Ellipsoidal harmonics and related theory

APPENDIX C

Ellipsoidal harmonics and related theory C.1. Ellipsoidal harmonics Solid harmonics The interior (regular) ellipsoidal solid harmonics Emn (x) are defined as [211] m m m Em n (x) = En (ρ) En (μ) En (ν) ,

(C.1)

where Enm is Lamé function and (ρ, μ, ν) are the ellipsoidal coordinates related to the Cartesian ones (x1 , x2 , x3 ) as H12 x21 = ρ 2 μ2 υ 2 ,  2

 2

 2







(C.2)

H22 x22 = ρ2 μ2 − h3 h23 − υ , H32 x23 = ρ32 h22 − μ2 h22 − υ 2 . In Eq. (C.2), ρ12 = ρ 2 , ρ22 = ρ 2 − h23 , and ρ32 = ρ 2 − h22 . Also, a1 > a2 > a3 are the semiaxes of the reference ellipsoid, h21 = a22 − a23 ,

h22 = a21 − a23 ,





h23 = a21 − a22 , and Hm = h1 h2 h3 /hm . (C.3)

The functions Emn (x) are the nth degree polynomials of (x1 , x2 , x3 ), regular in any finite point x. Their computation is discussed in [341]. The particular values are E10 (x) = 1, Ei1 (x) = Hi xi (i = 1, 2, 3) ,  3   x2k i   +1 E2 (x) = Li (i = 1, 2) , 2  − a i k k=1

where

(C.4)

 1 4 1 2 i = a21 − h2 + h23 ± h + h22 h23 , 3 3 1     Li = i − a21 i − a22 i − a23 .

Also, E32 (x) = h1 h2 h23 x1 x2 , E42 (x) = h1 h2 h23 x1 x2 , and E52 (x) = h1 h2 h23 x1 x2 . 569

570

Ellipsoidal harmonics and related theory

The exterior (irregular) ellipsoidal solid harmonics Fmn (x) are defined as m m m Fm n (x) = Fn (ρ) En (μ) En (ν) ,

(C.5)

Fnm (ρ) = (2n + 1) Enm (ρ) Inm (ρ)

(C.6)

where

is known as Lamé function of the second kind. In Eq. (C.6), 

Inm (ρ) = 

∞



dt

2

Enm (t) (t)

ρ



and (t) = t2 − h22 t2 − h23 . For computational purposes, the integral is conveniently written as 

Inm (ρ) = ρ

1

0



dt

. 

2  ρ 2 − h22 t2 ρ 2 − h23 t2 Enm (ρ/t)

Gauss–Legendre quadrature rule provides an accurate evaluation of this integral. The particular values are F10 (x) = I01 (ρ) ,

Fi1 (x) = 3I1i (ρ) Hi xi .

(C.7)

As x → ∞, Fmn (x) = O(r −n−1 ). In particular, Fm 1 (x) = Hm

xm + O(r −3 ). r3 

(C.8)



We note also that the Wronskian W Enm , Fnm is equal to [211] Enm (ρ)

d m d 2n + 1 Fn (ρ) − Fnm (ρ) Enm (ρ) =  .  dρ dρ ρ 2 − h2 ρ 2 − h2 2

(C.9)

3

Surface harmonics The ellipsoidal surface harmonics Snm (μ, ν) = Enm (μ) Enm (ν) obey the orthogonality property at the ellipsoidal surface ρ = const:

(C.10)

571

Ellipsoidal harmonics and related theory





Sρ

Sm (μ, ν) Snm (μ, ν) n  dS = ρ 2 − μ2 ρ 2 − ν 2





Sρ

Snm (μ, ν) Snm (μ, ν) d (μ, ν)

(C.11)

= γnm δnn δmm ,

where



 μ2 − ν 2 dμ dν   d (μ, ν) =  ,  μ2 − h23 h22 − μ2 h23 − ν 2 h22 − ν 2

γnm is the normalization constant and δnm is the Kronecker’s delta. In particular, γ1m = 43π Hm2 [211]. As a consequence of Eq. (C.11), any regular in a vicinity of x = 0 harmonic function  (x) can be expanded as  (x) =

∞ 2k +1  

Akl Elk (x) ,

(C.12)

k=0 l=1

where Akl =



1 Ekl (ρ0 ) γkl



 (x) Skl (μ, ν) d (μ, ν)

and integration is performed over the surface : ρ = ρ0 .

C.2. Differentiation and integration Derivatives of solid harmonics The gradient of the interior ellipsoidal harmonic Emn (x) is evaluated using the formula ∇Em n (x) =

  3 m  Em n (x) ∂ En ξj ∂ξj   ii , ∂ξj ∂ xi Em ξj i,j=1 n

where we denote ξ1 = ρ , ξ2 = μ, and ξ3 = υ . Taking the derivative     ∂ Enm ξj /∂ξj is elementary for Enm ξj is a polynomial function of ξj . The gradient of the exterior ellipsoidal harmonic Fmn (x) is evaluated using the formula ∇Fm Em n (x) n (x) m 1 = ∇Em

2 ∇F0 (x) . n (x) In (ρ) + m (2n + 1) En (ρ)

(C.13)

572

Ellipsoidal harmonics and related theory

Here, we made use of the formula ∂ m 1 In (ρ) = −

2 m ∂ρ E (ρ)  (ρ) n

following directly from Eq. (C.6). In particular,  xi ρ ii , 2 hρ  (ρ) i=1 ρi2 3

∇F10 (x) = −

where hρ is the ellipsoidal metric coefficient. When x → ∞, ∇Fmn (x) = O(r −n−2 ). In particular, ∇F10 (x) = −

3  xi

r3

ii + O(r −3 ),

i=1 3 

∇Fm 1 (x) = Hm

i=1

(C.14)

3xm xi − 5 ii + O(r −4 ). 3 r r

δmi

In view of ∂∂n = n · ∇ , taking the normal derivative of solid harmonics is elementary. The normal derivatives of ∇Emn and ∇Fmn are calculated using the differentiation rule ∂∂n = h1ρ ∂ρ∂ and the following formulas:

3 2 m 2  ∂ En (x) ∂ξj ∂∇Em ∂Em n (x) n (x) ∂ ξj + ii , = ∂ρ ∂ρ∂ξj ∂ xi ∂ξj ∂ρ∂ xi i,j=1 ∂∇Fm ∂∇Em ∇Em n (x) n (x) n (x) + Inm (ρ) = −

2 m ∂ρ ∂ρ (2n + 1) En (ρ)  (ρ)  

3  ρ ∂ 2ρ Em Enm (ρ) ∂ρ ρ n (x) − + + m ii . −

2 m ∂ρ∂ xi En (ρ) ∂ xi ρ22 ρ32 i=1  (ρ) En (ρ)

1

Numerical integration Integration over  in Eq. (C.11) and other similar formulas requires special treatment as the Cartesian coordinates admit variable signs in the eight Cartesian octants and the integration limits for each octant are essentially different [211]. We write the integral over  as a sum   (x) d (μ, ν) = I1 + I4 + I5 + I8 , 

573

Ellipsoidal harmonics and related theory

where 

Ik =



h3 −h3

h2

h3

  (x)k 

μ2 − h23



 μ2 − υ 2 dμ dν   h22 − μ2 h23 − υ 2 h22 − υ 2

(C.15)

and subindex k indicates the octant number. Introducing the integration     variables y1 = ν/h3 and y2 = 2μ − h3 − h2 / h2 − h3 yields 

Ik =

1 −1





dy1

1 − y21

⎡ 1

−1

⎢ ⎣ 



μ2

− ν2

⎤



 (x)k ⎥ dy2 , ⎦   2 μ + h3 μ + h2 h2 − ν 2 1 − y22

(C.16)

where the expression in brackets is regular throughout the entire integration domain including the end points. It is seen from Eq. (C.16) that Chebyshev’s quadrature rule [223] 

1

−1

 

N f y π    dx ≈ f yi , N i=1 1 − y2





yi = cos

2i − 1 π 2N

(C.17)

suits perfectly because Eq. (C.17) eliminates the square-root end singularity in both integrals of Eq. (C.16). The whole integral of Eq. (C.15) equals Ik ≈ 

f y1 , y2



N1  N2  π 2  

N



f y1i , y2j ,

(C.18)

i=1 j=1

   (x)k μ2 − ν 2 =  .   μ + h3 μ + h2 h22 − ν 2

The accuracy of Eq. (C.18) is governed by the number of integration points N1 and N2 .

C.3. Reexpansion formulas Let X denote the point outside the ellipsoidal surface ρ = a1 . The following reexpansion formula exists: Fm n (x + X) =

∞ 2k +1  

k=0 l=1

ml ηnk (X) Elk (x) ,

(C.19)

574

Ellipsoidal harmonics and related theory

ml where the series expansion coefficients ηnk are given by the formula ml ηnk (X) =



1 Ekl (ρ0 ) γkl

Sa1

l Fm n (x) Sk (μ, ν) d (μ, ν) .

(C.20)

In view of Eq. (C.11), derivation of Eqs. (C.19) and (C.20) is straightforward. Eq. (C.19) enables the series expansion of the perturbation field of an ellipsoidal inhomogeneity centered in the point x + X, in a vicinity of another inhomogeneity centered in the point x, provided they do not intersect. ml The expansion coefficients ηnk of Eq. (C.20) are found by numerical integration using the Chebyshev quadrature formula. For the well-separated ml is faciliinhomogeneities (x + X > a1 and x < X), evaluation of ηnk tated greatly by applying the following semianalytical procedure. Consider two auxiliary formulas, namely Fm n (x) =

∞  k 

l μml nk Yk (x) ,

x = R > a1 ,

(C.21)

k=n l=−k

and ymn (x) =

n 2k +1  

ml l νnk Ek (x) ,

(C.22)

k=0 l=1

respectively. Here, yst (x) and Yts (x) are the spherical solid harmonics from Eq. (A.10). ml The geometry-dependent complex constants μml nk and νnk in Eqs. (C.21) and (C.22) are given by the formulas 

μml nk



2k + 1 Rk+1 = (k + l)! 4π

and ml νnk =

1 Ekl (a1 ) γkl



2π  1

l Fm n (x) χk (η, ϕ) dη dϕ

(C.23)

ymn (x) Skl (μ, ν) d (μ, ν) ,

(C.24)

−1

0

 Sa1

respectively. In Eq. (C.23), χts (η, ϕ) is the spherical surface harmonic defined by Eq. (A.5). In deriving Eq. (C.23) we made use of the orthogonality of χts , Eq. (A.6). For a given geometry of an ellipsoid, these coefficients are calculated numerically only once. The analytical expressions are available

575

Ellipsoidal harmonics and related theory

for the low-order coefficients. In particular,

μ30 11 = H3 ,

−1,1 11 = −ν11 = ν11

03 = ν11

1 , H3

H1 , 2

H2 , 2 3,−1 20 31 μ10 11 = μ11 = μ11 = μ11 = 0,

1,−1 μ11 11 = −μ11 =

1 , 2H1

2,−1 μ21 11 = μ11 = −i

−1,2 12 ν11 = ν11 =

i , 2H2

−1,3 01 02 13 ν11 = ν11 = ν11 = ν11 = 0.

By combining Eqs. (C.21) and (C.22) with the reexpansion formula of Eq. (A.24) for the spherical solid harmonics, we find that Fm n (x + X) =

∞ 2k +1  

ml ηnk (X) Elk (x) ,

k=0 l=1

where ml ηnk (X) =

∞  i 

i=k j=−i

(−1)i+j νik jl

t ∞   t=n s=−t

s−j

μms nt Yt+i (X).

(C.25)