~rprice/area51/documents/3droswell.pdf 3D ROSWELL COORDINATES FOR TWO CENTERS RHP notes, revised and corrected; Notes added in September 2 by LMB; last on 1:pm September 3, 22 by LMB and RHP) I. Definitions of coordinates To create 3D Roswell coordinates, we introduce a third Roswell coordinate Φ. This is an azimuthal coordinate, ranging from to 2π, whose axis is the line between the Roswell centers. This is illustrated in Fig. a). χ =2 Φ= z Y y X Θ=.8 a) b) x Z We will need a disturbingly large number of coordinate systems due to the importance of two different axes: the line through the centers, and the rotation axes of the objects. We define the following coordinates: Name Symbols Character 3D Roswell coordinates χ, Θ, Φ Conform to centers Roswell-related Cartesians X, Y, Z The Z axis goes through the centers. Roswell-related Sphericals r, θ, φ Spherical polar coords related to X, Y, Z Ordinary Cartesians x, y, z The z axis is the rotation axis. Ordinary Sphericals r, θ, ϕ Spherical polar coords related to x, y, z. The two sets of Cartesians are illustrated in part b) of the figure. The ordinary Cartesians and Sphericals will be important for expressing the symmetry of the periodic solution; they will also be important for multipole structure of the radiation. The Roswell coordinates will be important because the solution will have very little multipole structure, in terms of these coordinates, in the nonlinear strong-field region near the centers. Note that multipoles in terms of θ, ϕ, and multipoles in terms of θ, φ are easily expressed in terms of each other through the use of the addition theorem of Spherical harmonics. This means that it will be relatively simple to work in Roswells, and to impose standing wave boundary conditions in terms of ordinary multipoles. 1
For definiteness, we present here the relations among the coordinates. Roswell-related Cartesians in terms of 3D Roswells: 1 Z = a 2 2 + χ 2 cos 2Θ + a 4 + 2a 2 χ 2 cos 2Θ + χ 4 ) 1) 1 X = a 2 2 χ 2 cos 2Θ + a 4 + 2a 2 χ 2 cos 2Θ + χ 4 ) cos Φ 2) 1 Y = a 2 2 χ 2 cos 2Θ + a 4 + 2a 2 χ 2 cos 2Θ + χ 4 ) sin Φ 3) 3D Roswells in terms of Roswell-related Cartesians: χ = r 1 r 2 = { Z a) 2 + X 2 + Y 2 Z + a) 2 + X 2 + Y 2} 1/4 Θ = 1 2 θ 1 + θ 2 ) = 1 ) 2Z X2 + Y 2 2 tan 1 Z 2 a 2 X 2 Y 2 Φ = tan 1 Y/X) 6) Ordinary Cartesians in terms of Roswell-related Cartesians and vice versa: Roswell-related Sphericals in terms of Roswell-related Cartesians: Z = x X = y Y = z 7) r = X 2 + Y 2 + Z 2 θ = tan 1 X2 + Y 2 /Z ) φ = tan 1 Y/X) 8) Roswell-related Cartesians in terms of Roswell-related Sphericals: Z = r cos θ X = r sin θ cos φ y = r sin θ sin φ 9) 4) 5) Ordinary Sphericals in terms of Ordinary Cartesians: ) r = x 2 + y 2 + z 2 θ = tan 1 x 2 + y 2 /z ϕ = tan 1 y/x) 1) Ordinary Cartesians in terms of Ordinary Sphericals: z = r cos θ x = r sin θ cos ϕ y = r sin θ sin ϕ 11) Relationship of Roswell-related Sphericals and Ordinary Sphericals: Of course, r = r. The relationship of { θ, φ} and {θ, ϕ} is given implicitly by Eqs. 7) 11). 2
II. 3D Wave equation in 3D Roswell coordinates Our scalar function has the form Ψr, θ, ϕ = φ Ωt). In these coordinates, the wave equation is simplified with the replacement As in the 2D case, we also have in the 3D case LΨ = 2 Ψ 1 c 2 2 Ψ t = 12) t = Ω ϕ. 13) ϕ = x y y x. 14) It is trivial to re-express this, in terms of the Roswell-related Cartesians, as so that the wave equation, with the symmetry embodied in it, becomes ϕ = Z X X Z, 15) LΨ = 2 Ψ Ω 2 1 Z c 2 X X ) 2 Ψ =. 16) Z The 2 part can easily be expressed in terms of the Roswell-related coordinates, so that our wave equation becomes LΨ = 2 Ψ X + 2 Ψ 2 Y + 2 Ψ 2 Z 1 2 Ω2 Z c 2 X X ) 2 Ψ =. 17) Z Equations 1) 6) can now be used to express the wave equation entirely in terms of the 3D Roswell coordinates. III. Simplifying the wave equation In terms of the Roswell coordinates, the wave equation can be written LΨ = A χχ 2 Ψ χ 2 + A ΘΘ 2 Ψ Θ 2 + A ΦΦ 2 Ψ Φ 2 + 2A χθ +B χ Ψ χ + B Θ Ψ Θ + B Φ 2 Ψ χ Θ + 2A χφ 2 Ψ χ Φ + 2A ΘΦ 2 Ψ Θ Φ Ψ Φ. 18) 3
The Laplacian is written as where 2 χ ) 2 χ X 2 + 2 χ + 2 χ Z 2 and so forth. We write the rotational term as = Āχχ 2 Ψ = Ψ, χχ χ ) 2 + Ψ,ΘΘ Θ ) 2 + Ψ,ΦΦ Φ ) 2 +2Ψ, χθ Θ χ ) + 2Ψ,χΘ Θ χ ) + 2Ψ,χΘ Θ χ ) 2 Ψ χ 2 + ĀΘΘ In terms of this notation, we have +Ψ χ 2 χ + Ψ Θ 2 Θ + Ψ Φ 2 Φ, 19) χ ) Θ ) χ X ) ) Θ + X 2 Ψ ϕ = Z 2 X X ) 2 Ψ Z 2 Ψ Θ 2 + ĀΦΦ 2 Ψ Φ 2 + 2ĀχΘ 2 Ψ χθ + 2ĀχΦ χ Y ) ) Θ + Y 2 Ψ χφ + 2ĀΘΦ χ Z 2 Ψ ΘΦ ) ) Θ Z + B χ Ψ χ + B Θ Ψ Θ + B Φ Ψ Φ. 2) A χχ = ) χ 2 Ω 2 c 2 Āχχ 21) A ΘΘ = ) Θ 2 Ω 2 c 2 ĀΘΘ 22) A ΦΦ = ) Φ 2 Ω 2 c 2 ĀΦΦ 23) A χθ = ) ) χ Θ Ω 2 c 2 ĀχΘ 24) A χφ = ) ) χ Φ Ω 2 c 2 ĀχΦ 25) A ΘΦ = Θ ) Φ ) Ω 2 c 2 ĀΘΦ 26) B χ = 2 χ ) Ω2 c 2 B χ 27) B Θ = 2 Θ ) Ω2 c 2 B Θ 28) B Φ = 2 Φ ) Ω2 c 2 B Φ. 29) 4
The dot products of the gradients are zero, since the coordinates are orthogonal. Furthermore as was checked with maple) the Φ coordinate is harmonic. We are thus left with a slightly simplified set of equations: IV. Explicit expressions for the wave equation A χχ = ) χ 2 Ω 2 c 2 Āχχ 3) A ΘΘ = ) Θ 2 Ω 2 c 2 ĀΘΘ 31) A ΦΦ = ) Φ 2 Ω 2 c 2 ĀΦΦ 32) A χθ = Ω2 c 2 ĀχΘ 33) A χφ = Ω2 c 2 ĀχΦ 34) A ΘΦ = Ω2 c 2 ĀΘΦ 35) B χ = 2 χ ) Ω2 c 2 B χ 36) B Θ = 2 Θ ) Ω2 c 2 B Θ 37) B Φ = Ω2 c 2 B Φ. 38) In explicit computations it is useful, as in the 2D case, to use the expression: Q With Maple we find the following expressions: a 4 + 2a 2 χ 2 cos 2Θ + χ 4 39) 2 χ = a2 + 2Q 4) χ 3 Q + a 2 + χ 2 cos2 Θ) 2 Q a 2 ) Θ = 41) Q a 2 χ 2 cos2 Θ) χ 4 2 Φ = 42) 5
χ χ = Q χ 2 43) Θ Θ = Q χ 4 44) Φ Φ = 2 Q + a2 + χ 2 cos2 Θ) χ 4 sin 2 2 Θ) The Ā and B terms are computed from the following: ) 2 ) 2 ) ) χ χ χ χ Ā χχ = Z 2 + X 2 2XZ X Z X Z ) 2 ) 2 ) ) Θ Θ Θ Θ Ā ΘΘ = Z 2 + X 2 2XZ X Z X Z ) 2 ) 2 ) ) Φ Φ Φ Φ Ā ΦΦ = Z 2 + X 2 2XZ X Z X Z ) ) ) ) ) ) ) ) χ Θ χ Θ χ Θ χ Θ Ā χθ = Z 2 + X 2 XZ + X X Z Z Z X X Z ) ) ) ) ) ) ) ) χ Φ χ Φ χ Φ χ Φ Ā χφ = Z 2 + X 2 XZ + X X Z Z Z X X Z ) ) ) ) ) ) ) ) Θ Φ Θ Φ Θ Φ Θ Φ Ā ΘΦ = Z 2 + X 2 XZ + X X Z Z Z X X Z B χ = Z 2 2 ) χ + X 2 2 ) χ 2 ) ) ) χ χ χ 2XZ X Z X 2 Z 2 X Z X Z B Θ = Z 2 2 ) Θ + X 2 2 ) Θ 2 ) ) ) Θ Θ Θ 2XZ X Z X 2 Z 2 X Z X Z B Φ = Z 2 2 ) Φ + X 2 2 ) Φ 2 ) ) ) Φ Φ Φ 2XZ X Z X 2 Z 2 X Z X Z 45) 46) 47) 48) 49) 5) 51) 52) 53) 54) 6
From Maple we find: Ā χχ = a4 sin 2 2Θ) cos 2 Φ χ 2 corrected Feb 12) 55) Ā ΘΘ = cos2 Φ χ 2 + a 2 cos2θ) 2 χ 4 ; 56) Ā ΦΦ = sin 2 Φ Q + a2 + χ 2 cos2θ) Q a 2 χ 2 cos2θ) 57) Ā χθ = a2 χ 2 + a 2 cos2θ) sin2θ) cos 2 Φ χ 3 58) Ā χφ = a2 Q + a 2 + χ 2 cos2θ) sin Φ cos Φ χ 3 59) Ā ΘΦ = sinφ) cosφ) a2 + χ 2 cos2 Θ) + Q χ 2 + a 2 cos2 Θ) χ 4 sin2 Θ) 6) B χ = a2 cos 2 Φ) {3a 2 cos 2 2Θ) Q 2a 2 + χ 2 cos2θ)} + Q + a 2 + χ 2 cos2θ) χ 3 61) B Θ = Q + a 2 + χ 2 cos2θ) χ 6 Q a 2 χ 2 cos2θ) c cos2 Φ + d) 62) where c a 2 χ 4 cos2 Θ) + 2 a 4 χ 2 + 4 a 6 cos2 Θ) + 4 a 4 χ 2 cos2 Θ)) 2 4 a 4 Q cos2 Θ) 2 a 2 Qχ 2 χ 6 63) d χ 4 a 2 cos2 Θ) + χ 2) 64) B Φ = 3Q + a2 + χ 2 cos 2Θ) sinφ) cosφ) Q a 2 χ 2 cos 2Θ 65) V. Choice of weight function In the 3D case, we write our spectral expansion as Ψχ, Θ, Φ) = lm a lm χ) Y lm Θ, Φ), 66) 7
and we substitute it in the wave equation 18) to get LΨ = lm 2 a lm χ 2 A χχ Y lm + a lm χ) + a lm χ A ΘΘ 2 Y lm Θ 2 + A ΦΦ 2 Y lm Φ 2 Y lm 2A χθ Θ + 2A Y lm χφ Φ + B χy lm 2 Y lm + 2A ΘΦ Θ Φ + B Y lm Θ Θ + B Y lm Φ Φ. 67) We know must consider how to project out the multipole equations. We note that the flat 3D metric in Roswell coordinates is ds 2 = χ2 Q dχ2 + χ4 Q dθ2 + 1 2 Note the limits. For χ a, Q χ 2 and the metric becomes For χ q, Q a 2 and the metric becomes χ 4 sin 2 2Θ) Q + a 2 + χ 2 cos 2Θ) dφ2. 68) ds 2 = + χ 2 dθ 2 + χ 2 sin 2 Θ) dφ 2. 69) ds 2 = χ2 a 2 dχ2 + χ4 a 2 dθ2 + χ4 sin 2 2Θ) 4a 2 dφ 2. 7) This last expression is the metric in spherical coordinates r, θ, Φ if we interpret χ = 2ar and Θ = θ/2. We want the extraction of multipoles to correspond to the spherical coordinates at both limits, so we choose to project with 2π dθ dφ g 71) Since the factors of χ don t matter, we choose them for convenience, and end up with the following projection 2π dθ dφ χ3 Q We multiply Eq. 67) by the weight factor sin 2Θ) 2Q + a 2 + χ 2 cos 2Θ)). 72) W χ, Θ) = χ3 Q sin 2Θ) 2Q + a 2 + χ 2 cos 2Θ)) = χ Q a2 χ 2 cos 2Θ) Q 2. 73) The second form on the right is better since it has no branch ambiguities. Note also, that the square root in that expression is X 2 + Y 2. 8
The weight function in Eq. 73) is inconvenient. It destroys the ability we had in 2D to relate all integrals to Legendre complete elliptic integrals. On the other hand it has some important or crucial features: i) It vanishes at Θ = π/2 when χ a, but does not vanish for Θ = π/2 when χ > a. ii) It reduces to the appropriate weight function for spherical harmonics in the χ >> a limit. Property i) is important because it means that the integration over the surface of the hole is not singular at the axis. Property ii), of course, is crucial. It does not appear possible at this point to construct an analytically simple replacement for W that has these properties. What we can do is to choose how much of the Q factor to put into our weight expressions. In the 2D case, the choice of the weight function had a clear motivation: it achieved a good separation of the highest derivatives of the multipoles; there is complete separation for Ω. We cannot do that in 3D, so the motivation is gone. The second χ derivatives will couple, but only in the transition region. This is obvious physically, since the coordinates go over to sphericals in both limits. It is also obvious in the details of Q. Both near the holes and far from the holes, Q is angle independent. For simplicity, and simplification of the integrals, we first on March 13) choose to have no factors of Q and to use the weight function Q a2 χ W χ, Θ) = 2 cos 2Θ). 74) 2 On March 13, there are vague reasons to think that this is not the best choice. The factor Q vanishes at the trouble point, the midpoint between the holes. We might want to have that Q in the denominator to increase the first term... the Ω-free term... in A χχ relative to the Ω term that strongly couples modes. We will start, on March 13, with the simplest case. We will take the weight function of Eq. 74) and we will set Ω =. VI. Coefficients The next step is to multiply by W χ, Θ) and by Y l m, and to integrate over Θ and Π. The result is our multipole equations where α l m lm = β l m lm = γ l m lm = 2π 2π 2π dφ dφ lm 2 a lm χ) a lm χ) α l m lm + β χ 2 l m lma lm χ) + γ l m lm χ dθ W χ, Θ) Y l m Θ, Φ)A χχy lm Θ, Φ) dθ W χ, Θ) Yl m Θ, Φ) 2 Y lm 2 Y lm A ΘΘ + A Θ 2 ΦΦ Φ 2 π dφ dθ W χ, Θ) Yl m Θ, Φ) Y lm 2A χθ Θ + 2A Y lm χφ Φ + B χy lm 9 =, 75) 2 Y lm + 2A ΘΦ Φ Θ + B Θ Y lm Θ + B Φ Y lm Φ. 76)
Note that the coefficients inherit all the symmetries that come from the relationship Thus, for example, α l, m,l, m = α l,m,l,m. Y lm θ, φ) = 1) m Ylm θ, φ). 77) VII. Explicit standing wave expressions for monopole and quadrupole, in the case Ω = For notational clarity, minuses are indicated as overbars. Integrals for l, m =, : We use a superscript N here to denote nonrotating. α N = 1 /2 QW dθ 78) χ 2 α N 2 = 5 2χ 2 /2 3 cos 2 Θ 1)QW dθ 79) β = β 2 = β 2 = β 2 2 = 8) γ N = 1 /2 a 2 + 2Q ) W dθ 81) χ 3 γ N 2 = 5 2χ 3 /2 a 2 + 2Q ) 3 cos 2 Θ 1)W dθ Corrected 1/χ 2 1/χ 3 ) March 29 82) γ N 22 = γ N 2 2 =. 83) Integrals for l, m = 2, : β N 2 = 3 5 2χ 4 /2 α N 2 = α N 2 = 5 2χ 2 /2 3 cos 2 Θ 1)QW dθ. 84) α22 N = 5 /2 3 cos 2 Θ 1) 2 QW dθ. 85) 4χ 2 α N 222 = αn 2 22 =. 86) Q + a 2 2 + χ 2 cos 2Θ) cos 2Θ) Q + Q a 2 χ 2 cos 2Θ) Q a2 ) sin 2Θ) W dθ 87) 1
β N 22 = 15 4χ 4 /2 Q + a 2 2 + χ 2 cos 2Θ) cos 2Θ) Q + Q a 2 χ 2 cos 2Θ) Q a2 ) sin 2Θ) 3 cos 2 Θ 1)W dθ 88) Integrals for l, m = 2, 2: γ N 2 = 5 2χ 3 γ N 22 = 5 4χ 3 /2 /2 β N 222 = β N 2 22 =. 89) a 2 + 2Q ) 3 cos 2 Θ 1) W dθ 9) a 2 + 2Q ) 3 cos 2 Θ 1) 2 W dθ 91) γ N 222 = γ N 2 22 =. 92) α N 22 = αn 222 = αn 2 222 =. 93) α2222 N = 15 /2 Q W sin 4 Θ dθ corrected March 18) 94) 8χ 2 β2222 N = 15 π/2 2Q cos 2Θ) 8 Q + a2 + χ 2 cos 2Θ) 8χ 4 sin 2 2Θ) β N 22 = β N 222 = β N 2 222 =. 95) sin 2 Θ corrected March 2) Q + a 2 + χ 2 cos 2Θ) + Q a 2 χ 2 cos 2Θ) Q a2 ) sin 2Θ) sin 2 Θ W dθ 96) γ N 2222 = 15 8χ 3 γ N 22 = γ N 222 = γ N 2 222 =. 97) /2 a 2 + 2Q ) sin 4 Θ W dθ 98) Integrals for l, m = 2, 2: These follow from the symmetry of the coefficients. In particular, all coefficients are zero except 11
α N 2 22 2 = αn 2222 β N 2 22 2 = βn 2222 γ N 2 22 2 = γn 2222 99) VIII. Expressions for Ω 2 mode computation; checked by Lior and Maria, April 9 All the coefficients will be written as sums of the Ω-independent part and a part that is multiplied by Ω 2. Thus we write α l m lm as α l m lm = αl N m lm Ω2 αl Ω m lm. 1) Here the superscript N denotes the part for Ω =, i.e., the nonrotating contributions that are given in the previous section. Integrals for l =, m = : α Ω = 2 a4 χ 2 α Ω 2 = 5 a 4 χ 2 /2 /2 sin 2 Θ cos 2 Θ W dθ 11) 3 cos 2 Θ 1) sin 2 Θ cos 2 Θ W dθ 12) β Ω = βω 2 =. 13) γ Ω = a2 2 χ 3 /2 3a 2 cos 2 2Θ + Q + 3χ 2 cos 2Θ ) W dθ. 14) γ Ω 2 = 5 4 Integrals for l = 2, m = : a 2 /2 3 cos 2 Θ 1) 3a 2 cos 2 2Θ + Q + 3χ 2 cos 2Θ ) W dθ. 15) χ 3 α Ω 2 = 5 a 4 χ 2 /2 3 cos 2 Θ 1) sin 2 Θ cos 2 Θ W dθ 16) α22 Ω = 5 a4 π/2 3 cos 2 Θ 1) 2 sin 2 Θ cos 2 Θ W dθ 17) 2 χ 2 12
+ sin 2Θ χ 6 β2 Ω = 3 5 π/2 {2 cos 2Θ χ2 + a 2 cos 2Θ) 2 4 χ 4 Q + a 2 + χ 2 cos 2Θ) Q a 2 χ 2 cos 2Θ) 2a 4 χ 2 + χ 6 + cos 2Θ 4a 6 + 3a 2 χ 4) +4a 4 χ 2 cos 2 2Θ 2a 2 Q χ 2 + 2a 2 cos 2Θ )} W dθ 18) + sin 2Θ χ 6 β Ω 22 = 15 8 /2 Q + a 2 + χ 2 cos 2Θ) Q a 2 χ 2 cos 2Θ) {2 cos 2Θ χ2 + a 2 cos 2Θ) 2 χ 4 2a 4 χ 2 + χ 6 + cos 2Θ 4a 6 + 3a 2 χ 4) +4a 4 χ 2 cos 2 2Θ 2a 2 Q χ 2 + 2a 2 cos 2Θ )} 3 cos 2 Θ 1) W dθ 19) γ2 Ω = a2 5 χ 3 4 /2 { 6 sin 2 2Θ χ 2 + a 2 cos 2Θ ) 3 cos 2 Θ 1) 3a 2 cos 2 2Θ + Q + 3χ 2 cos 2Θ } W dθ 11) γ22 Ω = a2 5 χ 3 8 /2 { 6 sin 2 2Θ χ 2 + a 2 cos 2Θ ) 3 cos 2 Θ 1) 3a 2 cos 2 2Θ + Q + 3χ 2 cos 2Θ } 3 cos 2 Θ 1) W dθ 111) VIII. Nonlinear and multiple-mode code Our starting point is Eq. 18), generalized to read LΨ = A χχ 2 Ψ χ 2 + A ΘΘ 2 Ψ Θ 2 + A ΦΦ +B χ Ψ χ + B Θ 2 Ψ Φ 2 + 2A χθ Ψ Θ + B Φ 13 Ψ Φ 2 Ψ χ Θ + 2A χφ 2 Ψ χ Φ + 2A ΘΦ 2 Ψ Θ Φ = F χ, Θ, Ψ). 112)
Here the coefficients A χχ,... B Φ are the same as those for Eq. 18) given in Eqs. 3) 38) and 55) 65). Ψχ, Θ, Φ) = lm a lm χ) Y lm Θ, Φ), 113) and we substitute it in the wave equation 112) to get lm d 2 a lm A χχ Y lm = lm + da lm dχ { a lm χ) A ΘΘ d 2 Y lm dθ 2 + A ΦΦ 2 Y lm Φ 2 } Y lm 2A χθ Θ + 2A Y lm χφ Φ + B χy lm + F χ, Θ, lm 2 Y lm + 2A ΘΦ Θ Φ + B Y lm Θ Θ + B Φ a lm χ) Y lm Θ, Φ) ) Y lm Φ 114). 115) The next step is to multiply by a chosen weight function W χ, Θ) and by Yl m, and to integrate over Θ and Φ. The result is our multipole equations where + da lm dχ lm d 2 a lm χ) α l m lm = R l m χ, a pn, da pn /dχ), 116) 2π α l m lm = dφ dθ W χ, Θ) Y 2π π R l m = dφ dθ W χ, Θ) Yl m Θ, Φ) lm +A ΦΦ 2 Y lm Φ 2 2 Y lm + 2A ΘΦ Θ Φ + B Y lm Θ Y lm 2A χθ Θ + 2A Y lm χφ Φ + B χy lm } l m Θ, Φ)A χχy lm Θ, Φ) 117) { 2 Y lm a lm χ) A ΘΘ Θ 2 Y lm Φ + F Θ + B Φ χ, Θ, lm a lm χ) Y lm Θ, Φ) The equations that we use follow by inverting the equations in 116) in the form ). 118) d 2 a lm χ) = lm α inv lml m R l m χ, a pn, da pn /dχ). 119) These can them be solved with Runge-Kutta or whatever. 14
IX. Specific 2 mode linear problem Angular mode for l = 2 As our first problem we consider only two modes. One, of course, is the monopole. The second is the relevant combination of l = 2, m = 2 modes referred to the rotation axis, z. We use the notation θ, ϕ for these angular coordinates. Our l = 2 mode of interest is The normalization is chosen so that Y 2 1 2 Y 2,2 θ, ϕ) + Y 2, 2 θ, ϕ) = 1 4 2π sin θdθ dϕ Y 2 ) 2 = 1. 15 π sin2 θ cos 2ϕ. 12) From the relationship of x, y, z and X, Y, Z given in Eq. 7) it is straightforward to show that and hence It is simple to show that sin 2 θ cos 2ϕ = 1 2 Y 2 = 1 8 15 π 3 cos 2 Θ 1 ) 1 2 sin2 Θ cos 2Φ 121) 3 cos 2 Θ 1) sin 2 Θ cos 2Φ 122) 3 Y 2 = 2 Y 2,Θ, Φ) 1 1 Y 2,2 Θ, Φ) + Y 2, 2 Θ, Φ), 123) 2 2 but we will use the expression in Eq. 122) in our computations. Note that Y 2 is not single valued at Θ = π/2. Damn!) Equations for 2 modes With our two modes, we now rewrite Eq. 114) as In place of Eq. 116) we now have Ψ = a χ)y + a 2 χ)y 2. 124) d 2 a a dχ + a d 2 a 2 2 2 = R χ, a, a 2, da /dx, da 2 /dx) 125) d 2 a a 2 dχ + a d 2 a 2 2 2 2 = R 2 χ, a, a 2, da /dx, da 2 /dx). 126) 15
Explicit αs Our explicit α expressions are α = 2π dφ dθ W χ, Θ) Yl m Θ, Φ)A χχy lm Θ, Φ) = 1 2π dφ dθ W χ, Θ) A χχ 127) 4π α 2 = α 2 = 1 2π dφ dθ W χ, Θ) Y 2 Θ, Φ)A χχ 128) 4π 2π α 2 2 = dφ dθ W χ, Θ) Y 2 Θ, Φ)) 2 A χχ 129) Since A χχ is perfectly well behaved, any weight function will do. Explicit Rs Our explicit expressions for the Rs are a 2 χ) R = 2π A ΘΘ 2 Y 2 Θ 2 da 2 dχ { π dφ dθ W χ, Θ) Y da dχ B χy + A 2 Y 2 ΦΦ Φ 2 2A χθ Θ + 2A χφ + 2A 2 Y 2 ΘΦ Θ Φ + B Θ Θ + B Φ Φ + B χy 2 } Φ 13) a 2 χ) R 2 = 2π A ΘΘ 2 Y 2 Θ 2 da 2 dχ { π dφ dθ W χ, Θ) Y 2 da dχ B χy + A 2 Y 2 ΦΦ Φ 2 2A χθ Θ + 2A χφ + 2A 2 Y 2 ΘΦ Θ Φ + B Θ Θ + B Φ Φ + B χy 2 } Φ 131) 16
Weight function The difficulty in the weight function occurs at Θ = π/2, and is associated with the expression τ Q a2 χ 2 cos 2Θ in the denominator. Near Θ = π/2 this factor is finite for χ > a, but for χ a behaves as τ 2 = Q a 2 χ 2 cos 2Θ 2χ4 a 2 χ 2 cos2 Θ 2χ4 a 2 χ 2 Θ π ) 2 2 Our early choice of weight function, in Eq. 74), contains τ, which helps to cancel out this term in the denominator. Problems still occur because ĀΦΦ and B Φ contain τ 2 in the denominator. This presents fatal troubles unless the singularities due to ĀΦΦ and B Φ cancel in the Φ integration. At this point June 14, 1:3am) it appears that they DO cancel in R, but not in R 2. In place of Eq. 74), we tentatively adopt the following weight function: W tent = Q a2 χ 2 cos 2Θ) 2 a 2 + χ 2 sin 2 Θ 132) Note that: i) The denominator has no real poles. ii) For χ < a, and Θ near π/2 the weight function is approximately τ 2 /2 a 2 + χ 2 ). iii) For χ a, the weight function is approximately χ sin Θ. We write R and R 2 as a 2 χ) Terms in R and R 2 R = 2π dθ W χ, Θ) { da dχ Y B χ Y 2 Y 2 Y A ΘΘ Θ + Y 2 Y 2 A 2 ΦΦ Φ +2 Y 2 Y 2 A 2 ΘΦ Θ Φ + Y B Θ Θ } + Y B Φ Φ a 2 χ) da 2 dχ R 2 = 2π 2 Y A χθ Θ +2 Y A χφ Φ + Y B χ Y 2 { π dθ W χ, Θ) da dχ Y 2B χ Y 2 Y 2 Y 2 A ΘΘ Θ + Y 2 Y 2 2A 2 ΦΦ Φ +2 Y 2 Y 2 2A 2 ΘΦ Θ Φ + Y 2B Θ Θ } + Y 2 B Φ Φ da 2 dχ 2 Y 2 A χθ Θ +2 Y 2A χφ Φ + Y 2B χ Y 2 17.
Here the notation means the average over Φ, that is f 1 f dφ 2π We now need to write down explicitly the 18 Φ averages in analytic form. We will break each of these down into two pieces, in the spirit of the β coefficients. Thus we have for example Below we use the notation R 5N Y B N Θ 2π Y B χ Y = Y B χ Y N Ω 2 Y B χ Y Ω. κ = 1 8 15 π R 1N Y B N χ Y N = 1 4π 133) a 2 + 2Q χ 3 134) R 2N Y A N 2 Y 2 ΘΘ Θ 2 N = 1 κ Q 6 cos 2Θ) 135) 4π χ4 = R 3N Y A N 2 Y 2 ΦΦ Φ 2 N = 136) R 4N Y A N 2 Y 2 ΘΦ Θ Φ N = 137) Q + a 2 + χ 2 cos2 Θ) Q a 2 ) 1 Θ N = κ 3 sin 2Θ) Q a 2 χ 2 cos2 Θ) χ 4 4π sin2θ) Q a 2 χ 2 cos2 Θ) R 6N Y B N Φ Q a 2 ) χ 2 1 4π κ 3 sin 2Θ) 138) Φ N = 139) R 7N Y A N χθ Θ N = 14) R 8N Y A N χφ Φ N = 141) R 9N Y B N χ Y 2 N = 1 4π κ a2 + 2Q χ 3 3 cos 2 Θ 1 142) 18
R2 1N Y 2 B N χ Y N = R 9N = 1 4π κ a2 + 2Q χ 3 3 cos 2 Θ 1 143) R2 2N Y 2 A N 2 Y 2 ΘΘ Θ 2 N = κ 2 Q χ cos 2Θ 19 cos 2 Θ 7 144) 4 R2 3N Y 2 A N 2 Y 2 ΦΦ Φ 2 N = 2κ 2 sin 4 Θ ) 2 Q + a2 + χ 2 cos2 Θ) χ 4 sin 2 2 Θ) = κ 2 sin2 Θ cos 2 Θ Q + a 2 + χ 2 cos 2 Θ χ 4 145) R2 5N Y 2 B N Θ κ 2 R2 4N Y 2 A N 2 Y 2 ΘΦ Θ Φ N = 146) Θ N = Q + a 2 + χ 2 cos2 Θ) Q a 2 χ 2 cos2 Θ) = 2κ 2 sin 2 Θ cos 2 Θ Q a 2 χ 2 cos 2Θ R2 6N Y 2 B N Φ Q a 2 ) 2χ 4 sin 2Θ 19 cos 2 Θ 7 Q a 2 ) χ 2 19 cos 2 Θ 7) 147) Φ N = 148) R2 7N Y 2 A N χθ Θ N = 149) R2 8N Y 2 A N χφ Φ N = 15) R2 9N Y 2 B N χ Y 2 N = κ 2 a2 + 2Q χ 3 3 cos 2 Θ 1 ) 2 1 + 2 sin4 Θ 151) 19
1 4π R 1Om Y Bχ Y Ω = a 2 2χ 3 3a 2 cos 2 2Θ + 3χ 2 cos 2Θ + Q 152) 2 Y 2 R 2Om Y Ā ΘΘ Θ 2 Ω = κ 7 χ 2 + a 2 cos 2Θ 2 cos 2Θ 153) 4π 4χ 4 2 Y 2 R 3Om Y Ā ΦΦ Φ 2 Ω = κ Q + a 2 + χ 2 ) cos2θ) sin 2 Θ 154) 4π Q a 2 χ 2 cos2θ) 2 Y 2 R 4Om Y Ā ΘΦ Θ Φ Ω = κ a 2 + χ 2 cos2 Θ) + Q χ 2 + a 2 cos2 Θ) 4π 2χ 4 155) R 5Om Y BΘ Θ Ω = κ 4π Q + a 2 + χ 2 cos2θ) sin 2Θ) 2χ 6 Q a 2 χ 2 cos2θ) 7 2 c + 6d = 2κ 4π with c and d given in Eqs. 63), 64). sin 2 Θ cos 2 Θ χ 4 Q a 2 χ 2 cos2θ)) 7 2 c + 6d 156) R 6Om Y BΦ Φ Ω = R 7Om Y Ā χθ Θ Ω = κ 3Q + a 2 + χ 2 cos 2Θ) 4π 2 Q a 2 χ 2 cos 2Θ) sin2 Θ 157) κ 7 4π 4 a 2 χ 2 + a 2 cos2θ) sin 2 2Θ) χ 3 158) R 8Om Y Ā χφ Φ Ω = κ a 2 Q + a 2 + χ 2 cos2θ) sin 2 Θ 4π 2χ 3 159) R 9Om Y Bχ Y 2 Ω = 1 a 2 κ 1 3 cos 2 Θ 1 ) h sin2 Θ j, 4π χ 3 2 4 16) where h and j are given in Eqs. 17),171). 2
R2 1Om Y 2 Bχ Y Ω = R 9Om. 161) R2 2Om Y 2 Ā ΘΘ 2 Y 2 Θ 2 Ω = κ 2 χ2 + a 2 cos 2Θ 2 2χ 4 cos 2Θ 11 25 cos 2 Θ ). 162) R2 3Om Y 2 Ā ΦΦ 2 Y 2 Φ 2 Ω = 2κ 2 Q + a 2 + χ 2 cos2θ) Q a 2 χ 2 cos2θ) ) sin 2 Θ cos 2 Θ 163) R2 4Om Y 2 Ā ΘΦ 2 Y 2 Θ Φ Ω = κ 2 a2 + χ 2 cos2 Θ) + Q χ 2 + a 2 cos2 Θ) 3 cos 2 Θ 1 2χ 4 164) R2 5Om Y 2 BΘ Θ Ω = Q + a 2 + χ 2 cos2θ) κ 2 sin 2Θ 2χ 6 Q a 2 χ 2 cos2θ) = 2κ 2 sin 2 Θ cos 2 Θ χ 4 Q a 2 χ 2 cos 2Θ) with c and d given in Eqs. 63),64). where and 25 c 2 cos2 Θ 11 ) + d 19 cos 2 Θ 7 ) 2 25 c 2 cos2 Θ 11 ) + d 19 cos 2 Θ 7 ) 165) 2 R2 6Om Y 2 BΦ Φ Ω = κ 2 3Q + a2 + χ 2 cos 2Θ) 2 Q a 2 χ 2 cos 2Θ) sin2 Θ 3 cos 2 Θ 1 ) 166) R2 7Om Y 2 Ā χθ Θ Ω = κ 2 a2 χ 2 + a 2 cos2θ) sin 2 2Θ) 4χ 3 25 cos 2 Θ 11 ) 167) R2 8Om Y 2 Ā χφ Φ Ω = κ 2 a2 Q + a 2 + χ 2 cos2θ) 2χ 3 sin 2 Θ 3 cos 2 Θ 1 ) 168) R2 9Om Y 2 Bχ Y 2 Ω = κ2 a 2 19 2χ 3 2 cos4 Θ 7 cos 2 Θ + 2) 3 h sin 2 Θ3 cos 2 Θ 1) j 169) h = 3a 2 cos 2 2Θ + Q + 3χ 2 cos 2Θ 17) j = 3a 2 cos 2 2Θ Q 2a 2 + χ 2 cos 2Θ. 171) 21
Explicit αs and solution α = 2π dθ W Y A χχ Y 172) α 2 = 2π dθ W Y A χχ Y 2 173) α 2 2 = 2π dθ W Y 2 A χχ Y 2 174) A N Y A χχ Y N = 1 4π A2 N Y A χχ Y 2 N = A22 N Y 2 A χχ Y 2 N = κ 2 Q χ 2 175) κ Q 4π χ 3 2 cos2 Θ 1) 176) Q 2χ 2 19 cos4 Θ 14 cos 2 Θ + 3) 177) A Om Y Ā χχ Y Ω = 1 4π 2a 4 sin 2 Θ cos 2 Θ χ 2 178) A2 Om Y Ā χχ Y 2 Ω = κ 4π a 4 sin 2 Θ cos 2 Θ χ 2 7 cos 2 Θ 3) 179) A22 Om Y 2 Ā χχ Y 2 Om = κ 2 2a4 sin 2 Θ cos 2 Θ χ 2 25 2 cos4 Θ 11 cos 2 Θ + 5 2 18) D α α 2 2 α 2 2 181) db dχ = 1 D α 2 2R α R 2 182) db 2 dχ = 1 D α R 2 α 2R 183) 22
X. 3 Dimensional rotation with 3 Modes We assume the form Basic expansion Ψ = a χ)y + a 2 χ)y 2 + a c χ)y c. 184) The notation here is rather inconsistent. The symbol Y represents Y = 1/ 4π, and Y 2 stands for the usual l = 2 spherical harmonic 5 3 cos 2 Θ 1) Y 2 Y 2 = 185) 4π 2 The function Y c is the normalized relevant real combination of l = 2, m = ±2: Y c = 1 4 15 π sin2 Θ cos 2Φ. 186) We now substitute this sum in the wave equation 112) to get a slight notational modification of Eq. refsum1) k=,2,c d 2 a k A χχ Y k = + da k dχ k=,2,c { a k χ) d 2 Y k A ΘΘ dθ + A 2 Y k 2 ΦΦ Φ 2 } Y k 2A χθ Θ + 2A Y k χφ Φ + B χy k + F χ, Θ, k + 2A 2 Y k ΘΦ Θ Φ + B Θ a k χ) Y k Θ, Φ) Y k Θ + B Φ ) Y k Φ. 187) We next multiply by a weight function W, then generate three equations by projecting respectively by Y, Y, Y c, and integrate over all Θ and Φ. These lead to the following equations: where and α pk = k=,2,c 2π α pk d 2 a k χ) = R p χ, a n, da n /dχ), 188) dφ dθ W χ, Θ) Y p Θ, Φ)A χχ Y k Θ, Φ) 189) 23
2π π R p = dφ dθ W χ, Θ) Y p Θ, Φ) k=,2,c + da k dχ +A ΦΦ 2 Y k Φ 2 2A χθ Y k Θ + 2A χφ + 2A 2 Y k ΘΦ Θ Φ + B Y k Θ Θ + B Φ } Y k Φ + B χy k + F { χ, Θ, k 2 Y k a k χ) A ΘΘ Θ 2 Y k Φ ) a k χ) Y k Θ, Φ) The equations that we use follow by inverting the equations in 188) in the form. 19) d 2 a k χ) = p α inv kp R pχ, a n, da n /dχ). 191) The explicit expressions of these equations involves the determinant D = α α 22 α cc α α 2c α c2 α 2 α 2 α cc + α 2 α c α c2 + α c α 2 α 2c α c α c α 22. 192) Decomposition in Φ parts To simplify expressions we introduce the following notation: Y c = Y c cos 2Φ Y c = 1 4 15 π sin2 Θ 193) A Ω χχ = A Ω χχ 1 + cos 2Φ) A Ω ΘΘ = AΩ ΘΘ 1 + cos 2Φ) AΩ A Ω χχ = a4 sin 2 2Θ) 2χ 2 194) ΘΘ = χ2 + a 2 cos2θ) 2 2χ 4 195) A Ω ΦΦ = AΩ ΦΦ 1 cos 2Φ) AΩ ΦΦ = 1 2 Q + a 2 + χ 2 cos2θ) Q a 2 χ 2 cos2θ) = 2χ 4 sin 2 Θ cos 2 Θ Q a 2 χ 2 cos2θ)) 2 196) A Ω χθ = AΩ χθ 1 + cos 2Φ) AΩ A Ω χφ = AΩ χφ sin 2Φ AΩ χθ = a2 χ 2 + a 2 cos2θ) sin2θ) 2χ 3 197) χφ = a2 Q + a 2 + χ 2 cos2θ) 2χ 3 198) 24
A Ω ΘΦ = A Ω ΘΦ sin 2Φ A Ω ΘΦ = a2 + χ 2 cos2 Θ) + Q χ 2 + a 2 cos2 Θ) 2 χ 4 sin2 Θ) 199) B Ω1 χ = a2 2χ 3 3a 2 cos 2 2Θ + Q + 3χ 2 cos 2Θ B Ω χ = BΩ1 χ + BΩ2 χ cos 2Φ 2) B Ω2 χ = a2 2χ 3 3a 2 cos 2 2Θ Q 2a 2 + χ 2 cos 2Θ 21) where B Ω Θ = BΩ1 Θ B Ω1 Θ = c + 2d) sin Θ cos Θ χ 4 Q a 2 χ 2 cos 2Θ) + BΩ2 Θ cos 2Φ 22) B Ω2 Θ = c sin Θ cos Θ χ 4 Q a 2 χ 2 cos 2Θ) 23) c a 2 χ 4 cos2 Θ) + 2 a 4 χ 2 + 4 a 6 cos2 Θ) + 4 a 4 χ 2 cos2 Θ)) 2 4 a 4 Q cos2 Θ) 2 a 2 Qχ 2 χ 6 24) d χ 4 a 2 cos2 Θ) + χ 2) 25) BΦ Ω = BΩ Φ sin 2Φ BΩ Φ = 3Q + a2 + χ 2 cos 2Θ) 2 Q a 2 χ 2 cos 2Θ) 26) The α s and the averages over Φ are related by: Expressions for αs α ij = α N ij Ω 2 α Ω ij 27) α = 2π dθ W Y A χχ Y 28) α 2 = α 2 = 2π dθ W Y A χχ Y 2 29) α 22 = 2π dθ W Y 2 A χχ Y 2 21) α c = α c = 2π dθ W Y A χχ Y c. 211) α 2c = α c2 = 2π dθ W Y 2 A χχ Y c. 212) 25
With x cos Θ the αs then follow explicitly from α cc = 2π dθ W Y c A χχ Y c 213) 1 W α ij = 4π 1 x 2 Aij dx 214) and from A Y A χχ Y = Y A N χχ Ω 2 A Ω χχ) Y 215) A2 Y A χχ Y 2 = Y A N χχ Ω 2 A Ω χχ) Y2 216) Ac = Y A χχ Y c = Ω2 2 Y A Ω χχ Y c 217) A22 Y 2 A χχ Y 2 = Y 2 A N χχ Ω 2 A Ω χχ) Y2 218) A2c = Y 2 A χχ Y c = Ω2 2 Y 2A Ω χχ Y c 219) Acc Y c A χχ Y c = 1 2 Y c ) A N χχ Ω 2 A Ω χχ Y c 22) 26
In general, the source terms are given by 2 Y k + Y p A ΦΦ Φ 2 + da k dχ R p = 2π dθ W χ, Θ) Explicit R c for linear case k=,2,c { a k χ) Y p A ΘΘ 2 Y k Θ 2 +2 Y 2 Y k pa ΘΦ Θ Φ + Y Y k pb Θ Θ + Y Y k pb Φ Φ 2 Y p A χθ Y k Θ +2 Y pa χφ Y k Φ + Y pb χ Y k In the case of p = c, this becomes with notation b i = da i /dχ and x = cos Θ): a c χ) a 2 χ) 1 R c = 4π dx W χ, Θ) 1 x 2 2 Y 2 Y c A ΘΘ Θ + Y cb 2 Θ Θ }. 221) Y c A ΘΘ 2 Y c Θ 2 + Y ca ΦΦ 2 Y c Φ 2 +2 Y ca ΘΦ 2 Y c Θ Φ + Y cb Θ Y c Θ + Y cb Φ Y c Φ b Y c B χ Y b 2 2 Y c A χθ Θ + Y cb χ Y 2 b c 2 Y c A χθ Y c Θ +2 Y ca χφ Y c Φ + Y cb χ Y c The following expressions are used for computation: Rca2 1 = Y c A ΘΘ 2 Y 2 Θ 2 = Ω2 2 Y c A Ω ΘΘ 222) 2 Y 2 Θ 2 223) Rca2 2 = Y c B Θ Θ = Ω2 2 Y c BΘ Ω2 Θ 224) 2 Y c Rcac 1 = Y c A ΘΘ Θ = 1 2 2 Y c A N ΘΘ Ω 2 A ) Ω 2 Yc ΘΘ 225) Θ 2 27
2 Y c Rcac 2 = Y c A ΦΦ = 2Y Φ2 c Rcac 3 = Y c A ΘΦ 2 Y c Θ Φ ) A N ΦΦ Ω 2 A Ω ΦΦ Y c 226) = 227) Y c Rcac 4 = Y c B Θ Θ = 1 2 Y c B N Θ Ω 2 B ) Θ Ω1 Yc Θ 228) Rcac 5 = Y c B Φ Y c Φ = 229) Rcb = Y c B χ Y = Ω2 2 Y c Bχ Ω2 Y 23) Rcb2 1 = Y c A χθ Θ = Ω2 2 Y c A Ω χθ Θ 231) Rcb2 2 = Y c B χ Y 2 = Ω2 2 Y c Bχ Ω2 Y 2 232) Rcbc 1 = Y c A χθ Y c Θ Rcbc 2 = Y c A χφ Y c Φ Rcbc 3 = Y c B χ Y c = 1 2 Y c = Ω2 2 Y c A Ω χθ Y c Θ 233) = 234) ) B N χ Ω 2 Bχ Ω1 Y c 235) 28
In the case of p = 2, Eq. 221) becomes a c χ) a 2 χ) Explicit R 2 for linear case R 2 = 4π 1 dx W χ, Θ) 1 x 2 2 Y 2 Y 2 A ΘΘ Θ + Y 2B 2 Θ Θ Y 2 A ΘΘ 2 Y c Θ 2 + Y 2A ΦΦ 2 Y c Φ 2 +2 Y 2A ΘΦ 2 Y c Θ Φ + Y 2B Θ Y c Θ + Y 2B Φ Y c Φ b Y 2 B χ Y b 2 2 Y 2 A χθ Θ + Y 2B χ Y 2 b c 2 Y 2 A χθ Y c Θ +2 Y 2A χφ Y c Φ + Y 2B χ Y c The following expressions are used for computation: 236) 2 Y 2 R2a2 1 = Y 2 A ΘΘ Θ = Y 2 2 A N ΘΘ Ω 2 A ) Ω 2 Y 2 ΘΘ 237) Θ 2 R2a2 2 = Y 2 B Θ Θ = Y 2 B N Θ Ω 2 B ) Θ Ω1 Θ 2 Y c R2ac 1 = Y 2 A ΘΘ = Ω2 Θ2 2 Y 2 A Ω ΘΘ 238) 2 Y c Θ 2 239) R2ac 2 = Y 2 A ΦΦ 2 Y c Φ 2 = 2Ω2 Y 2 A Ω ΦΦ Y c 24) R2ac 3 = Y 2 A ΘΦ R2ac 4 = Y 2 B Θ Y c Θ 2 Y c Θ Φ = Ω2 Y 2 A Ω ΘΦ = Ω2 2 Y 2 BΘ Ω2 Y c Θ Y c Θ 241) 242) R2ac 5 = Y 2 B Φ Y c Φ = Ω2 Y 2 B Ω Φ Y c 243) 29
R2b = Y 2 B χ Y = Y 2 B N χ ) Ω 2 Bχ Ω1 Y 244) R2b2 1 = Y 2 A χθ Θ = Ω2 Y 2 A Ω χθ Θ 245) R2b2 2 = Y 2 B χ Y 2 = Y 2 B N χ ) Ω 2 Bχ Ω1 Y2 246) R2bc 1 = Y 2 A χθ Y c Θ = Ω2 2 Y 2 A Ω χθ Y c Θ 247) R2bc 2 = Y 2 A χφ Y c Φ = Ω2 Y 2 A Ω χφ Y c 248) R2bc 3 = Y 2 B χ Y c = Ω2 2 Y 2 B Ω2 χ Y c 249) Summary To compute the αs, the following functions of χ and Θ must be put into the program: A N χχ, AΩ χχ, Y, Y 2, Y c 25) The αs are then computed from 1 αij = 4π W Aijdx 251) with the Aij given by Eqs. 215) 22). If only the c mode is to be propagated, then the only equation that must be solved is α c d 2 a + α d 2 a 2 c2 dχ + α d 2 a c 2 cc dχ = R 2 c. 252) If both the c and the 2 mode are propagated, but the mode is fixed, then the equations to be solved are α c d 2 a + α d 2 a 2 c2 dχ + α d 2 a c 2 cc = R c 253) d 2 a α 2 dχ + α d 2 a 2 2 22 3 + α d 2 a c 2c = R 2. 254)
To separate these we define so that d 2 a c = 1 d α 2 a 22 R c α c D 2c dχ α 2 cc d 2 a 2 = 1 D 2c D 2c α cc α 22 α 2c α c2 255) R 2 α 2 d 2 a ) d 2 ) a α c2 R 2 α 2 ) d 2 ) a α 2c R c α c 256). 257) WARNING! We will want to keep α ij distinct from α ji, since we may be using different weight functions to project out Eq. 253) and 254). To compute R c or R 2 we need the following functions of χ along with the following Θ dependent parts of the spherical harmonics Y a 2, a c, b, b 2, b c, 258) c, dyc dθ, d 2 Yc dθ, Y 2, 2 In addition, to compute R c we need the following Roswell functions dy 2 dθ, d 2 Y 2 dθ 2,. 259) A N ΘΘ, AΩ ΘΘ, AΩ χθ, AN ΦΦ, AΩ ΦΦ, To compute R 2 we need instead A N ΘΘ, AΩ ΘΘ, AΩ χθ, AΩ ΘΦ, AΩ ΦΦ, AΩ χφ, Finally R c is computed from BN Θ, BΩ1 Θ, BΩ2 Θ, BN Θ, BΩ1 Θ, BΩ2 Θ, BN χ, BΩ1 χ, BΩ2 χ. 26) BN χ, BΩ1 χ, BΩ2 χ, BΩ Φ. 261) 1 W R c = 4π dx total integrand 1 x 2 c 262) where total integrand c = a 2 Rca2 1 + Rca2 2 a c Rcac2 1 + Rcac2 2 + 2 Rcac2 3 + Rcac2 4 + Rcac2 5 And R 2 is computed from b Rcb b 2 2 Rcb2 1 + Rcb2 2 b c 2 Rcbc 1 + 2 Rcbc 2 + Rcbc 3. 263) R c = 4π 1 W 1 x 2 dx total integrand 2 264) 31
where total integrand 2 = a 2 R2a2 1 + R2a2 2 a c R2ac2 1 + R2ac2 2 + 2 R2ac2 3 + R2ac2 4 + R2ac2 5 b R2b b 2 2 R2b2 1 + R2b2 2 b c 2 R2bc 1 + 2 R2bc 2 + R2bc 3. 265) Relationship of Modes First we repeat the definitions of l = 2 spherical harmonics used before and we introduce some new definitions: From Eqs. 185) and 186) Y 2 Y 2 = 5 4π 3 cos 2 Θ 1) 2 Y c = 1 4 15 π sin2 Θ cos 2Φ. We now repeat the definition of the real, normalized, l = 2, m = ±2 multipole about the rotation axis in Eq. 122) Y 2 = 1 15 3 cos 2 Θ 1) sin 2 Θ cos 2Φ 8 π We now introduce the symbol Y to represent the l = 2, m = multipole about the rotation axis. In terms of Roswell coordinates, this is Y = 1 4 15 π We now write our solution in the form and we note the following relationships 1 3 cos 2 Θ) 3 sin 2 Θ cos 2Φ 266) Ψ = a Y + a 2 Y 2 + a c Y c = a Y + A 2 Y 2 + A c Y c 267) Y 2 = 1 2 3 Y2 Y ) Y 2 = 1 2 Y2 + 3 Y ) 268) and A = 1 2 a2 + 3 a c ) A 2 = 1 2 3 a2 a c ) 269) 32
X. 3 Dimensional Ingoing/Outgoing Waves with 4 Modes We assume the form Notes here and below primarily by LMB) Ψ = a χ)y + a 2 χ)y 2 + a C χ)y C + a S χ)y S. 27) The notation here is rather inconsistent. The symbol Y represents Y = 1/ 4π, and Y 2 stands for the usual l = 2 spherical harmonic Y 2 Y 2 = 1 8 5 π 1 3 cos 2 Θ 3 sin 2 Θ cos 2Φ ) 271) The functions Y C and Y S are given for the normalized) real and imaginary parts of the Y 2,2 function about the axis of rotation instead of around the Z axis in the Roswell coordinates), given in terms of the Roswell coordinates: Y C = 1 15 8 π 3 cos2 Θ 1) sin 2 Θ cos 2Φ, 272) and and Y S = 1 4 15 π As before, for the linearized case we can use the notation: Y C = Y C Y Φ C cos 2Φ Y C = 1 8 sin 2Θ cos Φ. 273) 15 π 3 cos2 Θ 1) Y Φ C = 1 8 Y S = Y Φ S cos Φ Y Φ S = 1 4 15 π 15 π sin2 Θ 274) sin 2Θ. 275) We abuse the notation one more time, and replace the caligraphic Y with a regular Y. That is, we denote Y C Y C and Y S Y S. Alternative notation might be Y R and Y I, to denote the real and imaginary parts.) We now substitute this sum in the wave equation 112) to get a slight notational modification of Eq. 114) k=,2,c,s d 2 a k A χχ Y k = + da k dχ k=,2,c,s { a k χ) d 2 Y k A ΘΘ dθ + A 2 Y k 2 ΦΦ Φ 2 } Y k 2A χθ Θ + 2A Y k χφ Φ + B χy k + F χ, Θ, k 33 + 2A 2 Y k ΘΦ Θ Φ + B Θ a k χ) Y k Θ, Φ) ) Y k Θ + B Φ Y k Φ. 276)
We next multiply by a weight function W, then generate four equations by projecting respectively by Y, Y 2, Y C, Y S, and integrate over all Θ and Φ. These lead to the following equations: where and R p = + da k dχ 2π α pk = k=,2,c,s 2π α pk d 2 a k χ) = R p χ, a n, da n /dχ), 277) dφ dθ W χ, Θ) Y p Θ, Φ)A χχ Y k Θ, Φ) 278) π dφ dθ W χ, Θ) Y p Θ, Φ) +A ΦΦ 2 Y k Φ 2 2A χθ Y k Θ + 2A χφ k=,2,c,s + 2A 2 Y k ΘΦ Θ Φ + B Y k Θ Θ + B Φ } Y k Φ + B χy k + F χ, Θ, k { a k χ) Y k Φ A ΘΘ 2 Y k Θ 2 a k χ) Y k Θ, Φ) ). 279) Expressions for αs We keep the notation... to denote the Φ integral for notational simplicity, also when we do not do the Φ integrals explicitly. The α s and the averages over Φ are related by: α ij = α N ij Ω2 α Ω ij 28) α = 2π dθ W Y A χχ Y 281) α 2 = α 2 = 2π dθ W Y A χχ Y 2 282) α C = α C = 2π dθ W Y A χχ Y C 283) α S = α S = 2π dθ W Y A χχ Y S 284) α 22 = 2π dθ W Y 2 A χχ Y 2 285) 34
α 2C = α C2 = 2π dθ W Y 2 A χχ Y C 286) α 2S = α S2 = 2π α CS = α SC = 2π With x cos Θ the αs then follow explicitly from and from dθ W Y 2 A χχ Y S 287) dθ W Y C A χχ Y S 288) α CC = 2π dθ W Y C A χχ Y C 289) α SS = 2π dθ W Y S A χχ Y S 29) 1 W α ij = 4π 1 x 2 Aij dx 291) A Y A χχ Y 292) A2 Y A χχ Y 2 293) AC Y A χχ Y C 294) A22 Y 2 A χχ Y 2 295) A2C Y 2 A χχ Y C 296) AS Y A χχ Y S 297) A2S Y 2 A χχ Y S 298) ACS Y C A χχ Y S 299) ACC Y C A χχ C S 3) 35
ASS Y S A χχ Y S sorry about that...) 31) which can be evaluated by hand for the linearized case if so desired. In general, the source terms are given by 2 Y k + Y p A ΦΦ Φ 2 + da k dχ R p = 2π dθ W χ, Θ) Explicit R p s k=,2,c,s { a k χ) Y p A ΘΘ 2 Y k Θ 2 +2 Y 2 Y k pa ΘΦ Θ Φ + Y Y k pb Θ Θ + Y Y k pb Φ Φ 2 Y p A χθ Y k Θ +2 Y pa χφ Y k Φ + Y pb χ Y k In the cases p = 2, C, S, this becomes with notation b i = da i /dχ and x = cos Θ): a C χ) a S χ) Y 2 A ΘΘ 2 Y C Θ 2 Y 2 A ΘΘ 2 Y S Θ 2 { a 2 χ) R 2 = 4π + Y 2A ΦΦ 2 Y C Φ 2 + Y 2A ΦΦ 2 Y S Φ 2 1 dx W χ, Θ) 1 x 2 2 Y 2 Y 2 A ΘΘ Θ + Y 2B 2 Θ Θ } + Y p F. 32) +2 Y 2A ΘΦ 2 Y C Θ Φ + Y 2B Θ Y C Θ + Y 2B Φ Y C Φ +2 Y 2A ΘΦ 2 Y S Θ Φ + Y 2B Θ Y S Θ + Y 2B Φ Y S Φ b Y 2 B χ Y b 2 2 Y 2 A χθ Θ + Y 2B χ Y 2 b C 2 Y 2 A χθ Y C Θ +2 Y 2A χφ Y C Φ + Y 2B χ Y C b S 2 Y 2 A χθ Y S Θ +2 Y 2A χφ Y S Φ + Y 2B χ Y S + Y 2 F } 33) 36
a C χ) a S χ) a C χ) a S χ) Y C A ΘΘ 2 Y C Θ 2 Y C A ΘΘ 2 Y S Θ 2 Y S A ΘΘ 2 Y C Θ 2 Y S A ΘΘ 2 Y S Θ 2 { a 2 χ) + Y CA ΦΦ 2 Y C Φ 2 + Y CA ΦΦ 2 Y S Φ 2 1 R C = 4π dx W χ, Θ) 1 x 2 2 Y 2 Y C A ΘΘ Θ + Y CB 2 Θ Θ +2 Y 2 Y C SA ΘΦ Θ Φ + Y Y C CB Θ Θ + Y Y C CB Φ Φ +2 Y CA ΘΦ 2 Y S Θ Φ + Y CB Θ Y S Θ + Y CB Φ Y S Φ b Y C B χ Y b 2 2 Y C A χθ Θ + Y CB χ Y 2 b C 2 Y C A χθ Y C Θ +2 Y CA χφ Y C Φ + Y CB χ Y C b S 2 Y C A χθ Y S Θ +2 Y CA χφ Y S Φ + Y CB χ Y S { a 2 χ) 1 R S = 4π + Y SA ΦΦ 2 Y C Φ 2 + Y SA ΦΦ 2 Y S Φ 2 + Y C F } 34) dx W χ, Θ) 1 x 2 2 Y 2 Y S A ΘΘ Θ + Y SB 2 Θ Θ +2 Y 2 Y C SA ΘΦ Θ Φ + Y Y C SB Θ Θ + Y Y C SB Φ Φ +2 Y SA ΘΦ 2 Y S Θ Φ + Y SB Θ Y S Θ + Y SB Φ Y S Φ b Y S B χ Y b 2 2 Y S A χθ Θ + Y SB χ Y 2 37
Y C b C 2 Y S A χθ Θ +2 Y Y C SA χφ Φ + Y SB χ Y C b S 2 Y S A χθ Y S Θ +2 Y SA χφ Y S Φ + Y SB χ Y S + Y S F } 35) In the 4 1 case, the mode is fixed, and the 2, the C, and the S mode are propagated. Then the equations to be solved are: d 2 a α 2 dχ + α d 2 a 2 2 22 dχ + α d 2 a C 2 2C α C d 2 a α S d 2 a To separate these we define + α d 2 a 2 C2 dχ + α d 2 a C 2 CC + α d 2 a 2 S2 dχ + α d 2 a C 2 SC + α d 2 a S 2S = R 2 36) + α d 2 a S CS = R C 37) + α d 2 a S SS = R S. 38) D 2CS α 22 α CC α SS + α C2 α SC α 2S + α S2 α 2C α CS α 2C α C2 α SS α 22 α SC α CS α S2 α CC α 2S 39) so that d 2 a 2 1 d 2 ) a = α SS α CC α SC α CS ) R 2 α 2 D 2CS d 2 ) a d 2 ) a + α 2S α SC α SS α 2C ) R C α C + α 2C α CS α CC α 2S ) R S α S 31) d 2 a C 1 d 2 ) a = α S2 α CS α SS α C2 ) R 2 α 2 D 2CS d 2 ) a d 2 ) a + α 22 α SS α S2 α 2S ) R C α C + α C2 α 2S α 22 α CS ) R S α S 311) d 2 a S 1 d 2 ) a = α C2 α SC α S2 α CC ) R 2 α 2 D 2CS d 2 ) a d 2 ) a + α S2 α 2C α 22 α SC ) R C α C + α CC α 22 α 2C α C2 ) R S α S. 312) WARNING! We will want to keep α ij distinct from α ji, since we may be using different weight functions to project out Eq. 36) 38). 38