define l as l (l+1):=λ
( ) ( ) ( ) 0111 2 =+−+=+−+ llll αααα
( ) ( )
−−=+±−=
++±−=
12
121
2
14112,1 l
llllα
( )
=+1
1l
l
r
rrR
59
SOLO
( )1sinsinsin
12
2
2+−=
∂∂+
∂∂
∂∂
llSS
S φθθ
θθ
θ
( ) ( )ϕθ ,SrR=ΦWe obtain:
Multiply this by S sin2θ and put to get:( ) ( ) ( )φθφθ ΦΘ=,S
01
sinsinsin1
2
22 =Φ
Φ+
+
Θ∂∂
Θ φθλ
θθ
θθ
d
d
d
d
Again, the first term, in the square bracket, and the last term must be equal and opposite constants, which we write m2, -m2. Thus:
( )
Φ−=Φ
=Θ
−++
Θ∂∂
22
2
2
2
0sin
1sinsin
1
md
d
mll
d
d
φ
θθθ
θθ
The Φ ( ) must be periodical in (a period of 2 π) and because ϕ ϕthis we choose the constant m2, with m an integer. Thus:
( ) φφφ mbma sincos +=Φ
Laplace Differential Equation in Spherical Coordinates
( ) φφφ mjmj ee +−=Φ ,or
With m integer, we have the Orthogonality Condition
21
21,
2
0
2 mmmjmj dee δπϕ
πφφ =⋅∫ +−
60
SOLO
( ) ( ) ( ) ( )φθφθ ΦΘ= rRrU ,,
We get:
or: ( ) Θ+−=Θ−Θ+Θ1
sincot
2
2
2
2
llm
d
d
d
d
θθθ
θ
( ) 0sin
1sinsin
12
2
=Θ
−++
Θ∂∂
θθθ
θθm
lld
d
Laplace Differential Equation in Spherical Coordinates
Change of variables: t = cos θ
θθ dtd sin−=
td
d
d
d Θ−=Θ θθ
sin
( )
td
dt
td
dt
td
d
d
d
td
d
td
d
td
d
td
d
d
d
d
d
d
d Θ−Θ−=Θ+Θ=
Θ−−=
Θ=Θ
−
2
22
cossin/1
2
22
2
2
1sin
sinsinsinsinθθ
θθθθθθθ
θθθ
( ) ( ) Θ
−
−++Θ−Θ−Θ=Θ
−++Θ+Θ=
=
2
2
2
2cos
2
2
2
2
11
sin1cot0
t
mll
td
dt
td
dt
td
dmll
d
d
d
d t θ
θθθ
θ
We obtain:( )
1cos
,2,1,001
122
2
2
2
≤⇒=
==Θ
−
−++Θ−Θ
tt
mt
mll
td
dt
td
d
θ
Associate Legendre Differential Equation
61
SOLO
( ) ( ) ( ) ( )φθφθ ΦΘ= rRrU ,,
Laplace Differential Equation in Spherical Coordinates
We obtain:
( )
1cos
,2,1,001
122
2
2
2
≤⇒=
==Θ
−
−++Θ−Θ
tt
mt
mll
td
dt
td
d
θ
Associate Legendre Differential Equation
Let start with m = 0 with:
( )
1cos
0122
2
≤⇒=
=Θ++Θ−Θ
tt
lltd
dt
td
d
θLegendre Differential Equation
They are named after Adrien-Marie Legendre. This ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates.
The Legendre polynomials were first introduced in 1785 by Adrien-Marie Legendre, in “Recherches sur l’attraction des sphéroides homogènes”, as the coefficients in the expansion of the Newtonian potential
Adrien-Marie Legendre(1752 –1833(
SOLO
62
Legendre Polynomials
Olinde Rodrigues (1794-1851)
Start from the function: ( ) .12 constktkyn
=−=
( ) 12 12:'−−== n
ttkntd
ydy
( ) ( ) ( ) 222122
2
11412:''−− −−+−== nn
ttnkntkntd
ydy
Let compute:
( ) ( ) ( ) ( ) ( ) '12211412''112222 ytntnttnkntknyt
nn −+=−−+−=− −
or: ( ) ( ) 02'12''12 =−−+− ynytnyt
Let differentiate the last equation n times with respect to t:
( )[ ] ( ) ( ) ( ) ( )
( ) ( ) ''1''2''1
00''13
''12
''11
''1''1
2
2
1
12
3
3
0
23
3
2
22
2
2
1
1222
ytd
dnny
td
dtny
td
dt
ytd
dt
td
dny
td
dt
td
dny
td
dt
xd
dny
td
dtyt
td
d
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
−
−
−
−
−
−
−
−
−
−
−++−=
+−
+−
+−
+−=−
( ) ( ) ( ) ( )
+−=
+−=− −
−
−
−
''12'1
'12'121
1
1
1
ytd
dny
td
dtny
td
dt
td
dny
td
dtnyt
td
dn
n
n
n
n
n
n
n
n
n
n
Derivation of Legendre Polynomials via Rodrigues’ Formula
Laplace Differential Equation in Spherical Coordinates
SOLO
63
Legendre Polynomials
Olinde Rodrigues (1794-1851)
Start from the function: ( ) .12 constktkyn
=−=
( ) ( ) 02'12''12 =−−+− ynytnytDifferentiate n times with respect to t:
( ) ( ) ''1''2''12
2
1
12 y
td
dnny
td
dtny
td
dt
n
n
n
n
n
n
−
−
−
−
−++−
( ) 02''121
1
=−
+−+ −
−
ytd
dny
td
dny
td
dtn
n
n
n
n
n
n
Define: a Polynomial( ) ( )[ ]n
n
n
n
n
ttd
dk
td
ydtw 1: 2 −==
( ) ( ) ( ) [ ] 02'121'2''12 =−+−+−++− wnwnwtnwnnwtnwt
( ) ( ) ( ) ( )[ ] 02121'2''12 =−−+−+−++− wnnnnnwtnttnwt
( ) ( ) 01'2''12 =+−+− wnnwtwt
This is Legendre’s Differential Equation. We proved that one of the solutions are Polynomials. We can rewrite this equation in a Sturm-Liouville Form:
( ) ( ) 0112 =+−
− wnnw
td
dt
td
d
Derivation of Legendre Polynomials via Rodrigues’ Formula
Laplace Differential Equation in Spherical Coordinates
SOLO
64
Legendre Polynomials
Olinde Rodrigues (1794-1851)
Let find k such that:
by choosing Pn (1) = 1
( ) ( )[ ]n
n
n
n
n
n ttd
dk
td
ydtP 12 −==
( ) ( )[ ] ( ) ( ) ( ) ( )
−+=
+−=−= ∑
>0
22 1!2111i
inn
v
n
u
n
n
nn
n
n
n ttatnktttd
dktk
td
dtP
!2
1
nk
n=
We obtain the Rodrigues Formula:( ) ( )[ ]n
n
n
nn ttd
d
ntP 1
!2
1 2 −=
Let use Leibnitz’s Rule (Binomial Expansion for the n Derivative of a Product - with u:=(t-1)n and v:=(t+1)n ):
( ) ( )( )
udvudvdnvddunn
vddunvdu
vdudmnm
nvud
nnnnn
n
m
mnmn
+++−++=
−=⋅
−−−
=
−∑1221
0
!2
1
!!
!
We have:
( ) ( )
1!2!2
11
1!20
12
0
21
00
==
+++−++==
=
−−− nkudvudvdnvddunn
vddunvdukxP n
xn
nnnnnn
n
We can see from this Formula that Pn (t) is indeed a Polynomial of Order n in t.
Derivation of Legendre Polynomials via Rodrigues’ Formula
Laplace Differential Equation in Spherical Coordinates
SOLO
65
Legendre PolynomialsThe first few Legendre polynomials are:
( )
( )( )
( )( )
( )( )
( )( )
( ) 256/63346530030900901093954618910
128/31546201801825740121559
128/35126069301201264358
16/353156934297
16/51053152316
8/1570635
8/330354
2/353
2/132
1
10
246810
3579
2468
357
246
35
24
3
2
−+++−+−+−
+−+−−+−
−+−
+−+−
−−
xxxxx
xxxxx
xxxx
xxxx
xxx
xxx
xx
xx
x
x
xPn n
Laplace Differential Equation in Spherical Coordinates
66
SOLO
Orthogonality of Legendre Polynomials
Define ( ) ( )tPwtPv nm == :&:
We use Legendre’s Differential Equations:
( ) ( ) 011 2 =++
− vmm
td
vdt
td
d
( ) ( ) 011 2 =++
− wnn
td
wdt
td
d
Multiply first equation by w and integrate from t = -1 to t = +1.
( ) ( ) 0111
1
1
1
2 =++
− ∫∫
+
−
+
−dtwvmmdtw
td
vdt
td
d
Integrate the first integral by parts we get
( ) ( ) ( ) 01111
1
1
1
2
0
1
1
2 =++−−− ∫∫+
−
+
−
+=
−=
dtwvmmdttd
wd
td
vdtw
td
vdt
t
t
In the same way, multiply second equation by v and integrate from t = -1 to t = +1.( ) ( ) 011
1
1
1
1
2 =++−− ∫∫+
−
+
−dtwvnndt
td
wd
td
vdt
Legendre Polynomials
67
SOLO
Orthogonality of Legendre Polynomials
( ) ( ) 0111
1
1
1
2 =++−− ∫∫+
−
+
−dtwvmmdt
td
wd
td
vdt
Subtracting those two equations we obtain
( ) ( ) 0111
1
1
1
2 =++−− ∫∫+
−
+
−dtwvnndt
td
wd
td
vdt
( ) ( )[ ] ( ) ( )[ ] ( ) ( ) 011111
1
1
1=+−+=+−+ ∫∫
+
−
+
−dttPtPnnmmdtwvnnmm nm
This gives the Orthogonality Condition for m ≠ n
( ) ( ) nmdttPtP nm ≠=∫+
−0
1
1
To find let square the relation and integrate between t = -1 to t = +1. Due to orthogonality only the integrals of terms having Pn
2(t) survive on the right-hand side. So we get
( )∫+
−
1
1
2 dttPn( )∑
∞
=
=+− 0
221
1
nn
n tPuutu
( )∑ ∫∫∞
=
+
−
+
−=
+− 0
1
1
221
1 221
1
nn
n dttPudtutu
Legendre Polynomials
68
SOLO
Orthogonality of Legendre Polynomials
( )∑ ∫∫∞
=
+
−
+
−=
+− 0
1
1
221
1 221
1
nn
n dttPudtutu
( ) ( )( ) 1
1
1ln1
1
1ln
2
121ln
2
1
21
12
21
1
21
1 2<
−+=
+−
−=−+
−=
−+
+=
−=
+
−∫ uu
u
uu
u
utuu
udt
tuu
t
t
( ) ( ) ( ) ( ) ( ) ( ) ( )∑∑∑∞ ++∞ +∞ +
+−−−=
+−−−
+−=−−+
0
11
0
1
0
1
11
1
11
1
11
11ln
11ln
1
n
uu
un
u
un
u
uu
uu
u
nnn
nn
nn
( ) ( ) ( )( ) ( ) ( )
( ) ∑∑∑∑∞∞ +∞ ++
+∞ ++
+=
+=
+−−−+
+−−−=
0
2
0
12
0
0
121212
0
12122
12
2
12
12
121
1
121
1 nnnn
nnn
n unn
u
un
uu
un
uu
u
Let compute first
Therefore
( )∑ ∫∑∫∞ +
−
∞+
−=
+=
+− 0
1
1
22
0
21
1 2 12
2
21
1dttPuu
ndt
utu nnn
Comparing the coefficients of u2n we get ( )12
21
1
2
+=∫
+
− ndttPn
Legendre Polynomials
( ) ( ) nmmn ndttPtP δ
12
21
1 +=∫
+
−Hence
SOLO
69
Associated Legendre Functions
Let Differentiate this equation m times with respect to t, and use Leibnitz Rule of Product Differentiation: ( ) ( )[ ] ( )
( ) ( )im
im
i
im
im
m
td
tgd
td
tsd
imi
mtgts
td
d−
−
=∑ −
=⋅0 !!
!
Start withLegendre DifferentialEquation:
( ) ( ) ( ) ( ) 1011 2 ≤=++
− ttwnntw
td
dt
td
dnn
or: ( ) ( ) ( ) ( ) ( ) 101212
22 ≤=++−− ttwnntw
td
dttw
td
dt nnn
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )twmmtwtmtwttwtd
dt
td
d mn
mn
mnnm
m
1211 1222
22 −−−−=
− ++
( ) ( ) ( ) ( ) ( )twmtwttwtd
dt
td
d mn
mnnm
m
+=
+1
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )twnntwmtwttwmmtwtmtwt mn
mn
mn
mn
mn
mn 122121 1122 ++−−−−−− +++
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) 011121 122 =+−+++−−= ++ twmmnntwtmtwt mn
mn
mn
Laplace Differential Equation in Spherical Coordinates
SOLO
70
Associated Legendre Functions
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) 011121 122 =+−+++−− ++ twmmnntwtmtwt mn
mn
mn
Define: ( ) ( ) ( )twty mn=:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) 011121 122 =+−+++−− tymmnntytmtyt
Now define: ( ) ( ) ( )tyttum
221: −=Let compute:
( ) ( ) ( )122122 11 ytyttm
td
ud mm
−+−−= −
( ) ( ) ( ) ( )1122222 111 ytyttm
td
udt
mm +−+−−=−
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )21221221221
2222222 1121111 ytyttmyttmyttmytmtd
udt
td
d mmmmm +− −+−+−−−−+−−=
−
( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) yt
tmmnnmmtymmnnytmytt
mm
−
+−+−+−++−+++−−−=2
22222
0
12222
111111211
We get:( ) ( ) 0
111
2
22 =
−
−++
− u
t
mnn
td
udt
td
d Associate Legendre Differential Equation
Laplace Differential Equation in Spherical Coordinates
SOLO
71
Associated Legendre Functions
Define: ( ) ( ) ( )twtd
dttu nm
mm
221: −=
We get: ( ) ( ) 01
112
22 =
−
−++
− u
t
mnn
td
udt
td
d
Start with Legendre Differential Equation:
( ) ( ) ( ) ( ) 1011 2 ≤=++
− ttwnntw
td
dt
td
dnn
Summarize
But this is the Differential Equation of Θ (θ) obtained by solving Laplace’s Equation
by Separation of Variables in Spherical Coordinates .
02 =Φ∇( ) ( ) ( ) ( )φθφθ ΦΘ=Φ rRr ,,
The Solutions Pnm(t) of this Differential Equation are called Associated Legendre
Functions, because they are derived from the Legendre Polynomials
( ) ( ) ( )tPtd
dttP nm
mmm
n221: −=
Laplace Differential Equation in Spherical Coordinates
SOLO
72
Associated Legendre Functions
Examples
( ) ( ) ( )tPtd
dttP nm
mmm
n221: −=
( ) ( ) 10 000 === tPtPn
( ) ( )( )
( )
( ) ( )
( ) ( ) ( ) θ
θ
θ
θ
θ
θ
sin1
cos:
sin111
cos2
121
11
1
cos
10
1
cos2
122
121
1
1
−=−−=−=
===
=−=−==
=−
=
=
t
t
t
tP
ttPtP
ttPtP
tttd
dttPn
( ) ( ) ( )( ) ( )tP
mn
mntP m
nmm
n !
!1
+−−=−
( ) ( )( )
( )
( ) ( )( )
( )
( ) ( )
( ) ( ) ( )( )
( ) ( ) ( ) ( ) θ
θθ
θ
θθ
θ
θ
θ
θ
θ
θ
2cos
222
222
cos2
1211
2
2cos2
202
cos2
12
2
2
121
2
2cos
22
2
2
2
222
2
sin8
11
8
1
!4
!01
2
cossin13
!3
11
2
1cos3
2
13
cossin3132
131
sin3132
1312
12
2
1
=−
=−
=
=
=
=−=−=
−=−−=
−=−==
=−=
−−=
=−=
−−==
t
t
tP
t
t
tP
t
tP
ttPtP
tttP
ttPtP
ttt
td
dttP
tt
td
dttPn
( ) 10 =tP ( )2
1
2
3 22 −= ttP( ) ttP =1
Laplace Differential Equation in Spherical Coordinates
SOLO
73
Associated Legendre Functions
Examples
( ) ( ) ( )tPtd
dttP nm
mmm
n221: −= ( ) ( ) ( )
( ) ( )tPmn
mntP m
nmm
n !
!1
+−−=−
Laplace Differential Equation in Spherical Coordinates
SOLO
74
Associated Legendre Functions
Orthogonality of Associated Legendre Functions
( ) ( ) ( )[ ]n
mn
mnm
nm
n ttd
dt
ntP 11
!2
1: 222 −−= +
+
Let Compute
( ) ( ) ( ) ( )[ ] ( )[ ]∫∫+
−+
+
+
+
+
+
−
−−−=1
1
2221
1
111!!2
1dtt
td
dt
td
dt
qpdttPtP
q
mq
mqp
mp
mpm
qpm
qm
p
Define X := x2 -1
( ) ( ) ( ) [ ] [ ]∫∫+
−+
+
+
+
+
+
−
−=1
1
1
1 !!2
1dtX
td
dX
td
dX
qpdttPtP q
mq
mqp
mp
mpm
qp
mm
qm
p
If p ≠ q, assume q > p and integrate by parts q + m times
[ ] [ ] mqidtXtd
dvdX
td
dX
td
du q
imq
imqp
mp
mpm
i
i
+==
= −+
−+
+
+
,,1,0
All the integrated parts will vanish at the boundaries t = ± 1 as long as there is a factorX = x2-1. We have, after integrating m + q times
( ) ( ) ( ) ( ) [ ]∫∫+
−+
+
+
+
+
++
−
−−=1
1
1
1 !!2
11dtX
td
dX
td
dX
qpdttPtP p
mp
mpm
mq
mqq
qp
mqmm
qm
p
SOLO
75
Associated Legendre Functions
( ) ( ) ( ) ( ) [ ] 1:!!2
11 21
1
1
1
−=
−−= ∫∫+
−+
+
+
+
+
++
−
xXdtXtd
dX
td
dX
qpdttPtP p
mp
mpm
mq
mqq
qp
mqmm
qm
p
Because the term Xm contains no power greater than x2m, we must haveq + m – i ≤ 2 m
or the derivative will vanish. Similarly,p + m + i ≤ 2 p
Adding both inequalities yieldsq ≤ p
which contradicts the assumption that q > p, therefore is no solution for i and the integral vanishes.
Let expand the integrand on the right-side using Leibniz’s formula
( )( )∑
+
=++
++
−+
−+
+
+
+
+
−++=
mq
i
pimp
impm
imq
imqqp
mp
mpm
mq
mqq X
td
dX
td
d
imqi
mqXX
td
dX
td
dX
0 !!
!
( ) ( ) qpdttPtP mq
mp ≠=∫
+
−
01
1
This proves that the Associated Legendre Functions are Orthogonal (for the same m).
Orthogonality of Associated Legendre Functions
SOLO
76
Associated Legendre Functions
( )[ ] ( ) ( )1:
!!2
11 21
12
21
1
2 −=
−−= ∫∫+
−+
+
+
++
−
xXdtXtd
dX
td
dX
ppdttP p
mp
mpm
mp
mpp
p
mpm
p
Let expand the integrand on the right-side using Leibniz’s formula
( )( )∑
+
=++
++
−+
−+
+
+
+
+
−++=
mp
i
pimp
impm
imp
imppp
mp
mpm
mp
mpp X
td
dX
td
d
impi
mpXX
td
dX
td
dX
0 !!
!
For the case p = q we have
Because X = x2 – 1 the only non-zero term is for i = p - m
( )[ ] ( ) ( )( ) ( ) ( )
( ) ( )
1:!2!!2
!1 21
1
!2
2
2
!2
2
2
22
1
1
2 −=⋅−+−= ∫∫
+
−
+
−
xXdtXtd
dX
td
dX
mmpp
mpdttP
p
pp
p
m
mm
mp
p
pm
p
( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )( )
( )( )!
!
12
2
sin1!!2
!2!11
!!2
!2!1
!12
!21
0
1222
cos1
1
222
212
mp
mp
p
dmpp
pmpdtx
mpp
pmp
p
p
pp
p
ptp
X
p
p
pp
−+⋅
+=
−−
+−=−−
+−=
+−
+=+
−+
∫∫
θθ
πθ
Orthogonality of Associated Legendre Functions
SOLO
77
Associated Legendre Functions
( ) ( ) ( ) ( ) ( )( ) qp
mq
mp
tm
qm
p mp
mp
pdPPdttPtP ,
0
cos1
1 !
!
12
2sincoscos δθθθθ
πθ
−+
+== ∫∫
=+
−
Therefore the Orthonormal Associated Legendre Functions is
( ) ( )( ) ( ) mnmP
mn
mnnΘ m
nm
n ≤≤−+−+= θθ cos
!
!
2
12cos
Orthogonality of Associated Legendre Functions
78
SOLO
Absolute Angular Momentum Relative to a Reference Point O
The Absolute Momentum Relative to a Reference Point O, of the particle of mass dmi at time t is defined as:
( ) ( ) iiOiiiOiiOiO dmVrdmVRRPdRRHd
×=×−=×−= ,, :
The Absolute Momentum Relative to a Reference Point O, of the mass m (t) is defined as:
( ) ( ) ∑∑∑===
×=×−=×−=N
ii
I
iOi
N
iiiOi
N
iiOiO dm
td
RdrdmVRRPdRRH
1,
11, :
By taking a very large number N of particles, we go from discrete to continuous
∫⇒∑∞→
=
NN
i 1
( )( )
( )( ) ( )
∫ ×=∫ ×−=∫ ×−=tm
Otm
Otv
OO dmVrdmVRRdvVRRH
,, ρ
The Absolute Momentum Relative to a Reference Point O, of the system (including the mass entering (+)/leaving (-) through surface S), at time t + Δt is given by:
( ) ( )∑∑ ∆
∆+×∆++
∆+×∆+=∆+
= openingsiflow
I
iflow
I
iflowOiflowOiflow
N
ii
I
i
I
iOiOiOO m
td
Rd
td
Rdrrdm
td
Rd
td
RdrrHH
,,1
,,,,
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
79
SOLO
Absolute Angular Momentum Relative to a Reference Point O (continue – 1)
By subtracting
I
O
tI
O
t
H
td
Hd
∆∆
=→∆
,
0
, lim
( ) ( )
t
dmtd
Rdrm
td
Rd
td
Rdrrdm
td
Rd
td
Rdrr
openings
N
ii
I
iiOiflow
I
iflow
I
iflowOiflowOiflow
N
ii
I
i
I
iOiOi
t ∆
×−∆
∆+×∆++
∆+×∆+
=
∑ ∑∑==
→∆
1,,
1,,
0lim
∑∑∑ ×+×+×=== openings
iflow
I
iflowOiflow
N
ii
I
iOiN
ii
I
iOi m
td
Rdrdm
td
Rd
td
rddm
td
Rdr
,1
,
12
2
,
Now let add the constraint that at time t the flow at the opening is such that
iopenS
( ) ( ) ( ) ( )trtrtRtR OiflowOiopeniflowiopen ,,
=→=
to obtain (next page)
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
( )( )
( )( ) ( )
∫ ×=∫ ×−=∫ ×−=tm
Otm
Otv
OO dmVrdmVRRdvVRRH
,, ρ
dividing by Δt, and taking the limit, we get
from ( ) ( )∑∑ ∆
∆+×∆++
∆+×∆+=∆+
= openingsiflow
I
iflow
I
iflowOiflowOiflow
N
ii
I
i
I
iOiOiOO m
td
Rd
td
Rdrrdm
td
Rd
td
RdrrHH
,,1
,,,,
80
SOLO
Absolute Angular Momentum Relative to a Reference Point O (continue – 2)
∑∑∑ ×+×+×=== openings
iflow
I
iflowOiopen
N
ii
I
i
I
OiN
ii
I
iOi
I
O mtd
Rdrdm
td
Rd
td
rddm
td
Rdr
td
Hd
,1
,
12
2
,,
( )∑ ×−+∑ ×
−+∑ ×=
== openingsiflow
I
iflowOiopen
N
ii
I
i
I
O
I
iN
ii
I
iOi m
td
RdRRdm
td
Rd
td
Rd
td
Rddm
td
Rdr
112
2
,
( )∑ ×−+∑×−∑ ×=== openings
iflow
I
iflowOiopen
N
ii
I
i
I
ON
ii
I
iOi m
td
RdRRdm
td
Rd
td
Rddm
td
Rdr
112
2
,
By taking a very large number N of particles, we go from discrete to continuous
∫⇒∑∞→
=
NN
i 1
( )( )∑ ×−+×−∫ ×=
openingsiflowiflowOiopenO
tmI
O
I
O mVRRPVdmtd
Rdr
td
Hd
2
2
,,
( ) ( ) ∑∑ −+=−−+=openings
OiopeniflowO
I
O
openingsOiopeniflowO
I
O rmVmtd
cdRRmVm
td
cdtP ,
,,
Substitute to obtain
( )( ) ( )∑ ×−+×
∑ −−++∫ ×=
openingsiflowiflowOiopenO
openingsiflowOiopenO
I
O
tmI
O
I
O mVRRVmRRVmtd
cddm
td
Rdr
td
Hd
,
2
2
,,
or
( )( )∑ −×+×+∫ ×=
openingsiflowOiflowOiopenO
I
O
tmI
O
I
O mVVrVtd
cddm
td
Rdr
td
Hd
,,
2
2
,,
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
81
SOLO
Absolute Angular Momentum Relative to a Reference Point O (continue – 3)
We obtained
( )( )
( )( ) ( )
∫ ×=∫ ×−=∫ ×−=tm
Otm
Otv
OO dmVrdmVRRdvVRRH
,, ρ
Substitute in the previous equation
OIO
O
OO
I
O
I
O
I
OO rtd
rdV
td
rd
td
Rd
td
RdVrRR ,
,,, :&
×++=+==+= ←ω
( )( ) ( )
∫
×++×=∫ ×−= ←
tmOIO
O
OOO
tmOO dmr
td
rdVrdmVRRH ,
,,,
ω
( )( )
( ) ( )∫
×+∫ ××+×
∫= ←tm O
OO
tmOIOOO
tmO dm
td
rdrdmrrVdmr ,,,,,
ω
We obtain
(a) (b) (c)
Let develop those three expressions (a), (b) and (c).
where is the angular velocity vector from I to O.IO←ω
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
82
SOLO
Absolute Angular Momentum Relative to a Reference Point O (continue – 4)
(a)( ) ( ) ( ) ( ) ( )
( ) OOC
tm
OC
tm
OC
tm
OC
tm
C
tm
O cmRRdmrdmrdmrdmrdmr ,,,,,,
=−===+= ∫∫∫∫∫
Where we used because C is the Center of Mass (Centroid) of the system.( )
0, =∫tm
C dmr
( )OOOOCO
tm
O VcVrmVdmr ×=×=×
∫ ,,,
( )( )
( )[ ]( )
IOOIOtm
OOOOtm
OIBO Idmrrrrdmrr ←←← ⋅=⋅∫ −⋅=∫ ×× ωωω ,,,,,,, 1(b)
where ( )[ ]( )∫ −⋅=
tm
OOOOO dmrrrrI ,,,,, 1:
2nd Moment of Inertia Dyadic of all the mass m(t) relative to O
We obtain (a) + (b) + (c)
( )( ) ( )
( )( ) ( )
∫
×+∫ ××+×
∫=∫ ×−= ←tm O
OO
tmOIOOO
tmO
tvOO dm
td
rdrdmrrVdmrvdVRRH ,,,,,, :
ωρ
( )∫
×+⋅+×= ←
tm O
OOIOOOO dm
td
rdrIVc ,,,,
ω
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
SOLO
Absolute Angular Momentum Relative to a Reference Point O (continue – 4)
( )[ ]( )∫ −⋅=
tm
OOOOO dmrrrrI ,,,,, 1:
2nd Moment of Inertia Dyadic of all the mass m(t) relative to O
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
=
O
O
O
O
z
y
x
r
,
,
,
,
( )[ ] [ ] [ ]OOO
O
O
O
O
O
O
OOOOOOO zyx
z
y
x
z
y
x
zyxrrrr ,,,
,
,
,
,
,
,
,,,,,,,
100
010
001
1
−
=−⋅
( )∫
+−−
−+−
−−+
=tm
OOOOOO
OOOOOO
OOOOOO
O dm
yxzyzx
zyzxyx
zxyxzy
I2
,2
,,,,,
,,2
,2
,,,
,,,,2
,2
,
, :