5 earth gravitation
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1) The document discusses concepts related to gravitation including Newton's law of universal gravitation, gravitational potentials and fields, and spherical harmonics. 2) Gravitation potentials can be used to describe the gravitational field generated by point masses or continuous mass distributions, such as a uniform spherical distribution. 3) Spherical harmonics allow the gravitational potential to be expressed as the sum of terms involving associated Legendre functions, enabling description of more complex mass distributions.
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5 earth gravitation
- 1. Earth Gravitation SOLO HERMELIN Updated: 17.01.13 16.12.14 1 http://www.solohermelin.com
- 2. 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 ( )Sr ρ 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 theorem Mac Cullagh’s Approximation World Geodetic System (WGS 84) Reference Ellipsoid Air Vehicle in Ellipsoidal Earth Atmosphere References
- 3. Gravity Nicolaus Copernicus (1473 – 1543) “De Revolutionibus Orbium Coelestium” (On the Revolutions of the Heavenly Spheres”, Nuremberg 1543) SOLO
- 4. 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
- 5. 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 dA h dt 2 = b •THIRD LAW THE SQUARE OF THE PERIOD OF A PLANET IS PROPORTIONAL TO THE CUBE OF ITS MEAN DISTANCE FROM THE SUN. a 2/322 a GMGM ab TP ππ == b Gravity SOLO
- 6. GALILEO GALILEI (1564-1642( “DISCOURSESANDMATHEMATICALDEMONSTRATIONS CONCERNINGTWONEWSCIENCES” 1636 THE FIRST TWO CHAPTERS DEAL WITH STRENGTHOFMATERIALS. THIS IS ESSENTIALLY THE FIRST NEW SCIENCE. THE THIRD AND FOURTH SUBJECT IS : • UNIFORMMOTION WITHCONSTANTACCELERATION ANDMOTION OFPROJECTILES •THE LAWOFINERTIA •THE COMPOSITION OFMOTIONSACCORDINGTO VECTORADDITION (“GALILEAN TRANSFORMATION”) AND •THE STUDY OFUNIFORMLYACCELERATEDMOTION Gravity SOLO
- 7. NEWTON’SLAWSOFMOTION “THE MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY”1687 • FIRSTLAW 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. • SECONDLAW THE RATE OF CHANGE OF MOMENTUM IS PROPORTIONAL TO THE FORCE IMPRESSED AND IS IN THE SAME DIRECTION AS THAT FORCE. • THIRDLAW TO EVERY ACTION THERE IS ALWAYS OPPOSED AN EQUAL REACTION. Gravity SOLO
- 8. 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 Gravity SOLO
- 9. THE LAGEOS SATELLITE MONITORS ITS POSITION RELATIVE TO THE EARTH USING 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 SEPARATIONS SMALLER 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: Rr r M GrV nn <<=+1 )( Gravity Henry Cavendish (1731 – 1810) Return to Table of Content SOLO
- 10. 10 SOLO Earth Gravitation Gravitation of a Point Mass M (Netwon Gravitation Law) SF SF SF SF SF FM rrr rr rr rr Mm GgmF −= − − − −== :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 ( ) ( )rrf r r r M Grg = −= 2 ( ) ( ) ( ) r r r rf r r rf rf ∂ ∂ =∇ ∂ ∂ =∇ Any function of the form has the property that( )rrf Because of this 1.we can write 2.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 on the values of the scalar function U (called Potential) at those two points. ( )rA ( )1rB
- 11. SOLO 11 Earth Gravitation Gravitation of a Point Mass M (Netwon Gravitation Law) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) r M G r M Grd r M Grd r r r M GrdrgrUrU 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 ( )Sr ρ The Gravitation Potential of the Distribution is obtaining by Integrating the Gravitation Potentials of the Point Masses dm over the Volume V ( ) ( ) ( ) ∫∫ − −= − −= V S S S M S S Vd rr r G rr rmd GrU ρ If we take the reference rf → ∞ we get 0. == fr M GConst ( ) r M GrU −= ( ) .Const r M GrU +−= Return to Table of Content
- 12. SOLO 12 Earth Gravitation ( ) S SS S rrMd r r r rr G rr Md GrUd > + − −= − −= 2 cos21 1 γ ( ) ( ) SSSSSSSS onDistributi Density SSS ddrdrrVdrMd ϕθθθϕρθϕρ sin,,,, 2 == ( ) ( ) S r SS SSSSSSSS rr r r r r ddrdrr r G rU S S S > + − −= ∫ ∫∫ϕ θ γ ϕθθθϕρ 2 2 cos21 sin,, ( ) S n n n SSS rrP r 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 n uP 1 !2 1 2 − = Gravitation of a Distribution of Mass Defined by the Density ( )Sr ρ ( ) ( ) ( ) ( ) 2/13&,1 2 210 −=== uuPuuPuPExamples: Using Spherical Coordinates (rS, φS, θS) The Potential Differential of any distribution of mass is given by:
- 13. SOLO 13 Earth Gravitation ( ) ( ) ( ) S r Md SSSSSSSS n n n S rrddrdrrP r r r G rU S S S > −= ∫ ∫ ∫ ∑ ∞ =ϕ θ ϕθθθϕργ sin,,cos 2 0 From Spherical Trigonometry: ( )SSS ϕϕθθθθγ −+= cossinsincoscoscos According to Addition Theorem for Spherical Harmonics: ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )∑= − + − += n m S m n m nSSnnn PPm mn mn PPP 1 coscoscos ! ! 2coscoscos θθϕϕθθγ ( ) ( ) ( )uP ud d uuP nm m mm n 2/2 1: −= Where Pn m (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 ( )Sr ρ ( ) ( )uPuP nn = 0 ( ) ( ) ( ) ( )22 2 2 1 21 2 13,13 ttPtttP −=−=They satisfy: The Potential of any distribution of mass is given by: See Presentation on “Legendre Functions”
- 14. SOLO 14 Earth Gravitation The Potential of any distribution of mass is given by: ( ) ( ) ( ) ( ) ( ) S n r Md SSSSSSSSn n S k S r Md SSSSSSSS k n n S rrddrdrrP a r r a r G rrddrdrrP r r r G rU 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 m S m n m nSSnnn PPm mn mn PPP 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 S n n n n m r SSSSSSSSSS m n n Sm n n n r SSSSSSSSSn n S n n M r Md SSSSSSSS S S S S S S S S S SS S S ddrdrrmP a r mP r a mn mn r G ddrdrrmP a r mP r a mn mn r G ddrdrrP a r P r a r G ddrdrrP a r r G rU ϕ θ ϕ θ ϕ θϕ θ ϕθθθϕρϕθϕθ ϕθθθϕρϕθϕθ ϕθθθϕρθθϕθθθϕργ Gravitation of a Distribution of Mass Defined by the Density ( )Sr ρ a – any reference length
- 15. SOLO 15 Earth Gravitation ( ) ( ) ( ) ( ) ( )( ) + + +−= ∑∑∑ ∞ = = ∞ = 1 11 0 sincoscoscos1 n n m nmnm m n n n nn n mSmCP r a CP r a r MG rU ϕϕθθ ( ) ( )∫∫∫ == S S Sr Md SSSSSSSSSn n S nn ddrdrrP a r M CC ϕ θ ϕθθθϕρθ sin,,cos 1 : 2 0 ( ) ( ) ( ) ( ) ( ) ( )∫ ∫∫ + − = S S Sr Md SSSSSSSS S S S m n n S nm nm ddrdrr m m P a r mn mn S C ϕ θ ϕθθθϕρ ϕ ϕ θ sin,, sin cos cos ! ! 2: 2 Gravitation of a Distribution of Mass Defined by the Density ( )Sr ρ The Potential of any distribution of mass is given by:
- 16. 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 ( ) ( ) ,2,10cossin 2 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. ( ) ( ) 0cos 1 sin,cos 1 : 2 cos 11 = = = ∫∫ ∫∫ m S S r md SSSSSSSS S md a r M ddrdrrP a r M C S S S S θϕθθθρθ ϕ 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 ( )Sr ρ In this case ( ) ( ) 0 1 :cos1 nn n nn n CJJP r a r MG rU −= −−= ∑ ∞ = θ
- 17. 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 S n ddPrdr a r M a C Gravitation of a Distribution of Mass Defined by the Density ( )Sr ρ ( ) r MG rU = In this case Return to Table of Content
- 18. 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 Distributed Mass the Gravitation Potential at a distance r > R is given by ( ) r MG r R GVd rr GrU S V S SS −=−= − −= ∫ 3/41 3 π ρρ The Gravitation Potential for a Static Spherical Volume with an Uniform Distributed Mass is equivalent to a Point Mass Gravitation Potential concentrated at the Center of the Sphere. The Equi-Potential Surfaces will be Concentric Spherical Surfaces outside the Spherical Mass.
- 19. SOLO 19 Earth Gravitation The Potential of the Reference Ellipsoid is given by: where: UG – Gravitational Potential (m2 /s2 ) GM – Earth’s Gravitational Constant r - Distance from the Earth’s Center of Mass rS - 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 md GrUd − = 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
- 20. SOLO 20 Earth Gravitation The Potential Equation of Gravity Field where: UG – Gravitational Potential (m2 /s2 ) GM – Earth’s Gravitational Constant r - Distance from the Earth’s Center of Mass rS - Distance from the Earth’s Center of Mass to dm ( ) ( ) ( ) ( ) ( )rG VdrrrG Vd rr rGrU S S V SSS V S S SS ρπ δπρ ρ 4 4 122 −= −−= − ∇−=∇ ∫∫∫ ∫∫∫ ( ) ( )∫∫∫∫ − −= − −= SV S S S M S rr Vd rG rr md GrU ρStart with: Inside the mass M we have: ( ) ( )rGrU ρπ42 −=∇Poisson Equation: ( ) 02 =∇ rU Laplace Equation: Outside the mass M we have: Siméon Denis Poisson 1781-1840 Pierre-Simon Laplace (1749-1827) Return to Table of Content
- 21. SOLO 21 Earth Gravitation Physical Meaning of the Low Degree and Order SHCs Gravitation of a Distribution of Mass Defined by the Density ( )Sr ρ 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 11 0 sincoscoscos1 n n m nmnm m n n n nn n mSmCP r a CP r a r MG rU ϕϕθθ We have (see Figure) = = S SS SS S s s s S r z y x r θ ϕθ ϕθ cos sinsin cossin ( ) 0& 1 cos 1 : 10 cos 1 1 110 === == ∫∫ S a z Mdz Ma MdP a r M CC G M S M S S S θ θ ( ) = = = = ∫∫ ∫ ay ax Md y x Ma Md a r M MdP a r MS C G G M S S M M S S S S S S S S / / 2 2 sin cos sin 2 sin cos cos 2 1 1 1 1 11 11 ϕ ϕ θ ϕ ϕ θ ( ) ( )2 1 21 1 1 ttP −=where we used Legendre Polynomials. GGG zyx ,, are the coordinates of the Mass Center of Gravity
- 22. SOLO 22 Earth Gravitation Physical Meaning of the Low Degree and Order SHCs Gravitation of a Distribution of Mass Defined by the Density ( )Sr ρ ( ) ( ) ( ) ( ) ( )( ) + + +−= ∑∑∑ ∞ = = ∞ = 1 11 0 sincoscoscos1 n n m nmnm m n n n nn n mSmCP r a CP r a r MG rU ϕϕθθ = = S SS SS S s s s S r z y x r θ ϕθ ϕθ cos sinsin cossin + −−= −= 2 1 2 31 2220 SSSS SSSSSS yyxx zzzzrr II I Ma II Ma C ( )∫ = M S S MdP a r MS C 0 1 cos 2 0 2 2 20 20 θ ( ) ( ) + −++= + −= −−− = − == 2 3 1 2 1 2 3 2 1cos3 coscos 2 222 2 22 2 22 22222 2 2 22 2 0 22 2 SS SSS SS S SSSSSS S S S S yx zyx a yx z aa zyxz a r P a r P a r θ θθ ( ) ∫∫∫ = = = = M zy zx SS SS M S S SSS M S S S S MaI MaI Md zy zx Ma Mdr Ma MdP a r MS C SS SS 2 2 2 2 2 1 2 2 21 21 / /1 sin cos cossin3 3 1 sin cos cos !3 12 ϕ ϕ θθ ϕ ϕ θ ( ) ( ) ( ) ( )∫∫∫ − = − = − ⋅ = = M yx xxyy SS SS M SS SS SS M S S S S MaI MaII Md yx yx Ma Mdr Ma MdP a r MS C SS SSSS 2 222 2 22 22 2 2 2 2 22 22 2/ 4/ 24 1 cossin2 sincos sin3 34 1 2sin 2cos cos !4 !12 ϕϕ ϕϕ θ ϕ ϕ θ ( ) ( ) ( ) ( )22 2 2 1 21 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
- 23. SOLO 23 Earth Gravitation Physical Meaning of the Low Degree and Order SHCs Gravitation of a Distribution of Mass Defined by the Density ( )Sr ρ
- 24. SOLO 24 Earth Gravitation Physical Meaning of the Low Degree and Order SHCs Gravitation of a Distribution of Mass Defined by the Density ( )Sr ρ Return to Table of Content
- 25. 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 ba f − =: R = 6,371.000 km a = 6,378.136 km b = 6,356.751 km Reference Earth Model Return to Table of Content
- 26. SOLO 26 Reference Ellipsoid Meridian Ellipse Equation: 12 2 2 2 =+ b z a p Slope of the Normal to Ellipse: 2 2 tan bp az zd pd =−=φ The Slope of the Geocentric Line to the same point p z O =φtan OO RzRp φφ sincos == Deviation Angle between Geographic and Geodetic At Ellipsoid Surface O b a φφ tantan 2 2 = = − O b a φφ tantan 2 2 1 ( ) ( ) φφφφ tan1tan1tantan 22 2 2 fe a b O −=−== a ba f − =: 2 2 22 2 2: ff a ba e −= − = Earth Gravitation
- 27. SOLO 27 Reference Ellipsoid Meridian Ellipse Equation: Oφφδ −= 12 2 2 2 =+ b z a p Slope of the Normal to Ellipse: 2 2 tan bp az zd pd =−=φ The Slope of the Geocentric Line to the same point p z O =φtan −= −+ − = + − = + − = 1 11 1 1 tantan1 tantan tan 2 2 2 2 2 2 2 2 2 22 22 2 2 b a a zp a p p a b a p z bp az p z bp az O O φφ φφ δ OO RzRp φφ sincos == ( ) ( ) ( )OO f O f ba R b ba a ba R ba ba φφφδ 2sin2sin 2 tan2sin 2 tan 1 2 11 12 22 22 1 ≈ + − = − = ≈≈<< −− Deviation Angle between Geographic and Geodetic At Ellipsoid Surface Earth Gravitation
- 28. 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 Geodetic At Altitude h from Ellipsoid Surface ( ) R h hR huu ≈ + ≈= − 11 sin sin sin sin δδπ Since u and δ1 are small: 1δ R h u ≈ The corrected value of δ is: ( )Of R h R h u φδδδ 2sin11 11 −= −=−= Therefore: ( )OO f R h φλδφφ 2sin1 −+=+= Earth Gravitation
- 29. 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: ( ) ( ) ( ) ( ) + + + = = φ φλ φλ sin cossin coscos HR HR HR z y x P M N N E E E E ( ) NhH e a RN += − = 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 ea RM − − = Earth Gravitation
- 30. SOLO 30 Reference Ellipsoid b a Oφ ?O – Geographic Latitude ? – Geodetic Latitude Equator North Pole Tangent to Ellipsoid p NR z P a r φ β Meridian Ellipse Ellipsoid Equation: ( ) 222 2 2 2 22 11 aeb b 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 dz 2 2 cot =−=φFrom the Figure: φtan2 2 p a b z = ( )φφ 222 4 2 22 4 2 2 2 tantan1 ba a p p a b a p +=+= φφ φ φφ φ 2222 2 2222 2 sincos sin , sincos cos ba b z ba a p + = + = Earth Gravitation
- 31. SOLO 31 Reference Ellipsoid ( ) φ φ φφ φ φ φ φφ φ 22 2 2222 2 222222 2 sin1 cos1 sincos sin sin1 cos sincos cos e ea ba b z e a ba a p − − = + = − = + = ( ) 222 1 aeb −= Earth Gravitation
- 32. SOLO 32 Reference Ellipsoid ( ) 222 1 aeb −= 12 2 2 2 =+ b z a p 0 0 sin cos φ φ rz rp = = 1 sincos 2 0 2 2 0 2 2 = + ba r φφ ( ) 2/1 2 0 2 2 0 2 0 sincos − += ba r φφ φ ( ) ( ) ( )0 22/1 0 2 2/1 0 2 2 2/1 0 2 2 222/1 0 2 2 2 0 2 0 sin1sin21 sin1sin1sincos φφ φφφφφ fafa b ba b a a ba a b ba a b a ar f −≈+≈ +− += − += += − − ≈ −− Earth Gravitation
- 33. 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: ( ) 222 2 2 2 2 11 aeb b z a p −==+ From this Equation, at any point (x,y) on the Ellipse, we have: φtan 1 2 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 φ φ 2 22 2 2 2 222 2 2 2 2 2 2 2 2 cos sin1 1 1 tan1111 e a p e e a p b a p z a p − = − −+= += ( ) ( ) ( ) 2/122 2 2 2 2/122 sin1 sin1 tan sin1 cos φ φ φ φ φ e ea p a b z e a p − − ==→ − = From the Figure on right: ( ) 2/122 sin1cos φφ e ap RN − == Earth Gravitation
- 34. SOLO 34 Reference Ellipsoid Let develop the RN and RM (continue): we have at any point (p,z) on the Ellipse: φtan 1 2 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 tan 1 11 : φφ φφ e ea e ea ea pd zd pd zd RM − − = − − − + = + = ( ) ( ) ( ) 2/122 2 2/122 sin1 sin1 sin1 cos φ φ φ φ e ea z e a p − − = − = ( ) ( ) 2/322 2 sin1 1 : φe ea RM − − = Radiuses of Curvature of the Ellipsoid Earth Gravitation
- 35. SOLO 35 Reference Ellipsoid ( ) ( ) 2/322 2 sin1 1 : φe ea RM − − = ( ) 2/122 sin1cos φφ e ap RN − == a ba f − =: 2 2 22 2 2: ff a ba e −= − =Using ( ) ( )[ ] ( ) ( ) [ ] ++−≈ +−++−≈ −− − = φφ φ 2222 2/322 2 sin321sin2 2 3 121 sin21 1 : ffaffffa ff fa RM ( )[ ] ( ) [ ]φφ φ 222 2/122 sin31sin2 2 3 1 sin21 faffa ff a RN +≈ +−+≈ −− = [ ]φ2 sin321 ffaRM +−≈ [ ]φ2 sin31 faRN +≈ We used and we neglect f2 terms ( ) ( ) + − ++= − !2 1 1 1 1 nn xn x n Radiuses of Curvature of the Ellipsoid Return to Table of Content Earth Gravitation
- 36. 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 ba f − =: Alexis Claude Clairaut )1713–1765( −+= φ2 sin 2 5 1 fmGg Earth Gravitation Return to Table of Content
- 37. SOLO 37 Reference Earth Model Mac Cullagh’s Approximation (1845) James MacCullagh (1809 – 1847) Mac Cullagh’s used the Approximation: ( ) ( ) ( ) ( ) ( ) − − + +−≈ ∫∫ ≈ M P SS P r a M M P S S P Md a r Mr a Md a r Mr a M r G rU S CM S θθ θ θ θθ θθ cos 22 cos 22 cos 1 cos 1 22 0 1 1 2 1cos31 2 1cos3 cos 1 cos1 ∫∫∫ + −++= + −= − = M SS SSS M SS S M SS dM yx zyx Ma dM yx z Ma dM a r M C 2 3 1 2 1 2 1cos31 : 2 222 2 22 2 2 2 2 2 20 θ 222220 2 1 2 31 J Ma ACII I Ma II Ma C SSSS SSSSSS yyxx zzzzrr −= − −= + −−= −= ( ) ( ) ( ) ( ) ( )SSSSSSSSSSSSSS zzyyxx M SSSrr M SSzz M SSyy M SSxx IIIMdzyxIMdyxIMdzxIMdzyI ++=++=+=+=+= ∫∫∫∫ 2 1 ::,:,: 222222222 Define Moments of Inertia 2 :,: SSSS SS yyxx zz II AIC + == ( ) ( ) Ma AC JJ r aMG M r G rU P 22 2 23 2 : 2 1cos3 2 − = − +−≈ θ θ Earth Gravitation Return to Table of Content
- 38. 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 ( ) isϕ derived 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 θ π ψϕλ −== 2 & Earth Gravitation
- 39. 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 yxV yx C Uyx yx yx Va ∇=+Ω= + + Ω= +Ω= 11 11 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 1 yxUC +Ω= By choosing Const. = 0, we obtain
- 40. 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 sincossin1 n n n m nmnmnm n G mSmCP r a r MG U λλψ where: UG – Gravitational Potential (m2 /s2 ) GM – Earth’s Gravitational Constant r - Distance from the Earth’s Center of Mass a - Semi-Major Axis of the WG 84 Ellipsoid n,m – Degree and Order respectively ψ– Geocentric Latitude λ – Geocentric Longitude = Geodetic Longitude Return to Table of Content
- 41. SOLO 41 Earth Gravitation The WGS 84 Gravity Model is defined in terms of normalized coefficients: ( ) ( ) ( ) ( ) ( )( ) + + +−= ∑∑∑ ∞ = = ∞ = 1 11 0 sincossinsin1 n n m nmnm m n n n nn n mSmCP r a CP r a r MG rU λλψψ ( )∫ == M Sn n S nn MdP a r M CC ψsin 1 :0 ( ) ( ) ( ) ( ) ( )∫ + − = M S S S m n n S nm nm Md m m P a r mn mn S C λ λ ψ sin cos sin ! ! 2: ( ) ( ) ( ) ( ) ( )( ) + + +−= ∑∑∑ ∞ = = ∞ = 1 11 0 sincossinsin1 n n m nmnm m n n n nn n mSmCP r a CP r a r MG rU λλψψ ( ) ( ) ( ) ( ) ( ) ≠ = = + − += 02 01 sin ! ! 12sin m m kP mn mn knP m n m n ψψ ( ) ( ) ( ) ≠ = = − + + = 02 01 ! ! 12 1 m m k 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)
- 42. 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
- 43. SOLO 43 World Geodetic System (WGS 84) Reference Earth Model Carlo Somigliana (1860 –1955) The Theoretical Gravity on the surface of the Ellipsoid is given by the Somigliana Formula (1929) 84 22 2 2222 22 sin1 sin1 sincos sincos WGS e pe e k ba ba φ φ γ φφ φγφγ γ − + = + + = where 1: −= e p a b k γ γ 2 22 : a ba e − = - 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 by P. Pizzetti (1894) Earth Gravitation
- 44. 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
- 45. SOLO 45 World Geodetic System (WGS 84) Reference Earth Model Earth Gravitation
- 46. 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 m Minimum = -106.99 m, Maximum = 85.39 m Earth Gravitation
- 47. 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
- 48. SOLO 48 World Geodetic System (WGS 84) Reference Earth Model a ba f − =:f - Ellipsoid Flattening (Ellipticity) a - Ellipsoid Semi-major Axis b - Ellipsoid Semi-minor Axis e - Ellipsoid Eccentricity 2 2 22 2 2: ff a ba e −= − = ( ) 2 11 eafab −=−= Reference Ellipsoid Earth Gravitation
- 49. 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: ( ) ( ) ( ) ( ) + + + = = φ φλ φλ sin cossin coscos HR HR HR z y x P M N N E E E E ( ) NhH e a RN += − = 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 ea RM − − = Radius of Curvature in the Prime Vertical Return to Table of Content Earth Gravitation
- 50. 50 SOLO Air Vehicle in Ellipsoidal Earth Atmosphere SIMULATION EQUATIONS
- 51. 51 SOLO 7. Forces Acting on the Vehicle (continue – 4) Gravitation Acceleration ( ) ( ) − − − == zg yg xg gg 100 0 0 0 010 0 0 0 001 χχ χχ γγ γγ σσ σσ cs sc cs sc cs scC EW E W ( ) gg − = γσ γσ γ cc cs s W 2sec/174.322sec/81.9 0 2 0 0 0 gg ftmg HR R == + = The derivation of Gravitation Acceleration assumes an Ellipsoidal Symmetrical Earth. The Gravitational Potential U (R, ) is given byϕ ( ) ( ) ( ) ( )φ φ µ φ , sin1, 2 RUg P R a J R RU E E n n n n ∇= −⋅−= ∑ ∞ = μ – The Earth Gravitational Constant a – Mean Equatorial Radius of the Earth R=[xE 2 +yE 2 +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 Function P (sin ) – Associated Legendre Polynomialsϕ Air Vehicle in Ellipsoidal Earth Atmosphere
- 52. 52 SOLO 7. Forces Acting on the Vehicle (continue – 5) Gravitation Acceleration Retaining only the first three terms of the Gravitational Potential U (R, ) we obtain:ϕ R z R z R z R a J R z R a J R g R y R z R z R a J R z R a J R g R x R z R z R a J R z R a J R g EEEE z EEEE y EEEE x E E E ⋅ +⋅−⋅ ⋅− −⋅ ⋅−⋅−= ⋅ +⋅−⋅ ⋅− −⋅ ⋅−⋅−= ⋅ +⋅−⋅ ⋅− −⋅ ⋅−⋅−= 34263 8 5 15 2 3 1 34263 8 5 15 2 3 1 34263 8 5 15 2 3 1 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 z R y R x E E E ( ) 2/1222 EEE zyxR ++= Air Vehicle in Ellipsoidal Earth Atmosphere
- 53. 53 SOLO 23. Local Level Local North (LLLN) Computations for an Ellipsoidal Earth Model ( ) ( ) ( ) ( ) ( )2 22 10 2 0 2 0 2 0 5 2 1 2 0 6 0 sin sin1 sin321 sin1 sec/10292116557.7 sec/051646.0 sec/780333.9 26.298/.1 10378135.6 Ae e p m e HR RLatgg g LatfRR LatffRR LatfRR rad mg mg f mR + + = += +−= −= ⋅=Ω = = = ⋅= − Lat HR V HR V HR V Ap East Down Am North East Ap East North tan + −= + −= + = ρ ρ ρ Lat Lat Down East North sin 0 cos Ω−=Ω =Ω Ω=Ω DownDownDown EastEast NorthNorthNorth Ω+= = Ω+= ρφ ρφ ρφ East North Lat Lat Long ρ ρ −= = • • cos ( ) ( ) ∫ ∫ • • += += t t dtLatLattLat dtLongLongtLong 0 0 0 0 SIMULATION EQUATIONS Return to Table of Content Air Vehicle in Ellipsoidal Earth Atmosphere
- 54. 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. 55 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA
- 56. 56 Vector AnalysisSOLO Vector Operations in Various Coordinate Systems Laplacian 2 ∇=∇⋅∇ •Cartesian: 2 2 2 2 2 2 2 z U y U x U U ∂ ∂ + ∂ ∂ + ∂ ∂ =∇ •Cylindrical: 2 2 2 2 22 2 2 2 2 2 2 2 1111 z UUUU z UUU U ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ =∇ θρρρρθρρ ρ ρρ •Spherical: ( ) φθθ θ θθ φθ 2 2 222 2 2 2 sin 1 sin sin 11 ,, ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ =∇ U r U rr U r rr rU
- 57. 57 SOLO Laplace Differential Equation in Spherical Coordinates 0 sin 1 sin sin 11 2 2 222 2 2 2 = ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ =∇ φθθ θ θθ U r U rr U r rr U Let solve this equation by the method of Separation of Variables, by assuming a solution of the form : ( ) ( ) ( )φθφθ ,,, SrRrU = Spherical Coordinates: θ ϕθ ϕθ cos sinsin cossin rz ry rx = = = In Spherical Coordinates the Laplace equation becomes: Substituting in the Laplace Equation and dividing by U gives: 0sinsin sin 11 2 2 22 2 2 = ∂ ∂ + ∂ ∂ ∂ ∂ + φθ θ θ θ θ SS Srrd Rd r 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: λ φθ θ θ θ θ λ −= ∂ ∂ + ∂ ∂ ∂ ∂ = 2 2 2 2 sinsin sin 1 1 SS S rd Rd r rd d R
- 58. 58 SOLO Laplace Differential Equation in Spherical Coordinates λ= rd Rd r rd d R 21 ( ) ( )ϕθ,SrR=Φ We get: 022 2 2 =−+ R rd Rd r rd Rd r 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 =−−− α λααα rC or: ( ) 01 =−+ λαα Let define l as l (l+1):=λ ( ) ( ) ( ) 0111 2 =+−+=+−+ llll αααα ( ) ( ) −− = +±− = ++±− = 12 121 2 1411 2,1 l llll α ( ) = +1 1 l l r r rR
- 59. 59 SOLO ( )1sinsin sin 1 2 2 2 +−= ∂ ∂ + ∂ ∂ ∂ ∂ ll SS S φθ θ θ θ θ ( ) ( )ϕθ,SrR=Φ We obtain: Multiply this by S sin2 θ and put to get:( ) ( ) ( )φθφθ ΦΘ=,S 0 1 sinsinsin 1 2 2 2 = Φ Φ + + Θ ∂ ∂ Θ φ θλ θ θ θ θ 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 ∂ ∂ 2 2 2 2 2 0 sin 1sin sin 1 m d d m ll d d φ θθ θ θθ The Φ ( ) must be periodical in (a period of 2 π) and becauseϕ ϕ this we choose the constant m2 , with m an integer. Thus: ( ) φφφ mbma sincos +=Φ Laplace Differential Equation in Spherical Coordinates ( ) φφ φ mjmj ee +− =Φ ,or With m integer, we have the Orthogonality Condition 21 21 , 2 0 2 mm mjmj dee δπϕ π φφ =⋅∫ +−
- 60. 60 SOLO ( ) ( ) ( ) ( )φθφθ ΦΘ= rRrU ,, We get: or: ( ) Θ+−=Θ− Θ + Θ 1 sin cot 2 2 2 2 ll m d d d d θθ θ θ ( ) 0 sin 1sin sin 1 2 2 =Θ −++ Θ ∂ ∂ θθ θ θθ m ll d d Laplace Differential Equation in Spherical Coordinates Change of variables: t = cos θ θθ dtd sin−= td d d d Θ −= Θ θ θ sin ( ) td d t td d t 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 2 2 cossin/1 2 2 2 2 2 1 sin sinsinsinsin θθ θ θθ θθθθ θθθ ( ) ( ) Θ − −++ Θ − Θ − Θ =Θ −++ Θ + Θ = = 2 2 2 2cos 2 2 2 2 1 1 sin 1cot0 t m ll td d t td d t td dm ll d d d d t θ θθ θ θ We obtain: ( ) 1cos ,2,1,00 1 12 2 2 2 2 ≤⇒= ==Θ − −++ Θ − Θ tt m t m ll td d t td d θ Associate Legendre Differential Equation
- 61. 61 SOLO ( ) ( ) ( ) ( )φθφθ ΦΘ= rRrU ,, Laplace Differential Equation in Spherical Coordinates We obtain: ( ) 1cos ,2,1,00 1 12 2 2 2 2 ≤⇒= ==Θ − −++ Θ − Θ tt m t m ll td d t td d θ Associate Legendre Differential Equation Let start with m = 0 with: ( ) 1cos 0122 2 ≤⇒= =Θ++ Θ − Θ tt ll td d t td d θ Legendre 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(
- 62. SOLO 62 Legendre Polynomials Olinde Rodrigues (1794-1851) Start from the function: ( ) .12 constktky n =−= ( ) 12 12:' − −== n ttkn td yd y ( ) ( ) ( ) 22212 2 2 11412:'' −− −−+−== nn ttnkntkn td yd y Let compute: ( ) ( ) ( ) ( ) ( ) '12211412''1 12222 ytntnttnkntknyt nn −+=−−+−=− − or: ( ) ( ) 02'12''12 =−−+− ynytnyt Let differentiate the last equation n times with respect to t: ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ''1''2''1 00''1 3 ''1 2 ''1 1 ''1''1 2 2 1 1 2 3 3 0 2 3 3 2 2 2 2 2 1 1 222 y td d nny td d tny td d t y td d t td dn y td d t td dn y td d t xd dn y td d tyt td d n n n n n n n n n n n n n n n n − − − − − − − − − − −++−= +− +− +− +−=− ( ) ( ) ( ) ( ) +−= +−=− − − − − ''12' 1 '12'12 1 1 1 1 y td d ny td d tny td d t td dn y td d tnyt td d n n n n n n n n n n n Derivation of Legendre Polynomials via Rodrigues’ Formula Laplace Differential Equation in Spherical Coordinates
- 63. SOLO 63 Legendre Polynomials Olinde Rodrigues (1794-1851) Start from the function: ( ) .12 constktky n =−= ( ) ( ) 02'12''12 =−−+− ynytnytDifferentiate n times with respect to t: ( ) ( ) ''1''2''1 2 2 1 1 2 y td d nny td d tny td d t n n n n n n − − − − −++− ( ) 02''12 1 1 =− +−+ − − y td d ny td d ny td d tn n n n n n n Define: a Polynomial( ) ( )[ ]n n n n n t td d k td yd tw 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 d t td d Derivation of Legendre Polynomials via Rodrigues’ Formula Laplace Differential Equation in Spherical Coordinates
- 64. SOLO 64 Legendre Polynomials Olinde Rodrigues (1794-1851) Let find k such that: by choosing Pn (1) = 1 ( ) ( )[ ]n n n n n n t td d k td yd tP 12 −== ( ) ( )[ ] ( ) ( ) ( ) ( ) −+= +−=−= ∑>0 22 1!2111 i inn v n u n n n n n n n ttatnktt td d ktk td d tP !2 1 n k n = We obtain the Rodrigues Formula: ( ) ( )[ ]n n n nn t td d n tP 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 ): ( ) ( ) ( ) udvudvdnvddu nn vddunvdu vdud mnm n vud nnnnn n m mnmn +++ − ++= − =⋅ −−− = − ∑ 1221 0 !2 1 !! ! We have: ( ) ( ) 1!2 !2 1 1 1!20 12 0 21 00 == +++ − ++== = −−− nkudvudvdnvddu nn vddunvdukxP n xn nnnnn n 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
- 65. SOLO 65 Legendre Polynomials The 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. 66 SOLO Orthogonality of Legendre Polynomials Define ( ) ( )tPwtPv nm == :&: We use Legendre’s Differential Equations: ( ) ( ) 011 2 =++ − vmm td vd t td d ( ) ( ) 011 2 =++ − wnn td wd t td d Multiply first equation by w and integrate from t = -1 to t = +1. ( ) ( ) 011 1 1 1 1 2 =++ − ∫∫ + − + − dtwvmmdtw td vd t td d Integrate the first integral by parts we get ( ) ( ) ( ) 0111 1 1 1 1 2 0 1 1 2 =++−−− ∫∫ + − + − += −= dtwvmmdt td wd td vd tw td vd t 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 vd t Legendre Polynomials
- 67. 67 SOLO Orthogonality of Legendre Polynomials ( ) ( ) 011 1 1 1 1 2 =++−− ∫∫ + − + − dtwvmmdt td wd td vd t Subtracting those two equations we obtain ( ) ( ) 011 1 1 1 1 2 =++−− ∫∫ + − + − dtwvnndt td wd td vd t ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) 01111 1 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 2 21 1 n n n tPu utu ( )∑ ∫∫ ∞ = + − + − = +− 0 1 1 22 1 1 2 21 1 n n n dttPudt utu Legendre Polynomials
- 68. 68 SOLO Orthogonality of Legendre Polynomials ( )∑ ∫∫ ∞ = + − + − = +− 0 1 1 22 1 1 2 21 1 n n n dttPudt utu ( ) ( ) ( ) 1 1 1 ln 1 1 1 ln 2 1 21ln 2 1 21 1 2 21 1 2 1 1 2 < − + = + − − =−+ − = −+ += −= + −∫ u u u uu u u tuu u dt tuu t t ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∑∑∑ ∞ ++∞ +∞ + + −− −= + − −− + −=−−+ 0 11 0 1 0 1 1 1 1 1 1 1 1 1 1 1ln 1 1ln 1 n uu un u un u u u u u u nn n n n n n ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∑∑∑∑ ∞∞ +∞ ++ + ∞ ++ + = + = + −− −+ + −− −= 0 2 0 12 0 0 1212 12 0 1212 2 12 2 12 1 2 12 1 1 12 1 1 n nnn n nn n u nn u un uu un uu u Let compute first Therefore ( )∑ ∫∑∫ ∞ + − ∞+ − = + = +− 0 1 1 22 0 2 1 1 2 12 2 21 1 dttPuu n dt utu n nn Comparing the coefficients of u2n we get ( ) 12 21 1 2 + =∫ + − n dttPn Legendre Polynomials ( ) ( ) nmmn n dttPtP δ 12 21 1 + =∫ + − Hence
- 69. 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 i m m td tgd td tsd imi m tgts td d − − = ∑ − =⋅ 0 !! ! Start with Legendre Differential Equation: ( ) ( ) ( ) ( ) 1011 2 ≤=++ − ttwnntw td d t td d nn or: ( ) ( ) ( ) ( ) ( ) 10121 2 2 2 ≤=++−− ttwnntw td d ttw td d t nnn ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )twmmtwtmtwttw td d t td d m n m n m nnm m 1211 122 2 2 2 −−−−= − ++ ( ) ( ) ( ) ( ) ( )twmtwttw td d t td d m n m nnm m += +1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )twnntwmtwttwmmtwtmtwt m n m n m n m n m n m n 122121 1122 ++−−−−−− +++ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) 011121 122 =+−+++−−= ++ twmmnntwtmtwt m n m n m n Laplace Differential Equation in Spherical Coordinates
- 70. SOLO 70 Associated Legendre Functions ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) 011121 122 =+−+++−− ++ twmmnntwtmtwt m n m n m n Define: ( ) ( ) ( )twty m n=: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) 011121 122 =+−+++−− tymmnntytmtyt Now define: ( ) ( ) ( )tyttu m 22 1: −= Let compute: ( ) ( ) ( )1221 22 11 ytyttm td ud mm −+−−= − ( ) ( ) ( ) ( )11 22222 111 ytyttm td ud t mm + −+−−=− ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )21 221221221 2222222 1121111 ytyttmyttmyttmytm td ud t td d mmmmm +− −+−+−−−−+−−= − ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]{ } ( ) ( ) y t tm mnnmmtymmnnytmytt mm − +−+−+−++−+++−−−= 2 22 222 0 12222 1 11111211 We get: ( ) ( ) 0 1 11 2 2 2 = − −++ − u t m nn td ud t td d Associate Legendre Differential Equation Laplace Differential Equation in Spherical Coordinates
- 71. SOLO 71 Associated Legendre Functions Define: ( ) ( ) ( )tw td d ttu nm mm 22 1: −= We get: ( ) ( ) 0 1 11 2 2 2 = − −++ − u t m nn td ud t td d Start with Legendre Differential Equation: ( ) ( ) ( ) ( ) 1011 2 ≤=++ − ttwnntw td d t td d nn 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 Pn m (t) of this Differential Equation are called Associated Legendre Functions, because they are derived from the Legendre Polynomials ( ) ( ) ( )tP td d ttP nm mm m n 22 1: −= Laplace Differential Equation in Spherical Coordinates
- 72. SOLO 72 Associated Legendre Functions Examples ( ) ( ) ( )tP td d ttP nm mm m n 22 1: −= ( ) ( ) 10 0 0 0 === tPtPn ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) θ θ θ θ θ θ sin1 cos: sin111 cos 2 1 21 1 1 1 cos 1 0 1 cos 2 1 22 1 21 1 1 −=−−=−= === =−=−== = − = = t t t tP ttPtP ttPtP tt td d ttPn ( ) ( ) ( ) ( ) ( )tP mn mn tP m n mm n ! ! 1 + − −= − ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) θ θθ θ θθ θ θ θ θ θ θ 2 cos 22 2 22 2 cos 2 1 211 2 2cos2 2 0 2 cos 2 1 2 2 2 1 21 2 2 cos 2 2 2 2 2 2 22 2 sin 8 1 1 8 1 !4 !0 1 2 cossin 13 !3 1 1 2 1cos3 2 13 cossin313 2 13 1 sin313 2 13 12 1 2 2 1 = − = − = = = =−=−= −=−−= − = − == =−= − −= =−= − −== t t tP t t tP t tP ttPtP tttP t tPtP tt t td d ttP t t td d ttPn ( ) 10 =tP ( ) 2 1 2 3 2 2 −= ttP( ) ttP =1 Laplace Differential Equation in Spherical Coordinates
- 73. SOLO 73 Associated Legendre Functions Examples ( ) ( ) ( )tP td d ttP nm mm m n 22 1: −= ( ) ( ) ( ) ( ) ( )tP mn mn tP m n mm n ! ! 1 + − −= − Laplace Differential Equation in Spherical Coordinates
- 74. SOLO 74 Associated Legendre Functions Orthogonality of Associated Legendre Functions ( ) ( ) ( )[ ]n mn mnm n m n t td d t n tP 11 !2 1 : 222 −−= + + Let Compute ( ) ( ) ( ) ( )[ ] ( )[ ]∫∫ + − + + + + + + − −−−= 1 1 222 1 1 111 !!2 1 dtt td d t td d t qp dttPtP q mq mq p mp mp m qp m q m p Define X := x2 -1 ( ) ( ) ( ) [ ] [ ]∫∫ + − + + + + + + − − = 1 1 1 1 !!2 1 dtX td d X td d X qp dttPtP q mq mq p mp mp m qp m m q m p If p ≠ q, assume q > p and integrate by parts q + m times [ ] [ ] mqidtX td d vdX td d X td d u q imq imq p mp mp m i i +== = −+ −+ + + ,,1,0 All the integrated parts will vanish at the boundaries t = ± 1 as long as there is a factor X = x2 -1. We have, after integrating m + q times ( ) ( ) ( ) ( ) [ ]∫∫ + − + + + + + ++ − −− = 1 1 1 1 !!2 11 dtX td d X td d X qp dttPtP p mp mp m mq mq q qp mqm m q m p
- 75. SOLO 75 Associated Legendre Functions ( ) ( ) ( ) ( ) [ ] 1: !!2 11 2 1 1 1 1 −= −− = ∫∫ + − + + + + + ++ − xXdtX td d X td d X qp dttPtP p mp mp m mq mq q qp mqm m q m p Because the term Xm contains no power greater than x2m , we must have q + m – i ≤ 2 m or the derivative will vanish. Similarly, p + m + i ≤ 2 p Adding both inequalities yields q ≤ 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 p imp imp m imq imq qp mp mp m mq mq q X td d X td d imqi mq XX td d X td d X 0 !! ! ( ) ( ) qpdttPtP m q m p ≠=∫ + − 0 1 1 This proves that the Associated Legendre Functions are Orthogonal (for the same m). Orthogonality of Associated Legendre Functions
- 76. SOLO 76 Associated Legendre Functions ( )[ ] ( ) ( ) 1: !!2 11 2 1 1 2 21 1 2 −= −− = ∫∫ + − + + + ++ − xXdtX td d X td d X pp dttP p mp mp m mp mp p p mp m p Let expand the integrand on the right-side using Leibniz’s formula ( ) ( )∑ + = ++ ++ −+ −+ + + + + −+ + = mp i p imp imp m imp imp pp mp mp m mp mp p X td d X td d impi mp XX td d X td d X 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 2 1 1 !2 2 2 !2 2 2 22 1 1 2 −=⋅ − +− = ∫∫ + − + − xXdtX td d X td d X mmpp mp dttP p p p p m m m m p p p m p ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )! ! 12 2 sin1 !!2 !2!1 1 !!2 !2!1 !12 !2 1 0 12 22 cos1 1 2 22 212 mp mp p d mpp pmp dtx mpp pmp p p pp p ptp X p p p p − + ⋅ + = − − +− =− − +− = + − + =+ − + ∫∫ θθ πθ Orthogonality of Associated Legendre Functions
- 77. SOLO 77 Associated Legendre Functions ( ) ( ) ( ) ( ) ( ) ( ) qp m q m p t m q m p mp mp p dPPdttPtP , 0 cos1 1 ! ! 12 2 sincoscos δθθθθ πθ − + + == ∫∫ =+ − Therefore the Orthonormal Associated Legendre Functions is ( ) ( ) ( ) ( ) mnmP mn mnn Θ m n m n ≤≤− + −+ = θθ cos ! ! 2 12 cos Orthogonality of Associated Legendre Functions
- 78. 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 i i I i Oi N i iiOi N i iOiO dm td Rd rdmVRRPdRRH 1 , 11 , : By taking a very large number N of particles, we go from discrete to continuous ∫⇒∑ ∞→ = NN i 1 ( )( ) ( )( ) ( ) ∫ ×=∫ ×−=∫ ×−= tm O tm O tv 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: ( ) ( )∑∑ ∆ ∆+×∆++ ∆+×∆+=∆+ = openings iflow I iflow I iflow OiflowOiflow N i i I i I i OiOiOO m td Rd td Rd rrdm td Rd td Rd rrHH ,, 1 ,,,, EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH
- 79. 79 SOLO Absolute Angular Momentum Relative to a Reference Point O (continue – 1) By subtracting I O t I O t H td Hd ∆ ∆ = →∆ , 0 , lim ( ) ( ) t dm td Rd rm td Rd td Rd rrdm td Rd td Rd rr openings N i i I i iOiflow I iflow I iflow OiflowOiflow N i i I i I i OiOi t ∆ ×−∆ ∆+×∆++ ∆+×∆+ = ∑ ∑∑ == →∆ 1 ,, 1 ,, 0 lim ∑∑∑ ×+×+×= == openings iflow I iflow Oiflow N i i I iOi N i i I i Oi m td Rd rdm td Rd td rd dm td Rd r , 1 , 1 2 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 SYSTEM SIMPLIFIED PARTICLE APPROACH ( )( ) ( )( ) ( ) ∫ ×=∫ ×−=∫ ×−= tm O tm O tv OO dmVrdmVRRdvVRRH ,, ρ dividing by Δt, and taking the limit, we get from ( ) ( )∑∑ ∆ ∆+×∆++ ∆+×∆+=∆+ = openings iflow I iflow I iflow OiflowOiflow N i i I i I i OiOiOO m td Rd td Rd rrdm td Rd td Rd rrHH ,, 1 ,,,,
- 80. 80 SOLO Absolute Angular Momentum Relative to a Reference Point O (continue – 2) ∑∑∑ ×+×+×= == openings iflow I iflow Oiopen N i i I i I Oi N i i I i Oi I O m td Rd rdm td Rd td rd dm td Rd r td Hd , 1 , 1 2 2 , , ( )∑ ×−+∑ × −+∑ ×= == openings iflow I iflow Oiopen N i i I i I O I i N i i I i Oi m td Rd RRdm td Rd td Rd td Rd dm td Rd r 11 2 2 , ( )∑ ×−+∑×−∑ ×= == openings iflow I iflow Oiopen N i i I i I O N i i I i Oi m td Rd RRdm td Rd td Rd dm td Rd r 11 2 2 , By taking a very large number N of particles, we go from discrete to continuous ∫⇒∑ ∞→ = NN i 1 ( ) ( )∑ ×−+×−∫ ×= openings iflowiflowOiopenO tm I O I O mVRRPVdm td Rd r td Hd 2 2 , , ( ) ( ) ∑∑ −+=−−+= openings OiopeniflowO I O openings OiopeniflowO I O rmVm td cd RRmVm td cd tP , ,, Substitute to obtain ( ) ( ) ( )∑ ×−+× ∑ −−++∫ ×= openings iflowiflowOiopenO openings iflowOiopenO I O tm I O I O mVRRVmRRVm td cd dm td Rd r td Hd , 2 2 , , or ( ) ( )∑ −×+×+∫ ×= openings iflowOiflowOiopenO I O tm I O I O mVVrV td cd dm td Rd r td Hd , , 2 2 , , EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH
- 81. 81 SOLO Absolute Angular Momentum Relative to a Reference Point O (continue – 3) We obtained ( )( ) ( )( ) ( ) ∫ ×=∫ ×−=∫ ×−= tm O tm O tv OO dmVrdmVRRdvVRRH ,, ρ Substitute in the previous equation OIO O O O I O I O I OO r td rd V td rd td Rd td Rd VrRR , ,, , :& ×++=+==+= ←ω ( )( ) ( ) ∫ ×++×=∫ ×−= ← tm OIO O O OO tm OO dmr td rd VrdmVRRH , , ,, ω ( ) ( ) ( ) ( ) ∫ ×+∫ ××+× ∫= ← tm O O O tm OIOOO tm O dm td rd rdmrrVdmr , ,,,, 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 SYSTEM SIMPLIFIED PARTICLE APPROACH
- 82. 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 ×=×=× ∫ ,,, ( ) ( ) ( )[ ]( ) IOOIO tm OOOO tm 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 O O tm OIOOO tm O tv OO dm td rd rdmrrVdmrvdVRRH , ,,,,, : ωρ ( ) ∫ ×+⋅+×= ← tm O O OIOOOO dm td rd rIVc , ,,, ω EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH
- 83. 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 SYSTEM SIMPLIFIED 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 I 2 , 2 ,,,,, ,, 2 , 2 ,,, ,,,, 2 , 2 , , :
Notes de l'éditeur
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