scratch 3

This is the Title PhL 1.11.15 Value of (I3-I1) for a slightly oblate or prolate spheroid of constant density which is valid for any perimeter equation r(θ). For the oblate perimeter of ** we can compute I3 - I1 = -2π!Syntax Error, Idθ sinθ P2(cosθ)!Syntax Error, Ir4dr ρ(r) = -2πρ !Syntax Error, Idθ sinθ P2(cosθ) (1/5)r(θ)5 Now recall r(θ) = a [ 1 - (2/3)ε P2(cosθ) ] so r(θ)5 ≈ a5 [ 1 - (10/3)ε P2(cosθ) ] Then I3 - I1 = -2πρ(1/5)!Syntax Error, Idθ sinθ P2(cosθ) a5 [ 1 - (10/3)ε P2(cosθ) ] = 2πρa5(1/5) (10/3)ε !Syntax Error, Idθ sinθP2(cosθ)P2(cosθ) = 2πρa5(1/5) (10/3)ε 1/ [ 2+1/2] = = 2πρa5(1/5) (10/3)ε (2/5) = = 2πρa5ε (20/75) = 2πρa5ε (4/15) = (8/15)π ρa5ε and then J2 = G(I3-I1) = (8/15)G π ρa5ε Now to zeroth order we know that ρ = M/[(4/3)πa3] so get J2 = G(I3-I1) = (8/15)G π ρa5ε = = (2/5)GM a2ε = But Texas says, But this is the wrong sign on the second term, and wrong coefficient since right answer is Source #1 (wrong) Source #2 (right) Source #3 (right) where C is the different one, so this source agrees with Texas I am dead in the water until this sign error is located. Time 11:30AM. 2 But apart from a minus sign, this is our desired integral J2 shown above, so we conclude that J2 = - (I3-I1) and then the gravitational potential of the oblate Earth is V(r) ≈ -G ME/r + G (I3-I1) P2(cosθ) / r3 The oblate spheroid perimeter function r(θ) of *** appears as the upper endpoint of the radial integration. We know that !Syntax Error, Idφ cos2φ = π and !Syntax Error, Ir4dr = (1/5)r(θ)5 so I3 - I1 =!Syntax Error, Idθ sinθ (π sin2θ - 2π cos2θ) (1/5)r(θ)5 = (π/5) !Syntax Error, Idθ sinθ ( 1 - 3cos2θ) r(θ)5 = -(2π/5) !Syntax Error, Idθ sinθ P2(cosθ) r(θ)5 Now r(θ) = a [ 1 - (2/3)ε P2(cosθ) ] so r(θ)5 ≈ a [ 1 - (10/3)ε P2(cosθ) ] Then I3 - I1 = -(2πa/5) !Syntax Error, Idθ sinθ P2(cosθ) [ 1 - (10/3)ε P2(cosθ) ] Now use z = rcosθ to write I3 - I1 = -(2πa/5) !Syntax Error, Idz P2(z) [ 1 - (10/3)ε P2(z) ] Recall from ** that !Syntax Error, Idz Pn(z)Pn'(z) = δn,n'/ (n+1/2) // orthogonality Since P2(z) = P2(z)P0(z), the first integral in vanishes leaving I3 - I1 = (2πa/5)(10/3)ε !Syntax Error, Idz P2(z)P2(z) = (2πa/5)(10/3)ε (2/5) = (2πa/5)(2/3)ε (2) = 8πaε/15 Because P2(z) = P2(z)P0(z), we know the first integral vanishes because the Pn are orthogonal as shown in something is of course wrong here. If ε = 0, I get I3 - I1 ≠ 0. and therefore I3 - I1 = !Syntax Error, Idθ sinθ (π sin2θ - 2π cos2θ)!Syntax Error, Ir2dr r2 = π !Syntax Error, Idθ sinθ If the potential of interest is azimuthally symmetric in φ (such as outside a spheroidal Earth), only those atomic forms with m = 0 can contribute, reducing the atomic form set to [rn , r-n-1] Pn(z) . The four lines above then become g(z) = Σn=0∞gn Pn(z) z = cosθ // expansion gn = (1/Kn0) ∫dz Pn(z) g(z) Kn0 = (n+1/2)-1 f(n,0) // projection !Syntax Error, Idz Pn(z)Pn'(z) = δn,n' Kn0 n,n' = 0,1,2.. // orthogonality Σn=0∞(1/Kn0) Pn(z') Pn(z) = δ(z'-z) f(n,0) = Γ(n+1)/Γ(n+1) = 1 // completeness Of particular interest is this expansion and projection of a potential V(z) V(z) = Σn=0∞Vn Pn(z) // expansion Vn = (n+1/2) !Syntax Error, Idz Pn(z) V(z) // projection (I.8.1) If V(z) is even in z, only even values of n appear in the expansion. We put this Legendre transform on hold for a while to consider an oblate spheroid. Equation of a slightly oblate ellipse and spheroid Consider an ellipse of the form x2/A2 + z2/B2 = 1. If we measure polar angle θ down from the z axis, then x = rsinθ and z = rcosθ and the ellipse equation becomes r2 = For an "oblate" (wide) ellipsoid we have A > B. If the ellipse is only slightly oblate, write A = B+δ . In this case we get r2 = ≈ ≈ ≈ A2 [ 1-2(δ/B)cos2θ ] ≈ A2 [ 1-2(δ/A)cos2θ ] Then r ≈ A [ 1-(δ/A)cos2θ ] The "eccentricity" e of the ellipse is e2 ≡ (A2-B2)/A2 = (A+B)(A-B)/A2 ≈ 2Aδ/A2 = 2δ/A Then have r = A [ 1- (1/2)(2δ/A)cos2θ ] = [ 1- (1/2)e2cos2θ ] On the other hand, the "ellipticity" of an ellipse is ε ≡ (A-B)/A = δ/A // = e2/2 and then we have r = A [ 1- (1/2)(2f)cos2θ ] = A[ 1- ε cos2θ ] . We now form a slightly oblate spheroid by rotating our slightly oblate ellipse about the vertical axis, introducing an azimuthal coordinate φ. One can then ask : what is the average value of r over this spheroid? The answer is <r> = A[ 1- ε <cos2θ> ] where, approximating the spheroid here as a sphere, <cos2θ> = ≈ = = = 1/3 We then find that <r> = A[ 1- (1/3)ε ] ≡ a Then, r = A[ 1- ε cos2θ ] = [ 1- ε cos2θ ] ≈ a(1+ε/3)(1- ε cos2θ) ≈ a [ 1 + ε/3 - ε cos2θ ] = a [ 1 - (ε/3)(3cos2θ - 1) ] = a [ 1 - (2/3)ε P2(cosθ) ] // since P2(cosθ) = (1/2)(3cos2θ - 1) or r(θ) = a [ 1 - (2/3)ε P2(cosθ) ] (I.8.2) One can regard r(θ) as a function defining the perimeter of the oblate spheroid. If ε = 0, one of course finds that r = a for a sphere. We have assumed that ε << 1 so we are assuming slight oblateness. The reader seeking verification will find(I.8.2) to be the very first equation on this nice web page http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node108.html which we have used as a guide. These are notes by Richard Fitzpatrick