Aspherical Gravitational Field II

Transcription

1 Aspherical Gravitational Field II p. 1/19 Aspherical Gravitational Field II Modeling the Space Environment Manuel Ruiz Delgado European Masters in Aeronautics and Space E.T.S.I. Aeronáuticos Universidad Politécnica de Madrid April 008

2 Applications of Gravitational Expansion Aspherical Gravitational Field II p. /19 McCullagh s approximation for large distances Physical meaning of the coefficients Torque over an finite-size body Geoid Far field of a revolution ellipsoid VOP effects of gravitational field

3 McCullagh s Formula (1855) Aspherical Gravitational Field II p. 3/19 z P Potential of a particle distribution in a far away point (unit mass) P(0, 0,z), with z r i : V = N i=1 Gm i x i + yi + (z i z) = O m i = N i=1 Gm i x i + y i + z i z iz + z = = N i=1 Gm i z [ 1 z ] 1/ i z + r i z As z i z + r i 1, we use the expansion: z (1 + ǫ) 1/ = 1 1ǫ ǫ +...

4 McCullagh s Formula (1855) Aspherical Gravitational Field II p. 4/19 Keeping only cuadratic terms, we obtain V = N i=1 Gm i z Adding it up: mi z i = mz g [ 1 1 = ( z ) i z + r i z N i=1 Gm i z + 3 ( 4z i 8 z 4z iri z 3 [ 1 + z i z + 1 3zi r i mi = ri m 3( x i + ) y i i Potential can then be written as: V = G ( m + mz g + I O z z z 3 ( 3z i r i z = I O 3I z ) k I O k z +... ) + r4 i z 4 ) +... ] +... ] =

5 McCullagh s Formula (1855) Aspherical Gravitational Field II p. 5/19 O P V = G z ( m + mz g z + I O z 3 In spherical coordinates: ) k I O k z +... r g = r g u r ; I r = u r I O u r ; u r = {cosφcos λ, cos φ sin λ, sin φ} m i leading to McCullagh s Formula: V = µ r [ ( R 1+ r ) ( ) rg u r R 1 R + r mr (I O 3 ) ] u r I O u r +... r g = x g y g z g I x Pxy Pxz ; I O = Pxy I y Pyz ; I O = I x + I y + I z Pxz Pyz I z

6 Physical Meaning of the Coefficients Aspherical Gravitational Field II p. 6/19 McCullagh s Formula can be compared with the multipolar expansion up to order : V = µ { ( ) R 1+ [ J 1 P 1 (sinφ)+p 11 (sin φ) (C 11 cosλ+s 11 sinλ)]+ r r ( ) R + [ J P (sin φ)+p 1(sinφ) (C 1 cosλ+s 1 sin λ)+ r Where: +P (sinφ) (C cos λ+s sinλ)]} Order 0: Kepler term P 1 = sin φ P 11 = cos φ P = ( 3 sin φ 1 ) / P 1 = 3 cos φ sin φ P = 3 cos φ

7 Physical Meaning of the Order 1 Coefficients Aspherical Gravitational Field II p. 7/19 Comparing term by term, we obtain: z g R = J 1 x g R = C 11 y g R = S 11 Taking G as origin of coordinates, all are 0: J 1 = C 11 = S 11 = 0 Then expansion can start with order : l= ( R r ) l { J l P l + l m=0 P lm (C lm...) }

8 Physical Meaning of the Order Coefficients Aspherical Gravitational Field II p. 8/19 z z z x,y x y ( J = mr 1 I z I ) x + I y C 1 = P xz mr S 1 = P yz mr Oblate Earth, I z > I x,i y, therefore J > 0 If Oz is a principal axis of inertia, C 1 = S 1 = 0 N-S axis not exactly principal: not 0, but very small. In the JGM- Model: C 0 = 484,165; C 1 = 0, ; S 1 = 0, (All 10 6 )

9 Physical Meaning of the Order Coefficients Aspherical Gravitational Field II p. 9/19 y y x x C = I y I x 4mR Related to triaxiality: not a revolution ellipsoid. S = P xy mr Ox y Oy not principal inertia axes. In JGM-, C =, In JGM-, S = 1, All 10 6

10 Torque over a finite-size body V = Gm r [ ( R 1+ r ) ( ) rg u r R 1 R + r mr (I O 3 ) ] u r I O u r +... Potential of the action-reaction system: m P V O O P Attaction of the fixed point P over a mobile finite-size body O Work on a solid: There is work if: Change of distance: dr Rotation: du r = ω u r dt dw = dv = F dr + M G O ωdt Computing dv and comparing the coefficients of du r (rotation work), we obtain: M G O = 3Gm P r 3 u r I O u r Aspherical Gravitational Field II p. 10/19 P

11 Geoid Aspherical Gravitational Field II p. 11/19 Circle: Simplest representation of Earth Ellipsoid: Revolution ellipsoid approximating the shape of the Earth. International Reference Ellipsoid (IGU) Earth Surface: Irregular, with all mountains and valleys. Geoid: Mean surface of the oceans (if free flow). Includes gravity and rotation. N Normal δ Vertical g φ c φ g φ a Geoid Horizontal Elipsoid Vertical: Direction of a plumb: normal to Geoid. Undulation: Difference between Geoid and Ellipsoid. Deflection: Difference between vertical and normal to ellipsoid a a Ref: Geodesy for the Layman, DMA Technical Report TR , LIB/Geodesy4Layman/geo4lay.pdf.

12 EGM96 Geoid Undulation/WGS-84 Ellipsoid Aspherical Gravitational Field II p. 1/19 Courtesy National Geospatial-Intelligence Agency, Office of GEOINT Sciences

13 Approximate Computation of Ellipsoid/Geoid Aspherical Gravitational Field II p. 13/19 z Equipotential surface through Equator r = R, φ = 0 Gravitational potential up to J : symmetry of revolution φ Centrifugal potential x µ r [ 1 J ( R r ) ] 3 sin φ 1 ω r cos φ = µ R ( 1+ J ) ω R Meridian plane: sin φ = z/r and cosφ = x/r ( ) [ R r 1 J ( ) 3 ) R 1 + (3 ] z r r 1 + ω R 3 µ [ x ] R 1 = J = 1, , ω R 3 µ = 3,

14 Approximate Computation of Ellipsoid/Geoid Aspherical Gravitational Field II p. 14/19 The first term is small too: R r 1 = R r r = R r r(r + R) R x z R Within the brackets not small R can be substituted for r: R x z R J ] [ z R 1 + ω R 3 [ x ] µ R 1 ( R 1 ω R 3 ) x (1 ω R 3 ) + y (3J + 1) µ µ 0 Which is an ellipse with major semiaxis a = R in the meridian plane

15 Semiaxes and Flattening Aspherical Gravitational Field II p. 15/19 1 ω R ( ) 3 1 ω R 3 µ b = R 3J + 1 R µ ( ) (3J + 1) 1 + ω R 3 µ 1 R = 0,996R 1 + 3J + ω R 3 µ Flattening: f = a b a WGS84 flattening: 98, = ǫ ǫ 1 98, Difference?: approximations and truncated expansion

16 Semiaxes and Flattening Aspherical Gravitational Field II p. 15/19 1 ω R ( ) 3 1 ω R 3 µ b = R 3J + 1 R µ ( ) (3J + 1) 1 + ω R 3 µ 1 R = 0,996R 1 + 3J + ω R 3 µ Flattening: f = a b a WGS84 flattening: 98, = ǫ ǫ 1 98, Difference?: approximations and truncated expansion Very small: f < 1/50 can hardly be noticed in the figure

17 Far Field of an Ellipsoid Aspherical Gravitational Field II p. 16/19 McCullagh s approximation: homogeneous revolution ellipsoid Moments of inertia: J : J = 1 Ma I x = I y = M ( a + b ) ( I z I ) x + I y = a b 5a = f 5 5 a (a b) a 1/3 more than the real J = 0, = I z = Ma 5 = a b 5a f( f) 5 a + b a = = 0, Difference: Unhomogeneous Earth, massive nucleus, J < (J ) el

18 VOP Effects of Perturbations Aspherical Gravitational Field II p. 17/19 Gravity 3rd Body Atm Drag Rad Press Zonal Sect/Tess a P P P P S P e P P P P S P i P P P P S P Ω P S P P S P P S ω P S P P S P P S M 0 P S P P S P P S P: Periodic S: Secular Also: coupling effects Source: Vallado

19 Aspherical Gravitational Field II p. 18/19 VOP Secular Effects of J Analytical results with a truncated field (n = ) Ω = 3nR J p cosi ȧ = 0 ω = 3nR J 4p ( 4 5 sin i ) ė = 0 M 0 = 3nR J 1 e ( 3 sin 4p i ) i = 0 Only secular effects Periodic effects removed through averaging

20 COWELL with gravitation perturbation Aspherical Gravitational Field II p. 19/19 Begin ẏ = f(y, t) Input data KB/File Initializations Load grav. coeffs. Common block ITRF / GMST r GCRF rit RF ODE Integrator Call Int step Call Derivs Grav Accel Save Data FILE Other Accel End