潮汐と自転による天体の変形および それを用いた衛星の内部構造の推定 ...

Tidal and Rotational Deformation of a Celestial Bodyand its Implication tothe Estimation of Internal Structures of Satellites Katsuaki Hiyama 2004/01/30

Murray and Dermott (1999):Solar System Dynamics.Cambridge UniversityPress 4 Tides, Rotation, and Shapes 4.14.7 4.14.7() 2 1 Darwin-Radau relation 2 1990

1 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 42.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 113.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 164.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Darwin-Radau relation . . . . . . . . . . . . . . . 184.2.1 Darwin-Radau relation . . . . . . . . . . . . . . . . . . . . . . . . 184.2.2 . . . . . . . . . . . . 184.3 . . . . . . . . . . . . . . . . . . . . . 204.3.1 . . . . . . . . . . . . . . . . . . . . . . . 204.3.2 . . . . . . . . . . . . . . . . . 224.3.3 . . . . . . . . . . . . . . . 235 255.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.1.1 . . . . . . . . . . . . . . . . . . . . . . 255.1.2 . . . . . . . . . . . . . . . . . . . 255.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 28 29

A 31A.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31A.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31A.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36A.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41A.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49A.6 Darwin-Radau Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . 53A.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 60

1. 11 1.1 2 1.2 Murray and Dermott (1999):Solar System Dynamics.Cambridge UniversityPress ( MD1999) 4 Tides, Rotation, and Shapes 4.14.7 4.14.7 2

1. 2Darwin-Radau relation (2 ) + (3 )‖ J 2 () Darwin-Radau relation (4.2 ) (ρ) (σ)(A/B) (2 ) + (3 )‖H h () (4.3 ) (ρ) (σ)(A/B)

1. 32 3 4 2 1 Darwin-Radau relation 2 5 6 MD 1999 4.14.7

2. 42 2.1 () 2.2 m p m s 2 〈F〉 〈F〉 = G m pm s(2.1)r 2r 2 (2.1)P 1 C 1 a p F (2.2)〈F〉 〈F〉 = ≠ F (2.2)F tidal F tidal = F − 〈F〉 (2.3)

2. 52.1: a p ,a s a = a p + a s (MD 1999 )2.2: a p P 1 ,P 2 C 1 ,C 2 (MD 1999 )

2. 62.3 P V V = −G m s(2.4)∆ ∆ P 2.3∆ = a[1 − 2( Rpa)cos ψ +ψ P R p /a ≪ 1 6,378 km 384,400 km (2.5) (2.4) V = −G m [s1 +a a≈ V 1 + V 2 + V 3( Rp)cos ψ +( Rpa( Rpa) 2] 12) ]212 (3 cos2 ψ − 1) + · · ·(2.5)(2.6)(2.6) 1 V 1 = −G(m s /a)P V 1P (2.6) 2 V 2 = −G(m s /a 2 )R p cos ψ P ∂V 2−∂(R p cos ψ) = G m sa = 〈F〉(2.7)2 m p2.3: R p aP ∆ V 2 = −G(m s /a 2 )R p cos ψ = constant (MD 1999 )

2. 7 (2.6) 3 V 3 (ψ) = −G m sa 3 R2 pP 2 (cos ψ) (2.8)P 2 (cos ψ) = 1 2 (3 cos2 ψ − 1) (2.9)cos ψ 2 A.3 F tidalm p= F m p− 〈F〉m p= −∇V − 〈F〉m p≈ −∇V 3 (ψ) (2.10)V 3 (ψ) V 3 (ψ) V 3 (ψ) = ζgP 2 (cos ψ) (2.11)ζ = m ( ) 3s RpR p (2.12)m p ag = G m pR 2 p(2.13)ζP 2 (cos ψ) ψ (2.12) m Sunm Moon(aMoona Sun) 3≈ 0.46 (2.14)ζ = 0.36 mζ = 0.16 m P 2 (cos ψ) ψ = 0, π ψ = π/2, 3π/2 24 1 1 2

2. 82.4: A ρ B σ AB (MD 1999 )2.4 σ A ρ µ B (2.4) µ 2.3 2 ψ (core boundary)R cb (ocean surface)R os R cb (ψ) = A[1 + S 2 P 2 (cos ψ)] (2.15) R os (ψ) = B[1 + T 2 P 2 (cos ψ)] (2.16) S 2 T 2 S 2 T 2 S 2 T 2 2 (i) (ii) S 2 T 2 2 (i) S 2 T 2 1 V os (r, ψ)

2. 9ρ − σ() V os (r, ψ) ()V os (r, ψ) = −ζgP 2 (cos ψ) − 4 3 πB2 σG1 − 2 5 T 2P 2 (cos ψ)− 4 (13 π A3B (ρ − σ)G − T 2 P 2 (cos ψ) +5( 3 ) A 2S2P 2 (cos ψ))B(2.17) ψ ψ (2.17)Gmp= 4B 3 πGρB2 S 2 T 2 1 [ ( )ζ c 2A = σ A 3 (5 ρ + 1 − σ ) ] T 2 − 3 ( ) A 5 (1 − σ )S 2 (2.18)B ρ 5 B ρ m c ζ c = m ( )s A 3A(2.19)m c ag c = Gm c(2.20)A 2ζ c (2.12) ζ ζgA 2 = ζ c g c B 2 (2.21)(2.18) S 2 T 2 1 (ii) XP 2 (cos ψ) X = 2 (5 ρg cA 1 − σ )( 5 ζ cρ 2 A − S 2 + 3 )σ2 ρ (T 2 − S 2 )(2.22) Love(1944) ∆R(ψ) = 5 A19 µ XP 2(cos ψ) (2.23)

2. 10 (2.16) 2 AS 2 P 2 (cos ψ) S 2 T 2 2 S 2 = 1˜µ(1 − σ )( 5 ζ cρ 2 A − S 2 + 3 )σ2 ρ (T 2 − S 2 )(2.24) ˜µ .˜µ = 19µ2ρg c A(2.25)S 2 T 2 (2.18) (2.24) S 2 T 2 AS 2 = F (5/2)ζ c1 + ˜µ BT 2 = H 5 2 ζ (2.26) F H F =(1 + ˜µ)(1 − σ/ρ)(1 + 3/2α)1 + ˜µ − σ/ρ + (3σ/2ρ)(1 − σ/ρ) − (9/4α)(A/B) 5 (1 − σ/ρ) 2 (2.27)H = 2〈ρ〉5ρα = 1 + 5 ( )ρ A 3 (1 − σ )2 σ B ρ( 1 + ˜µ + (3/2)(A/B) 2 )F δ(1 + ˜µ)(δ + 2σ/5ρ)( ) A 3 ( δ = 1 − σ )B ρ(2.28)(2.29)〈ρ〉 ˜µ = 0 H H h H h = 2〈ρ〉 (1 + (3δ/5γ)(A/B) 2 )5ρ δ + 2σ/5ρ − (9δσ/25γρ)(A/B) 2γ = 2 5 + 3σ5ρ(2.30)(2.31) (Dermott 1979a)H H h

3. 113 3.1 10 2 (2.4) a () b = c ψ 2P 2 (cos ψ) 3.5: Ω P (x − z ) θ z( ) r (MD 1999 )

3. 123.2 3.5 Ω P a cf,x = Ω 2 r sin θˆx x = r sin θ a cf,x = Ω 2 xˆxy − z a cf,y = Ω 2 yŷ (x, y, z) a cf = Ω 2 (xˆx + yŷ) (3.32) (x, y, z) m F cf F cf = ma cf = mΩ 2 (xˆx + yŷ) (3.33)3.3 V cf θ , a cf = −∇V cf V cf (r, θ) = − 1 2 Ω2 r 2 sin 2 θ (3.34) P 2 (cos θ) = 1(2 − 3 2 sin2 θ) V cf = 1 3 Ω2 r 2 [P 2 (cos θ) − 1] (3.35)2 3.4 (flattening) f f f = r equatorial − r poler equatorial(3.36) r equatorial r pole

3. 13V total (r, θ) = − Gm pr+ V cf (r, θ) (3.37)r ocean = a + δr(θ) (3.38)a a = r equatorial (equatorial radius) (3.37) (3.38) V total (surface) ≈ − Gm pa+ Gm pa 2 δr − 1 2 Ω2 a 2 sin 2 θ − Ω 2 a sin 2 θδr (3.39)Ω 2 a ≪Gm p /a 2 (3.39) δr ≈ constant + Ω2 a 4sin 2 θ (3.40)2Gm p (3.40) f ≈ q/2 q = Ω2 a 3Gm p(3.41)2 (a = b)1 (c) V gravity (r, θ) = − Gm [∞∑( ) a nPn1 − J n (cos θ)](3.42)rn=2r m a J n P n (cos θ) n n = 1 J n J n J 2 x y z 3 A, B, C

3. 14A = B J 2 (Cook 1980)J 2 = C − A(3.43)ma 2(3.35) V total (r, θ) = − Gm pr+[ Gmp a 2r 3 J 2 + 1 3 Ω2 r 2 ]P 2 (cos θ) − 1 3 Ω2 r 2 (3.44)J n r =a + δr(θ) (3.44) δr = constant −[J 2 + 1 ]3 q aP 2 (cos θ) (3.45)f = 3 2 J 2 + 1 2 q (3.46)f ≈ q/2 f (Yoder 1995) f calc = 0.003349f obs = 0.003353 f calc = 0.06670f obs = 0.06487 (3.46) f J 2 J 2 q = Ω 2 a 3 /Gm J 2 1/r J 2 J 2 (3.43) J 2 C − A , J 2 5.1C A C A (C − A)/C (Cook

3. 151980) 26000 C A

4. 164 4.1 • 〈ρ〉 =3m4πB 3 (4.47)B 〈ρ〉 = ( A ) 3(ρ − σ) + σ (4.48)B A/Bσρ • r ρ(r) I = A + B + C3= 8π 3∫ R0ρ(r)r 4 dr (4.49)A , B 2 C Rρ m = 4 3 πρR3 I = 0.4mR 2 (4.50)C A/Bσρ CmB = 2 [ ( σ2 5 〈ρ〉 + 1 −〈ρ〉)( σ ) ] A 2B(4.51)0.4 0.4 A/B σ ρ 3

4. 17 1 (4.48) 2 (4.51) 3.4 J 2 /f C A/Bσρ3 (4.48) (4.51) 2 3 1 2

4. 184.2 Darwin-Radau relation 4.2.1 Darwin-Radau relationDarwin-Radau relation C/mR 2 (m R )qfJ 2 Cook(1980) CmR = 2 [1 − 2 ( ) ]5 q 1/2 2 3 5 2 f − 1(4.52)¯C ¯C =CmR 2 (4.53)(3.46) J 2 qf (4.52) Darwin-Radau relationJ 2f = − 3 10 + 5 2 ¯C − 15 8 ¯C 2 (4.54)4.2.2 Darwin-Radau relation ¯C J 2 /f 2 Dermott(1979b) J 2 /fH h J 2f = 2 (1 − 2 )3 5H h1(2.29) (2.30) (2.31) (4.55)J 2f = 2 3 + ¯C − 2 5 (A/B)21 − (A/B) 2 + 8 − 20(A/B) 2 + 10 ¯C[5(A/B) 3 − 2]12[(A/B) 5 − 1] + 15 ¯C[2 − 5(A/B) 3 + 3(A/B) 5 ](4.56)2 1 (2.29) δ = ( )A 3 ( ) (B 1 −σρ (2.30) Hh = 2〈ρ〉5ρ)1+(3δ/5γ)(A/B) 2δ+2σ/5ρ−(9δσ/25γρ)(A/B) 2(2.31) γ = 2 5 + 3σ5ρ

4. 194.6: ¯C J 2 /f (i) ( ), (ii)A/B =0.5 ( )(iii) Darwin-Radau relation ( )J 2 /f(MD 1999 )• A/B → 0 ( )J 2 /f → ¯C (4.57)• A/B → 1 J 2 /f → − 310 + 5 2 ¯C − 15 8 ¯C 2 (4.58)Darwin-Radau relation 4.6 Darwin-Radau A/B = 0.5 ¯C J 2 /f J 2 /f ¯C J 2 /f ¯C J 2 /f J 2 /f (3.43) C − A C A

4. 204.3 4.3.1 .n (r, θ, ψ) V rotational = 1 3 Ω2 r 2 P 2 (cos θ) (4.59) ((3.35)2 ) θ P 2 (cos θ) =(1/4)(3 cos 2θ + 1) 2 V rotational x − y ψ z V tidal = − Gm pa 3 r2 P 2 (cos ψ) (4.60) ((2.8) 3 ) m p V tidal z θ x Gm3 p= an 2a 2n = Ω V tidal = −Ω 2 r 2 P 2 (cos ψ) (4.61) V rotational V tidal 3 2.4 4.7 2 (3.35) V cf = 1 3 Ω2 r 2 [P 2 (cos θ) − 1]3 (2.8) V 3 (ψ) = −G m saRpP 2 3 2 (cos ψ)

4. 214.7: (a) z (b) x (MD1999 )(4.7a) z (4.7b) x () (2.16) () 4 −1/3 a b c ( xyz ) B T 2 a b θ = π/2 P 2 (cos θ) c θ = 0 a r = B(1 + T 2 /6) , b r = B(1 + T 2 /6) , c r = B(1 − T 2 /3) (4.62)a ψ = 0 P 2 (cos ψ) b c ψ = π/2 a t = B(1 + T 2 ) , b t = B(1 − T 2 /2) , c t = B(1 − T 2 /2) (4.63)4 (2.16) R os (ψ) = B[1 + T 2 P 2 (cos ψ)]

4. 22a = B(1 + 7T 2 /6) , b = B(1 − T 2 /3) , c = B(1 − 5T 2 /6) (4.64)b − c = 1 (a − c) (4.65)4(Dermott1979b)4.3.2 4.1 A/B σ ρ 3 2 1 1 4.2.3 (4.48) 2 5(2.12) ζ = m p( B ) 3B (4.66)m s a(3.41) 6 3 = an 2 n = Ω Gm pa 2q = Ω2 a 3= m p( B ) 3(4.67)Gm s m s aζ/B = q (4.68)(4.64) a − c = 2BT 2 (4.69) (2.26) BT 2 = H 5ζ 2 5 (2.12) ζ = m s6 (3.41) q = Ω2 a 3Gm p( ) 3Rp Rpm p aH h = a − c5qB(4.70)

4. 23a − cBq H h (2.30) H h A/B σρ 3 (4.48) H h (4.70) 4.3.3 V gravity (r, θ, ψ) = Gm [∞∑ n∑) R n(Cnm1 +cos mψ + S nm sin mψ)P nm (cos θ)](4.71)rn=2 m=0(r(r, θ, ψ) R P nm (cos θ) C nm S nm 3 V gravity (r, θ, ψ) = Gm [1 − 1 ( ) R 2(3 ( ) R 2r 2 J 2 cos 2 θ − 1) + 3C 22 sin 2 θ cos 2ψ](4.72)rr(n,m)=(2,1) J 2 = −C 20 C 22 C 22 = B − A4mR 2 (4.73)A B (4.59) V rotational = 1 3 Ω2 r 2 P 2 (cos θ)(4.61) V tidal = Ω 2 r 2[ 12 P 2(cos θ) − 3 4 (sin2 θ cos 2ψ) ] (4.74)V rotational + V tidal = 5 6 Ω2 r 2 P 2 (cos θ) − 3 4 sin2 θ cos 2ψ (4.75)

4. 24(4.72)J 2 ∝ 5/6Ω 2 r 2 , C 22 ∝1/4Ω 2 r 2 J 2 C 22 J 2 = 10 3 C 22 (4.76) (Anderson et al 1996a,1998a) C 22 (3.41) q C 22 = 3 αq (4.77)4 α α = 1/2 α Darwin-Radau relation ¯C = 2 3[1 − 2 5( ) ]4 − 3α 1/21 + 3α(4.78)

5. 255 5.1 5.1.1 1990 4 1000 km 2.3 GHz J 2 C 22 J 2 = 10C 3 22. 4.3.3 Darwin-Radau relation 5.1 5.10 (Anderson et al 1996a,1996b,1998a,1998b) m (10 23 kg) 0.893 0.480 1.482 1.076 B (km) 1821 1565 2634 2403 〈ρ〉 (g cm −3 ) 3.53 2.99 1.94 1.85¯C 0.378 0.348 0.311 0.3585.1: (Anderson et al 1996a,1996b,1998a,1998b)5.1.2 1 ¯C = 0.378 ± 0.007 (A/B = 0.36) (A/B= 0.52) (Anderson et al 1996b)

5. 265.8: (MD 1999 )H 2 O ¯C = 0.347 ± 0.014 (A/B = 0.5) H 2 O (80 170 km) (Anderson et al 1998b)( R.Reynold and S.Squyres 1982). 2631 km 2440 km 1.94 g cm −3 ¯C = 0.311 ± 0.003 . (400 km) (1300 km) (800 km) 3 (Anderson et al 1996a). (Squyres et al 1983). ¯C = 0.406 ± 0.030 (Anderson et al 1997b)¯C = 0.358 ± 0.004 (Anderson et al 1998a)

5. 275.2 Dermott Thomas(1988) (4.64) 0.25 (b − c)(a − c) = 0.27 ± 0.04 Dermott Thomas B = 198.8 km Kozai(1957) 〈ρ〉 = 1.137 ± 0.018 g cm −3 20.3 ± 0.3 km a − c = 16.9 ± 0.7 km a − c = 16.9 ± 0.7 km σ = 3.0 g cm −3 (4.68) A/B = 0.44 ± 0.09 σ = 0.96 ± 0.08 g cm −3 2 1.88 g cm −3 1997 2004 7 (Rappaport et al. 1997)

6. 286 ¯C J 2 1989 14 2003 9 22 1997 2004 7 2008 2010 2011

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A. 31AMurray and Dermott (1999):Solar System Dynamics.Cambridge University Press 4 Tides, Rotation, and Shapes 4.14.7 A.1 (tide) tidal bulgeA.2 m p m s 2 〈F 〉 〈F 〉 = G m pm sr 2 , (A.1)

A. 32A.1: a p ,a s a = a p + a s r A.1a s /a p = m p /m s ,(A.2)a = a p + a s P 1 C 1 a p P 2 C 1 C 2 P 1 P 2 C 1 C 2 A.2F 〈F〉 〈F〉 = (centrifugal force) ≠ F.(A.3)F tidal F tidal = F − 〈F〉.(A.4)A.7 P V ()V = −G m s∆ ,(A.5)

A. 33A.2: a p P 1 ,P 2 C 1 ,C 2 ∆ P A.3∆ = a[1 − 2( Rpa)cos ψ +( Rpa) 2] 12, (A.6)ψ R p /a ≪ 1 6,378 km 384,400 km (A.6) 7 V = −G m [s1 +a a≈ V 1 + V 2 + V 3 ,( Rp)cos ψ +( Rpa) ]212 (3 cos2 ψ − 1) + · · ·(A.7)(A.7) 1 V 1 = −G(m s /a)F/m p = −∇V (A.7) 2 V 2 = −G(m s /a 2 )R p cos ψP ( ) ( ) 2 7 Rx = 1 − 2pcos ψ + aR pam sV = −G m s∆ = −Ga(1 + x) 1 2= −G m sa (1 + x)(− 1 2 )(1 + x) (− 1 2 ) = 1 − 1 2 x + 3 8 x2 · · ·x 3 (A.7)

A. 34A.3: R p aP ∆ V 2 = −G(m s /a 2 )R p cos ψ = constant 2 ∂V 2−∂(R p cos ψ) = G m sa = 〈F 〉 .2 m p(A.8)(A.7) 3 V 3 (ψ) = −G m sa 3 R2 pP 2 (cos ψ),P 2 (cos ψ) = 1 2 (3 cos2 ψ − 1)(A.9)(A.10)n = 2cosψ A.3 F tidalm p= F m p− 〈F〉m p= −∇V − 〈F〉m p≈ −∇V 3 (ψ), (A.11)V 3 (ψ) V 3 (ψ) = ζgP 2 (cos ψ),(A.12)ζ = m ( ) 3s RpR p (A.13)m p ag = G m pR 2 p(A.14)ζP 2 (cos ψ) ψ (equilibrium tide) P 2 (cos ψ)ψ = 0, π ψ = π/2, 3π/2

A. 35A.4: V 3 (ψ) (a) z Ω n I (b) (θ M , φ M )P P (θ P , φ P ) φ P ,φ M 24 1 2 ( ) ( ) A.4a z Ω nI A.4b (θ M , φ M )P P (θ P , φ P ) φ P ,φ M OP OM ψ 8cos ψ = cos θ P cos θ M + sin θ P sin θ M cos(φ P − φ M ).(A.15)8 . | OM || OP | cos ψ = x P x M + y P y M + z P z M

A. 3612 (3 cos2 ψ − 1) = 1 2 (3 cos2 θ P − 1) 1 2 (3 cos2 θ M − 1)+ 3 4 sin2 θ P sin 2 θ M cos 2(φ P − φ M )+ 3 4 sin 2θ P sin 2θ M cos(φ P − φ M ). (A.16)θ P P φ P θ M φ M (A.16) 1 cos 2 θ M = (1/2)(1 + cos 2θ M )2n (fortnightly tide) 2 2(Ω − n) ≈ 2Ω (semidiurnal tide) 3 (Ω − n) ≈ Ω (diurnal tide) 3 sin 2θ M 2n Sun 2(Ω − n Sun ) ≈ 2Ω(Ω − n Sun ) ≈ Ω n Sun (A.13) m Sunm Moon(aMoona Sun) 3≈ 0.46.(A.17)ζ = 0.36 mζ= 0.16 m A.3 γ C r δr δm(Ramsey 1940)r ′ V ext (r ′ ) = − Gδm(A.18)r ′V ext (C) = − G ∑ δmC= − 4 3 πγGC2 . (A.19)

A. 379 (inverse square law) V int (r) = − G ∑ δm= −4πγGrδr. (A.20)r C r V int (C, r) = −4πγG∫ Crrdr = −2πγG(C 2 − r 2 ).(A.21)r (r < C) (r > C) V int (r) = − 4 3 πγGr2 − 2πγG(C 2 − r 2 ) = − 2 3 πγG(3C2 − r 2 ), (A.22)V ext (r) = − 4 C3πγG3 r .(A.23)Ramsey(1940),MacRobert(1967),Bullen(1975),and Blakely(1995) V ∇ 2 V = 0.(A.24)(Euler’s equation)V n (homogeneousof degree n) x ∂V∂x + y ∂V∂y + z ∂V∂z = nV.(A.25)10 (spherical solid harmonics) 3 r,θ,φ 3 1 (Blakely(1995) ) (r,θ,φ) r θ φ (∂r 2 ∂V )+ ∂ ((1 − µ 2 ) ∂V )1 ∂ 2 V+= 0, (A.26)∂r ∂r ∂µ ∂µ (1 − µ 2 ) ∂φ2 9 2 2 2 10

A. 38 µ = cos θ V = r n S n (µ, φ) (A.26) S n (µ, φ) r (∂r 2 ∂V )= n(n + 1)r n S n , (A.27)∂r ∂r (A.26) ∂∂µ((1 − µ 2 ) ∂S n∂µ)+1 ∂ 2 S n(1 − µ 2 ) ∂φ + n(n + 1)S 2 n = 0. (A.28)(Legendre’s equation) S n (spherical surface harmonic) n −(n + 1) n(n + 1) ∞∑ (V = An r n + B n r −(n+1)) S n (µ, φ). (A.29)n=0n n + 1 (solid harmonic) (Ramsey 1940)(1 − µ 2 ) ∂2 P n (µ)− 2µ ∂P n(µ)+ n(n + 1)P∂µ 2 n (µ) = 0, (A.30)∂µP n (µ) [1 · 3 · 5 . . . (2n − 1)P n (µ) = µ n n(n − 1)−n!2(2n − 1) µn−2(Rodrigues’s formula)n(n − 1)(n − 2)(n − 3)+2 · 4 · (2n − 1)(2n − 3) µn−4 + · · ·](A.31)P n (µ) = 1 d n (µ 2 − 1) n(A.32)2 n n! dµ nθ (zonal harmonics) 5 P 0 (µ) = 1, (A.33)P 1 (µ) = µ = cos θ, (A.34)P 2 (µ) = 1 2 (3µ2 − 1) = 1 (3 cos 2θ + 1), (A.35)4P 3 (µ) = 1 2 (5µ3 − 3µ) = 1 (5 cos 3θ + 3 cos θ),8(A.36)P 4 (µ) = 1 8 (35µ4 − 30µ 2 + 3) = 1 (35 cos 4θ + 20 cos 2θ + 9).64(A.37)

A. 39(surface harmonics) (r = 1) sin θdθdφ = −dµdφ Y m (µ, φ) , S n (µ, φ) m n (m ≠ n)∫ 2π ∫ +10 −1Y m (µ, φ)S n (µ, φ)dµdφ = 0.(A.38) 2 n 1 P n (µ) ∫ 2π ∫ +10−1S n (µ, φ)P n (µ)dµdφ =4π2n + 1 S n(1), S n (1) P n (µ) S n (µ, φ) (A.39)2 (θ ′ , φ ′ ) (θ, φ)2 ψ ∫ 2π ∫ +10 −1S n (θ ′ , φ ′ )P n (cos ψ)dµ ′ dφ ′ .(A.40)(θ, φ) (Θ ′ , Φ ′ ) Θ ′ = ψ S n (θ ′ , φ ′ ) Y n (Θ ′ , Φ ′ ) (A.39) ∫ 2π ∫ +10−1∫ 2π ∫ +10Y n (Θ ′ , Φ ′ )P n (cos Θ ′ )d(cos Θ ′ )dΦ ′ =−1Y n (1) = S n (θ, φ).S n (µ ′ , φ ′ )P n (cos ψ)dµ ′ dφ ′ =4π2n + 1 Y n(1).4π2n + 1 S n(µ, φ)(A.41)(A.42)(A.43) S n (µ, φ) (µ, φ) S n (µ ′ φ ′ ) (µ ′ , φ ′ ) (MacRobert 1967)P R(θ ′ ) = C [ 1 + ɛ 2 P 2 (cos θ ′ ) ] ,(A.44) ɛ 2 (≪ 1) C P (r < C) (r > C)(r, µ, φ) µ = cos θ θ

A. 40A.5: P C (A.5)P 2 1 (A.22) (A.23) P ′ (r ′ , µ ′ , φ ′ ) ɛ 2 CP 2 (µ ′ )ɛ 2 C 3 P 2 (µ ′ )dµ ′ dφ ′ P P P ′ = ∆ 1∆ = ( C 2 + r 2 − 2Cr cos ψ ) −1/2.(A.45) r < C 1∆ = 1 [1 + ( r ) 2 ( r ) ] −1/2− 2µ , (A.46)C C Cr/C 1∆ = 1 [1 + ( r ) ( r ) 2 ( 1µ + −C C C 2 + 3 2 µ2) + ( r ) 3 ( 3 −C 2 µ + 5 ]2 µ3) + · · · , (A.47)(A.34) (A.37) (A.47) 1∆ = 1 C∞∑ ( r ) nPn(cos ψ) + O(ɛ 2 ). (A.48)n=0CP V nc,int = −γGC 2 ∑ ∞ɛ ( r ) ∫ ∫ 22P 2 (µ ′ )P n (cos ψ)dµ ′ dφ ′ . (A.49)Cn=0

A. 41(A.43) ∞∑ ( r ) ∫ ∫ nn=0CP 2 (µ ′ )P n (cos ψ)dµ ′ dφ ′ = 4π 5( r ) 2P2(cos θ) (A.50)CP V nc,int = − 4π 5 γGr2 ɛ 2 P 2 (cos θ),(A.51) P (A.22) V int (r, θ) = − 4 [ 3C 2 − r 23 πC3 γG + 3 r 2]2C 3 5 C ɛ 2P 3 2 (cos θ) . (A.52) (r > C) 1∆ = 1 r∞∑ ( C ) nPn(cos ψ) + O(ɛ 2 ) (A.53)n=0rV ext (r, θ) = − 4 [ 13 πC3 γGr + 3 C 2]5 r ɛ 2P 3 2 (cos θ) . (A.54)A.4 h(ψ) (local height) −ζgP 2 (cos ψ) V total (r, ψ) = − Gm pB + gh(ψ) − ζgP 2(cos ψ), (A.55) B ψ h(ψ) = ζP 2 (cos ψ) ψ 2 B σ A ρ µ (Street 1925, Dermott 1979a,A.6) µ

A. 42 2 (second-order surface harmonic) 1 (self gravitation) (Street 1925)2 2 (Love 1911)A.2 ψ 2 (core boundary) (ocean surface) R cb (ψ) = A[1 + S 2 P 2 (cos ψ)](A.56) R os (ψ) = B[1 + T 2 P 2 (cos ψ)](A.57)S 2 T 2 S 2 T 2 (i) (ii) V o (r, ψ) (i) V 3 (r, ψ) = − Gm sa 3 r2 P 2 (cos ψ) = −ζg ( r ) 2P2(cos ψ), (A.58)B (A.9) 11 (A.12) 12 (ii) V int (r, ψ)(iii) V ext (r, ψ) V o (r, ψ) = −ζg ( r ) 2P2(cos ψ)B− 4 ( 3B 2 − r 23 πB3 σG + 3 r 2)2B 3 5 B T 2P 3 2 (cos ψ)− 4 ( 13 πA3 (ρ − σ)Gr + 3 A 2)5 r S 2P 3 2 (cos ψ) . (A.59)11 (A.9) : V 3 (ψ) = − Gmsa 3 R 2 pP 2 (cos ψ)12 (A.12) : V 3 (ψ) = −ζgP 2 (cos ψ)

A. 43A.6: A ρ B σ AB ρ − σ() V os (r, ψ) r = B[1 + T 2 P 2 (cos ψ)] (A.59) ζ/BS 2 T 2 2 13 V os (r, ψ) = −ζgP 2 (cos ψ) − 4 3 πB2 σG(1 − 2 5 T 2P 2 (cos ψ)− 4 (13 π A3B (ρ − σ)G − T 2 P 2 (cos ψ) +5( 3 ) A 2S2P 2 (cos ψ)). (A.60)B ψ 14ψ [ ( )ζ c 2A = σ A 3 (5 ρ + 1 − σ ) ] T 2 − 3 ( ) A 5 (1 − σ )S 2 .B ρ 5 B ρ)(A.61)()13 2 − 4 3 πB2 1σG2 − 2 5 T 2P 2 (cos ψ)14 Gmp= 4 3 πGρB2 B

A. 44 m c ζ c = m ( )s A 3A,(A.62)m c ag c = Gm cA 2(A.63)ζ c (A.13) 15 ζ ζgA 2 = ζ c g c B 2 .(A.64)(A.61) S 2 T 2 1 2 V c (r, ψ) V 3 (r, ψ) V c (r, ψ) = −ζ c g ( r ) 2P2c (cos ψ)A− 4 ( 3B 2 − r 23 πB3 σG + 3 r 2)2B 3 5 B T 2P 3 2 (cos ψ)− 4 ( 3A 2 − r 23 πA3 (ρ − σ)G + 3 r 2)2A 3 5 A S 2P 3 2 (cos ψ) . (A.65)r V c (r, ψ) P 2 (cos ψ) V c (r, ψ) = −Zr 2 P 2 (cos ψ),(A.66)Z = g (c ζcA A + 3 σ5 ρ (T 2 − S 2 ) + 3 )5 S 2 . (A.67)Chree(1896a) (yielding)r = A ρZA 2 P 2 (cos ψ) (B − A) ≪ B g (i)ψ (ii)ψ ( 15 (A.13) : ζ = m s Rpm p a) 3Rp

A. 452 P o (ψ) = gσB(T 2 − S 2 )P 2 (cos ψ)P c (ψ) = g c ρAS 2 P 2 (cos ψ).(A.68)(A.69) g 16 P o (ψ) =∫ RosR cbσ(r) ∂V o(r, ψ)∂rdr,(A.70)P o (ψ) = σ[V o (R os , ψ) − V o (R cb , ψ)].(A.71)()V o (R os , ψ) V cb (ψ) R cb = A[1 + S 2 P 2 (cos ψ)] (A.59) (A.65) ζ/BS 2 T 2 2 ( ζcV cb (ψ) = constant − Ag cA + 3 σ5 ρ (T 2 − S 2 ) − 2 )5 S 2 P 2 (cos ψ), (A.72) ψ 2 P o (ψ) ψ ( ζcP o (ψ) ψ = σAg cA + 3 σ5 ρ (T 2 − S 2 ) − 2 )5 S 2 P 2 (cos ψ). (A.73)17XP 2 (cos ψ) X = ρA 2 Z − P o (ψ) ψ − ρg c AS 2= 2 (5 ρg cA 1 − σ )( 5ρ 2dPζ cA − S 2 + 3 2)σρ (T 2 − S 2 ) . (A.74)16 dr= −ρg17 1 ρA 2 Z Chree(1896a)

A. 46σ → ρ X → 0 Love(1944) ∆R(ψ) = 5 A19 µ XP 2(cos ψ), (A.75) AS 2 P 2 (cos ψ) S 2 T 2 2 S 2 = 1˜µ(1 − σ )( 5 ζ cρ 2 A − S 2 + 3 )σ2 ρ (T 2 − S 2 )(A.76) ˜µ (effective rigidity) ˜µ = 19µ2ρg c A .(A.77)˜µ ≪ 1 ˜µ ≫ 1 σ = ρ S 2 = 0 σ = 0 (A.76) AS 2 = (5/2)ζ c1 + ˜µ . (A.78)AS 2 = F (5/2)ζ c1 + ˜µ BT 2 = H 5 2 ζ,(A.79) F H (A.61) (A.76) T 2 F =(1 + ˜µ)(1 − σ/ρ)(1 + 3/2α)1 + ˜µ − σ/ρ + (3σ/2ρ)(1 − σ/ρ) − (9/4α)(A/B) 5 (1 − σ/ρ) 2 (A.80)H = 2〈ρ〉5ρα = 1 + 5 ( )ρ A 3 (1 − σ )2 σ B ρ( 1 + ˜µ + (3/2)(A/B) 2 )F δ(1 + ˜µ)(δ + 2σ/5ρ)( ) A 3 ( δ = 1 − σ )B ρ(A.81)(A.82) 〈ρ〉 (

A. 47(synchronous) )˜µ = 0 H 18(Dermott 1979a)H h = 2〈ρ〉 (1 + (3δ/5γ)(A/B) 2 ), (A.83)5ρ δ + 2σ/5ρ − (9δσ/25γρ)(A/B) 2γ = 2 5 + 3σ5ρ(A.84)A =Bζ c = ζ〈ρ〉 = ρ (A.61) ζA = 2 (5 S 2 + 1 − 3 )σ(T 2 − S 2 ). (A.85)5 ρ(A.61) (A.76) S 2 A(T 2 − S 2 ) =ζ ˜µ1 − σ/ρ + ˜µ(1 − 3σ/5ρ)(A.86)(Chree 1896b)˜µ → 0 A(T 2 − S 2 ) → 0 σ = ρ A(T 2 − S 2 ) = 5 2 ζ(A.87) ˜µ (A.13) 19 5/2 (A.86) T 2 − S 2 (A.61) AS 2 = 5 []2 ζ (1 − σ/ρ). (A.88)1 − σ/ρ + ˜µ(1 − 3σ/5ρ)σ = 0 AS 2 = (5/2)ζ1 + ˜µ , (A.89) Lord Kelvin (Thompson 1863) Kelvin AS 2 ≈ 0.6 ζ ∼ 1.2 × 10 11 N m −2 ∼ 50% (Bullen 1975)2〈ρ〉5σ18 ( 19 (A.13) : ζ = m s Rpm p a) 3Rp

A. 48A.7: d 2 (tidalwave) U u ρ µ σ (A.86) (A.88) (ocean basin) Proudman(1953) d 2 U = 2πA/T E ≈ 500 m s −1 T E U u (A.7)ζ (U − u)(d + ζ) = Ud (A.90) ζ ≪ d u = ζ d U(A.91)∼ 4 km u ∼ 0.1 m s −1 12 (U − u)2 + gζ + Ψ = constant, (A.92)

A. 49 Ψ Ψ = −g¯ζ U 2 u 2 ≪ uU Uu = g(ζ − ¯ζ).(A.93)(A.91) u ζ =¯ζ1 − U 2 /gd . (A.94) d res = U 2 /g ≈ 22 km d res (A.94) A.5 A.4 (A.6) a () b = c ψ 2 P 2 (cos ψ) Ω (A.8)Pa cf,x = Ω 2 r sin θˆx x = r sin θ a cf,x = Ω 2 xˆx y − z a cf,y = Ω 2 yŷ (x, y, z) a cf = Ω 2 (xˆx + yŷ).(A.95)V cf a cf = −∇V cf V cf (r, θ) = − 1 2 Ω2 r 2 sin 2 θ.(A.96)V total (r, θ) = − Gm pr+ V cf (r, θ). (A.97)

A. 50A.8: Ω P (x − z ) θ z( ) r r ocean = a + δr(θ)(A.98) a = r equatorial (equatorial radius) V total (surface) ≈ − Gm pa+ Gm pa 2 δr − 1 2 Ω2 a 2 sin 2 θ − Ω 2 a sin 2 θδr. (A.99)Ω 2 a ≪ Gm p /a 2 () (A.99) δr ≈ constant + Ω2 a 4sin 2 θ.(A.100)2Gm p(oblateness)(flattening) f = r equatorial − r pole. (A.101)r equatorialf ≈ q/2 q = Ω2 a 3Gm p(A.102)

A. 51q → 1 20(A.102) Ω max ≈( Gmpa 3 ) 1/2≈ 2(G〈ρ〉) 1/2 , (A.103) 〈ρ〉 〈ρ〉 = 5.52 g cm −3 Ω max ≈ 1.2 ×10 −3 rad s −1 P min = 1.4 h P min ≈ 2.9 h 9.9 h ( ) ( 2 (a = b) 1 (c) 3 )21V gravity (r, θ) = − Gm [∞∑( ) R nPn1 − J n (cos θ)], (A.104)rn=2r m R(= a a ) J n A.3 P n (cos θ) n n = 1 J n J 2 3 A, B, CMacCullagh’s 22 (Cook 1980)J 2 = C − 1 (A + B)2≈ C − Ama 2 ma , (A.105)2A ≈ B J n J n = + 1 ∫ R ∫ +1r n PmR n n (µ)ρ(r, µ)2πr 2 dµdr,(A.106)0 −120 (A.103) a(G〈ρ〉) 1/2 21 (A.104) P 2 (cos θ) 22 I = ∑ δmR 2 sin 2 θV = − Gm sr−G(A + B + C − 3I)2r 3

A. 52 µ = cos θρ(r, µ) P n (µ) n J 3 = J 5 = J 7 = · · · = 0 J 3 q J n ∝ q n/21 q ≪ 1 J n J 2 = q/2 (A.96) V cf = 1 3 Ω2 r 2 [P 2 (µ) − 1].(A.107) µ V cf J 2 V total (r, θ) = − Gm [p Gmp a 2+ Jr r 3 2 + 1 ]3 Ω2 r 2 P 2 (µ), (A.108) J 4 J 6 r = a + δr(θ) (A.108) δr = constant −[J 2 + 1 ]3 q RP 2 (µ).(A.109)δr (A.101) f f = 3 2 J 2 + 1 2 q,(A.110)f ≈ q/2 Yoder(1995) f f calc = 0.003349f obs = 0.003353 f calc = 0.06670f obs = 0.06487 (A.110)2 A.4 (i) (ii) A.7 , .

A. 53 J 2 J 2 J 2 (A.105) J 2 2 C − A C A (C − A)/C (Cook 1980)(luni-solar precession)C A A.6 Darwin-Radau RelationDarwin-Radau relation( Cook 1980) C/mR 2 (m R ) qfJ 2 Clairaut(1743) Radau(1885) Darwin(1899) Cook(1980) CmR = 2 [1 − 2 ( ) ]5 q 1/2 2 3 5 2 f − 1 . (A.112)¯C ¯C =CmR 2(A.113)(A.110)J 2 qf (A.112)Darwin-Radau relationJ 2f = − 310 + 5 2 ¯C − 158 ¯C 2 . (A.114)

A. 54Darwin-Radau relation ¯C J 2 /f A.4 Dermott(1979b) ¯C = 2 5[ ( σ〈ρ〉 + 1 −〈ρ〉)( σ ) ] A 2. (A.115)B(A.79)23 H (A.83) 24 H h Dermott(1979b)J 2 /f H h J 2f = 2 (1 − 2 ). (A.116)3 5H hδ γ 25 A/B J 2 /f H h J 2f = 2 3 + ¯C − 2 5 (A/B)21 − (A/B) 2 + 8 − 20(A/B) 2 + 10 ¯C[5(A/B) 3 − 2]12[(A/B) 5 − 1] + 15 ¯C[2 − 5(A/B) 3 + 3(A/B) 5 ] .(A.117) 2 • A/B → 0 J 2 /f → ¯C.(A.118)• Darwin-Radau relation A/B → 1 J 2 /f → − 310 + 5 2 ¯C − 158 ¯C 2 . (A.119)A.9 Darwin-Radau A/B = 0.5 ¯C J 2 /f 23 (A.79) AS 2 = F (5/2)ζ c1+˜µ BT 2 = H 5 2 ζ()H = 2〈ρ〉 1 + ˜µ + (3/2)(A/B) 2 F δ5ρ (1 + ˜µ)(δ + 2σ/5ρ)24 (A.83)()H h = 2〈ρ〉 1 + (3δ/5γ)(A/B) 25ρ δ + 2σ/5ρ − (9δσ/25γρ)(A/B) 225 (A.82) δ = ( )A 3 ( )B 1 −σρ (A.84) γ =25 + 3σ5ρ

A. 55A.9: ¯C J 2 /f (i) ( ), (ii)A/B =0.5 ( )(iii) Darwin-Radau relation ( )J 2 /fJ 2 /f ¯C J 2 /f ¯C J 2 /f J 2 /f (A.105) 26 C − A C A 26 (A.105)J 2 = C − 1 2(A + B)ma 2 ≈ C − Ama 2

A. 56A.7 n (r, θ, ψ) V rotational = 1 3 Ω2 r 2 P 2 (cos θ)(A.120)((A.107) 27 ) θ P 2 (cos θ) =(1/4)(3 cos 2θ + 1) 2 V rotational x − y ψ z V tidal = − Gm pa 3 r2 P 2 (cos ψ) (A.121)((A.9) 28 ) m p V tidal z θ x Gm3 p= an 2 a 2n = Ω V tidal = −Ω 2 r 2 P 2 (cos ψ)(A.122) V rotational V tidal 3 A.4 A.10 (A.10a) z (A.10b) x () (A.57) () 29 −1/3 27 (A.107) V cf = 1 3 Ω2 r 2 [P 2 (µ) − 1]28 (A.9) V 3 (ψ) = −G msaRpP 2 3 2 (cos ψ)29 (A.57) R os (ψ) = B[1 + T 2 P 2 (cos ψ)]

A. 57A.10: (a) z (b) x a b c( xyz ) θ ψ B T 2 a b θ = π/2 P 2 (cos θ) c θ = 0 a r = B(1 + T 2 /6) , b r = B(1 + T 2 /6) , c r = B(1 − T 2 /3) . (A.123)a ψ = 0 P 2 (cos ψ) b c ψ = π/2 a t = B(1 + T 2 ) , b t = B(1 − T 2 /2) , c t = B(1 − T 2 /2) . (A.124)a = B(1 + 7T 2 /6) , b = B(1 − T 2 /3) , c = B(1 − 5T 2 /6) . (A.125)

A. 58A.11: Anderson et al.(1996a,1996b,1997a,1997b)b − c = 1 (a − c)4 (A.126)(Dermott1979b)(A.13) 30 (A.102) 31 , ζ/B = q 32 33 (A.79) BT 2 a − c = 2BT 2 = 5H h qB.(A.127)a − cBq 〈ρ〉 (A.83) H h 34 A/B(A )σρ() Dermott(1979b) A.5 A.6 (a) (b) 30 (A.13) ζ = ms( ) 3Rp Rpm p a31 (A.102) q = Ω2 a 3Gm p32 ζ/B = 3q/4 (A.127) 15 4 H hqB 33 (A.79) AS 2 = F (5/2)ζc1+˜µ BT 2 = H 5 2 ζ34 (A.83)()H h = 2〈ρ〉 1 + (3δ/5γ)(A/B) 25ρ δ + 2σ/5ρ − (9δσ/25γρ)(A/B) 2

A. 59Dermott Thomas(1988) 2 (a second-order version of the shapetechnique) (A.126) 0.25 (b − c)(a − c) = 0.27 ± 0.04 Dermott ThomasB = 198.8 km Kozai(1957) 〈ρ〉 = 1.137 ± 0.018 g cm −3 20.3 ± 0.3 km a − c = 16.9 ± 0.7 km A/B = 0.44 ± 0.09 σ = 0.96 ± 0.08 g cm −3 35 () 1000 km Anderson et al(1996a,1996b,1997b) ( ¯C = 0.378 ± 0.007) ( ¯C = 0.347 ± 0.014) ( ¯C = 0.311 ± 0.003) ( ¯C = 0.406 ± 0.030) ¯C = 0.4 ¯C (Anderson et al. 1997b)(Anderson et al. 1998)A.11 A.11 ¯C J 2 (Rappaportet al. 1997)35 ρ = 0.96 ± 0.08 g cm −3 σ

. 60 EPnetFaN 4