流体力学的散逸の基礎理論について - 地球惑星科学科 - 北海道大学

The basic theory ofHydrodynamic Escape 22060169 UMEMOTO Takafumi Department of Earth Sciences, Undergraduate school of Science,Hokkaido UniversityPlanetary and Space Group 2010 3 11

26 256.1 . . . . . . . . . . . . . . . . . . 266.2 . . . . . . . . . . . . . . . . . . . 276.3 Γ . . . . . . . . . . . . . . . . . . . . . . . 286.4 . . . . . . . . . . . . . . 297 31 A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 33A.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33A.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 B Γ 80B.1 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80B.2 H 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83B.3 H 2 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 87umemoto˙B.tex2010/03/11()

1 31 1.1 . , , 460 . . , , . 400 , . .. . , ., , .1.2 , 2 . 3 , . 4 . , . . 5 . 6 4 , . 7 . Henny J.G.L.M.Lamer and Joseph P.Cassinelli, 1999, p.60-111 Γ .umemoto˙B.tex2010/03/11()

2 5, , . , . 5 .2.2 . , . . , .2.2.1 , . , . , .2.2.2 , . . , .2.2.3 , , . , . ..umemoto˙B.tex2010/03/11()

2 62.2.4 , . , .2.3 Impact ErosionImpact Erosion . , . , 6500 , 10 1 ., , . , . , . , . Impact Erosion ,, . , , .umemoto˙B.tex2010/03/11()

3 73 3.1 , . λ .λ ≡ GMmrkT(r) = GMµrRT(r) . (3.1)G , k , m , R ., , . , , .3.2 , . , , .. . .( ) Γ−1 ρT = T 0 (3.2)ρ 0p = p 0( ρρ 0) Γ(3.3), Γ . Γ r . 1, .umemoto˙B.tex2010/03/11()

3 8, . ., p , ρ .(3.4) (3.3) , , 1 dpρ dr + GM ∗r 2 = 0 (3.4)p = ρRTµ(3.5), .{ Γ − 1(p = p 0Γ λ r0) } Γ0r − 1 Γ−1+ 1(3.6), 0 ..(3.2), (3.3), (3.6) , { Γ − 1(T = T 0Γ λ r0) }0r − 1 + 1{ Γ − 1ρ = ρ 0Γ λ 0{ Γ − 1λ =Γ( r0r − 1 )+ 1+ r ( 1− Γ − 1r 0 λ 0 Γ} 1Γ−1(3.7)(3.8))} −1 (3.9), r p > 0 . (4.22) p > 0 , .λ 0

4 94 , . , , . , . , . , . , , . , () .. , r 0 , . , , , r . , .4.1 , . , , , . . 3.2 ., .umemoto˙B.tex2010/03/11()

4 104.1.1 , ..4πr 2 ρv = constant . (4.1)v dvdr + 1 dpρ dr + GM ∗r 2 = 0 . (4.2), M ∗ , v, ρ, T, p , , , , . R, µ , . (4.1) , r . (4.2) , (), () . , T(r) = T = constant . (4.2) (3.5) .1 dpρ dr = R dTµ dr + RT dρµρ dr = RT 1 dρµ ρ dr . (4.3)(4.3) (4.1) .1 dρρ dr = −1 dvv dr − 2 r . (4.4)(4.2), (4.3), (4.4) .{1 dv 2a2v dr = − GM }∗r r 2 /{v 2 − a 2 }. (4.5) (4.5) r .r c ≡ GM ∗2a 2 . (4.6)r c , r c .. r c > r 0 . c (4.5) 1 ,, r dv/dr > 0 . umemoto˙B.tex2010/03/11()

4 11, , .(4.5) v(r) = a , . r = r c , v = a , (4.4) 0 . v = a , r = r c , (4.5) ±∞ . , dv/dr > 0 (4.5) = 0, = 0 r = r c v(r c ) = a = v esc(r c )2(4.7). . v esc (r c ) = √ 2GM ∗ /r c . , r = r c v(r c ) = a . (4.5) . (4.5) , .v 2 2 − a2 ln(v) = 2a 2 ln(r) + GM ∗r+ constant (4.8) 1 . , (4.7) . , . , v(r c ) = a , r < r c , r > r c . v = a , .umemoto˙B.tex2010/03/11()

4 1243v(r)/a2100 1 2 3 4 5r/r crit 1 .r c = 5r 0 . , .umemoto˙B.tex2010/03/11()

4 134.1.2 , ,e(r) = v2 2 − GM ∗r+ γ RTγ − 1 µ(4.9). , .e(r) = e(r 0 ) + v2 − v 2 02+ GM ∗r 0(1 − r 0r). (4.10), 2 , , v 2 < a 2 < v 2 esc a2 ≪ v 2 esc , v 2 > a 2 v 2 > GM ∗ /r v ≫ a , e(r 0 ) ≃ −GM ∗ /r e(r → ∞) ≃ v 2 /2 . . , , .4.1.3 , .(4.8) (r c , v c ) , .) (vexp(− v2 rc) 2{2a 2 = a exp − 2r cr r + 3 }2r = r 0 , v 0 ≪ a ,(rcv 0 ≃ a) 2expr 0(vesc (r 0 )= a2a{− 2r c}+ 3 r 0 2) 4exp{− v2 esc(r 0 )2a 2 + 3 2}(4.11)(4.12) v 0 r → ∞ (4.8) v(r → ∞) ≃2a √ ln(r/r 0 ) , . .umemoto˙B.tex2010/03/11()

4 14(4.11) (4.12) , (4.1) r 2 vρ = r 2 0 v 0ρ 0 v , .⎧ ⎛ρ ⎪⎨expρ 0 ⎪⎩ +1 v 0 ρ 0 r022⎜⎝aρr⎞⎟⎠⎫ 2 {⎪⎬⎪⎭ 2 = exp − GM (∗ 1a 2 − 1 )}r 0 r(4.13), . (3.4) (3.5) .r 2 ρdρdr = −GM ∗a 2 (4.14)a = (RT/µ) 1/2 . ,ρ(r)ρ 0{= exp − GM (∗ 1a 2 − 1 )}r 0 r(4.15)v 0 ≪ a , (4.15) (4.13) . . . (4.2) vdv/dr (1/ρ)dp/dr . , , .4.1.4 , , Ṁ . (4.13) r = r c ,ρ c = ρ 0 exp{− GM (∗ 1a 2 − 1 )− 1 }r 0 r 2.Ṁ = 4πr 2 caρ(r c )= 4πρ 0 ar 2 0 exp {− GM ∗= 4πρ 0 ar 2 0{vesc (r 0 )2a}( 1a 2 − 1 )− 1 r 0 r c 2} 4exp{− v2 esc(r 0 )2a 2 + 3 2(4.16)}. (4.17)umemoto˙B.tex2010/03/11()

4 15, v 0 ≪ a . .4.1.5 r c = GM/2a 2 > r 0 , .λ 0 > 2 . (4.18), . , (4.18) .(3.11), (4.18) , (4.18) .4.2 . , . r , r , .4.2.1 , (3.5), (4.1), (4.2) (3.2), (3.3) .(4.2) , (3.3) d ln p = Γlnρ , (3.5), (4.1) 1 dpρ dr = p d ln p= ΓRT {− 1 dvρ dr µ v dr − 2 }r(4.19), (4.2) .v dvdr + ΓRT {− 1 dvµ v dr − 2 }+ GM ∗r r 2 = 0 . (4.20)umemoto˙B.tex2010/03/11()

4 16{1 dv 2Γa2v dr = − GM }∗r r 2 / { v 2 − Γa 2} . (4.21) r c , v(r c ) (4.21) =0, =0 .r c =GM ∗2Γa 2 (r c ) . (4.22)v 2 (r c ) = Γa 2 (r c ) = GM ∗= v2 esc(r c ). (4.23)2r c 4, a √ Γa . , a r .4.2.2 (4.20) p ∝ (p/ρ) Γ/Γ−1 .1 dpρ dr = p d ln p= Γ p d ln(p/ρ)= Γ d(p/ρ) . (4.24)ρ dr Γ − 1 ρ dr Γ − 1 dr Γ 1 , . .v dvdr + Γ ( )d p+ GM ∗Γ − 1 dr ρ r 2 = d { ( ) v2dr 2 + Γ RTΓ − 1 µ − GM }∗= 0 . (4.25)r, e Γ .( )e Γ ≡ v2 2 + Γ RTΓ − 1 µ − GM ∗r. (4.26)e Γ . , . (), (), () ., . , v ∞ . (4.1) ρ(r → ∞) = 0umemoto˙B.tex2010/03/11()

4 17. T ∝ ρ Γ−1 T(r → ∞) = 0 . , , e Γ r → ∞ .e Γ = e(r → ∞) = v2 ∞2(4.27).e Γ (4.26) r c , (4.22), (4.24) .( ) ( ) ( ) 5 − 3Γe Γ = · Γa2 c 5 − 3ΓΓ − 1 2 = · v2 c 5 − 3ΓΓ − 1 2 = GM∗. (4.28)Γ − 1 4r c e Γ > 0 , Γ 1 < Γ < 5/3 .. (4.9) e Γ .e(r) = v2 2 − GM ∗r+ γ RTγ − 1 µ = e γ − Γ RT(r)Γ −. (4.29)(γ − 1)(Γ − 1) µ.∆e = e(∞) − e(r 0 ) =γ − Γ RT(r 0 ). (4.30)(γ − 1)(Γ − 1) µΓ 1 (4.31) , .dedr = − γ − Γ R dT(γ − 1)(Γ − 1) µ dr . (4.31)Γ = γ , . . , e(r 0 ) , . 1 < Γ ≤ γ , , . Γ = 1 , . , . (4.31) , Γ , .umemoto˙B.tex2010/03/11()

4 184.2.3 (4.20) (4.26) , . (4.26) a 2 = a 2 c(T/T c ) = a 2 c(ρ/ρ c ) Γ−1 , r v .e Γ = v2 2 + (4.24) .Γ ( ) 2−2Γ ( ) 1−Γ r vΓ − 1 a2 c− v2 esc(r c ) r cr c v c 2 r . (4.32)x = r/r c , w = v/v c (4.33)v 2 c2 w2 + 1Γ − 1 v2 c x 2−2Γ w 1−Γ − 2v2 cx = e Γ . (4.34) (4.28) , x w .w Γ+1 − w Γ−1 ( 4x + 5 − 3ΓΓ − 1w ≪ 1 , .w ≃)+ 2Γ − 1 x2−2Γ = 0 . (4.35){ ( 2Γ − 1 x3−2Γ / 4 + 5 − 3Γ )} 1/(Γ−1)Γ − 1 x . (4.36). , x → ∞ , Γ > 1 w → √ (5 − 3Γ)/(Γ − 1) , ,( ) ( ) 5 − 3Γ 5 − 3Γv 2 ∞ = v 2 GM∗c =(4.37)Γ − 1 Γ − 1 2r c. 2 Γ = 7/6,4/3,3/2 (4.35) , . M(r) = v(r)/ √ Γa(r) , umemoto˙B.tex2010/03/11()

4 19. T(r) (3.3) r u .T(r)T(r c ) = ( vv c) 1−Γ ( rr c) 2−2Γ. (4.38)1 < Γ < 5/3 . Γ , , . , . v(r) 1 < Γ < 3/2 , 3/2 < Γ < 5/3 . Γ = 3/2 . . de/dr , , . T r −1 , T ∝ ρ Γ−1 ρ ∝ r 1/(1−Γ) . v = , ρ ∝ r −2 . ρ 1/(1 − Γ) = −2, Γ = 3/2 ., 1 < Γ < 5/3 . 3/2 < Γ < 5/3 , . Γ = 3/2 , . .1 < Γ < 3/2 (4.39)4.2.4 ρ 0 , T 0 , v 0 Ṁ . (4.38) (4.33) . (4.40) (4.24) ,ΓRT 0µ(4.35) (4.41) x 0 , w 0 .T c = T 0 w Γ−10x 2Γ−20(4.40)· w Γ−10x 2Γ−20= GM ∗2r 0· x 0 (4.41)umemoto˙B.tex2010/03/11()

4 20M=v/c sv/v crit4.03.02.01.03.0 0.02.01.0Γ =7/6Γ =4/3Γ =3/2log T/T crit1.0 0.00.50.0-0.5-1.0-1.0 -0.5 0.0 0.5 1.0log r/r crit 2 Γ = 7/6, 4/3, 3/2 x = r/r c . , Γ = 3/2 km s −1 , 3/2 < Γ < 5/3() .umemoto˙B.tex2010/03/11()

4 21(4.24) v c = √ ΓRT c /µ , (4.40) v 0 .v 0 = w 0 v c√ΓRT c= w 0µ√ΓRT 0= · w (Γ+1)/2µ0x0 Γ−1 . (4.42), Ṁ = 4πr 2 0 ρ 0√ΓRT 0· w (Γ+1)/2µ0x0 Γ−1(4.43)., , w 0 ≪ 1 (4.36) w 0 x 0 .{ }8ΓRT0 /µx 0 ≃v 2 − 4Γ + 4 /(5 − 3Γ) . (4.44)esc(r 0 ).r c =(5 − 3Γ)r 0{4 − 4Γ + 8ΓRT0 /µv 2 esc(r 0 ) } . (4.45) 0 < x 0 < 1 , (4.44) .Γ − 12Γ < RT 0µv 2 esc(r 0 ) < Γ + 18Γ . (4.46) w 0 (4.41) .{ v2} 1/(Γ−1)w 0 = esc (r 0 )· x (3−2Γ)/(Γ−1)4ΓRT 0 /µ0. (4.47)(4.44), (4.47) (4.43) , .√ {Ṁ ≃ 4πr0 2 ρ ΓRT 0 v2} (Γ+1)/2(Γ−1) [{ }esc (r 0 )(5−3Γ)/2(Γ−1) 8ΓRT0 /µ0µ 4ΓRT 0 /µv 2 − 4Γ + 4 /{5 − 3Γ}](4.48) .esc(r 0 ) γ , Γ .umemoto˙B.tex2010/03/11()

4 224.2.5 , (4.39) . r c = GM/2Γa 2 (r) > r 0 , , T c /T 0 < 1 λ 0 > 2Γ (4.49). (4.49) (3.11) 2Γ < λ 0

5 235 , . , , . ., .5.1 . , . r e .σ∫ ∞r en(r)dr = 1 . (5.1) n r , σ . , 1 .5.2 , ., , ,p(r e ) = mgσn(r e ) =mgσkT(r e )(5.2)(5.3)umemoto˙B.tex2010/03/11()

5 24.. v v + dv f (v) .(f (v)dv = 4πm2πkT(r e )) 3/2 )exp(− mv2 v 2 dv . (5.4)2kT(r e )r = r e . , v v + dv , *1 , .(14 n(r e) f (v)vdv = πn(r e )m2πkT(r e )) 3/2 )exp(− mv2 v 3 dv . (5.5)2kT(r e ) v e < v < ∞ , F e . F e .F e = 1 √2Gρσ 3 (λ(r e) + 1) √ λ(r e )exp(−λ(r e )) . (5.6), ρ , . .(5.6) , r e .Ṁ = 4πr2 eµσ√2Gρ3 (λ + 1) √ λexp(−λ) (5.7), 1 .*1 1/2. r → r/cosθ , 1/2 . n(r e )/4 .umemoto˙B.tex2010/03/11()

6 256 4 , , , , , , 6 ( v 0 ≪ v c ), H, H 2 H 2 O , T = 300, 1000, 2000, 3000K ρ 0 , H 2 O H, H 2 H 2 O , 5.0 × 10 3 kg/m 3 6.1 µ, 6.2 Γ, 6.3 T 0 (4.17) (4.48) , , 6.4 (5.7) umemoto˙B.tex2010/03/11()

6 266.1 T = 1000K (4.15) µ = 1.0 × 10 −3 , 2 × 10 −3 , 1.8 × 10 −2 kg/mol , 3 , 10 20Escape flux[kg/s]10 1510 1010 5µ=1µ=210 0 µ=181.0×10 23 1.0×10 24 1.0×10 25Mass [kg] 3T = 1000K, Γ = 1 (4.18) () (4.15) v 0 ≪ v c , , 6.2, 6.3, 6.4 , (4.15), (4.17) , 3 umemoto˙B.tex2010/03/11()

6 276.2 µ = 2.0 × 10 −3 kg/mol, Γ = 1() T = 300, 1000, 2000, 3000K , 4 , 10 20Escape flux[kg/s]10 1510 1010 5 T 0 =300KT 0 =1000KT 0 =2000KT10 0 0 =3000K1.0×10 23 1.0×10 24 1.0×10 25 1.0×10 26Mass [kg] 4µ = 2g/mol, Γ = 1 , , , (4.18) , 4 umemoto˙B.tex2010/03/11()

6 286.3 Γ µ = 2.0 × 10 −3 kg/mol, T = 1000K Γ = 1, 1.05, 7/6, 4/3, 7/5 , Γ 5 , 10 20 1.0×10 24 1.0×10 25escape flux[kg/s]10 1510 1010 510 0Γ=1Γ=1.05Γ=7/6Γ=4/3Γ=7/5mass [kg] 5 Γ µ = 2g/mol, T = 1000K Γ , . , , Γ 5 (3.11) , Γ umemoto˙B.tex2010/03/11()

6 296.4 µ = 2, T = 1000K , Γ = 1, 1.05, 7/6, 4/3, 7/5 , 6 , 10 10 1.0×10 24 1.0×10 25 1.0×10 26Escape time [myr]10 510 010 -510 -10Γ=1Γ=1.05Γ=7/6Γ=4/3Γ=7/5jeansMass [kg] 6 . µ = 2g/mol, T =1000K ., 10 −10 , , , , Γ umemoto˙B.tex2010/03/11()

6 30, umemoto˙B.tex2010/03/11()

7 317 , . , , . , . , . . , . , , . 1 < Γ < 3/2 ., , . Γ , . Γ , .umemoto˙B.tex2010/03/11()

7 32, ., ., ., ..umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 33 AHenny J.G.L.M.Lamer and Joseph P.Cassinelli(1999) p.60-111 A.1 , .. , ., .. , , 1 . , . , .A.1 . A.2, A.3, A.4 . r −2 vdv/dr , . A.5 .A.1.1, . , . . 1 , . .umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 34A.1.1.1, r . .Ṁ = 4πr 2 ρ(r)v(r) = 4πF m(A.1) F m . f = m × a f = ρdv/dt . f , f /ρ . , r t . ,dv(r,t)dt= ∂v(r,t)∂t+ ∂v(r,t)∂rdr(t)dt= v(r) dvdr(A.2), , , ∂v(r,t)/∂t = 0 . , .v dvdr + 1 dpρ dr + GM ∗r 2 = 0 (A.3) *2 . . 1 , ( 2 ) ( 3 ) . , T(r) = T = . . . .p = RρT/µ(A.4) R , µ m H *3 . µ , µ = 0.602 . *2 M ∗ *3 m H umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 35, .1 dpρ dr = R dTµ dr + RT dρµρ dr = RT 1 dρµ ρ dr(A.5) ( A.1) , .1 dρρ dr = −1 dvv dr − 2 r(A.6)( A.6) ( A.5) ( A.3) ,. {− 1 dvv dr − 2 rv dvdr + RTµ{1 dv 2a2v dr = − GM }∗r r 2 /{v 2 − a 2 }a =( RTµ}+ GM ∗r 2 = 0 (A.7)(A.8)) 1/2(A.9), . ( A.8) r = r 0 , v(r 0 ) = v 0 . , r 0. ( A.8) v(r) = a . , . . ( A.8) . r −1 r −2 , r = r c ≡ GM ∗ /2a 2(A.10). r .GM ∗ /2a 2 > r 0 GM ∗ /2r 0 ≡ v2 esc4 > a2 (A.11), r c > r 0 . r c < r 0 , , . , . v(r c ) = a, . , v = a r = r c , = 0 ±∞ . , umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 36. r c = GM ∗2a 2 v(r c) = a(A.12). ,v(r c ) = a = v esc(r c )2(A.13). v esc = √ 2GM ∗ /r c . v(r) = a. v(r) = v esc (r) . , 1958 Eugene Parker Parker Point . , .. .( A.8) v(r 0 ) (A.1) . r < r c ( A.8) , v(r) < a v(r) > a . r > r c ( A.8) , v(r) < a , v(r) > a . 1 , , . 2 , . 3 v 0 . , r c . 4 v 0 . 5 6 . ( A.8) , v = a dv/dr → ∞ . r c , ( A.8) f (r), g(r) , ( A.8) . , .1v.( ) dv={− 2a2drr crc2 + 2GM ∗rc3}{ } −1 2vdv= a2drr crc2{ } −1 vdvdrr c(A.14)( dv=dr)r ±2a3cGM ∗(A.15)umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 3743v(r)/a2100 1 2 3 4 5r/r crit A.7. r c = 5R ∗ . , ., (dv/dr) 2 ., X (??). , . , . v 0 (crit) . , ρ 0 .Ṁ = 4πr 2 0 ρ 0v 0 (crit)(A.16) (ρ 0 , T 0 ) , . , . , . , . . , , . .umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 38A.1.1.2, T(r) = T . e(r) = v2 2 − GM ∗r+ 5 2RTµ(A.17). 1 , 2 , 3 3RT/2µ RT/µ . v 2 /2 = a 2 /2,GM ∗ /r c = 2a 2 , . , . (A.17) .e(r) = e(r 0 ) + v2 − v 2 02− GM ∗r(1 − rr 0)(A.18), 2 3 . . , *4 . v 2 < a 2 < v 2 esc , a2 ≪ v 2 esc , e(r 0 ) ≃ − GM ∗r 0(A.19). v 2 > a 2 v 2 > GM ∗ /r , v ≫ a e(r → ∞) ≃ v2 2(A.20). e(r 0 ) e(∞) , ..*4 umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 39A.1.1.3, . , . ( A.8) ( A.7) 1 2 d(v 2 /2)/dr −a 2 d(lnv)/dr . .v 2 2 − a2 ln(v) = 2a 2 ln(r) + GM ∗r+ constant (A.21) v(r c ) = a . .) (vexp(− v2 rc) 2{2a 2 = a exp − 2r cr r + 3 }(A.22)2, r c = GM ∗ /2a 2 . ( A.22) ( A.21) a 2 , . r 0 ( A.22) . v 0 ≪ a < v esc , .(rcv 0 ≃ a) 2expr 0(vesc (r 0 )= a2a{− 2r c}+ 3 r 0 2) 4exp{− v2 esc(r 0 )2a 2 + 3 2}(A.23). , ( A.23) , . r 0 ρ 0 , . Ṁ . .)v(exp(− v2 r0) 2{ ( GM∗ 1v 0 2a 2 = expr a 2 − 1 )}r 0 rr ≫ r 0 , .(A.24)v(r → ∞) ≃ 2a √ ln(r/r 0 )(A.25). . , umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 40. . , . ( A.1) , .⎧ ⎛ρ ⎪⎨expρ 0 ⎪⎩ +1 v 0 ρ 0 r022⎜⎝aρr⎞⎟⎠⎫ 2 {⎪⎬⎪⎭ 2 = exp − GM (∗ 1a 2 − 1 )}r 0 r(A.26), ρ(r)/ρ 0 . 1 (3.2) . a 2 = 0.05GM ∗ /r 0 . r c = 10r 0 . . .1 dpρ dr + GM ∗r 2 = 0 (A.27) ( A.5) ..ρ(r)ρ 0= expr 2 ρdρdr = −GM ∗a 2{− GM (∗ 1a 2 − 1 )} {= exp − (r − r }0) r 0r 0 rH 0 r(A.28)(A.29),H 0 = RT/µg 0 (g 0 = GM ∗ /r 2 0 )(A.30). ( A.29) . ( A.26) v 0 , ( A.29) . (3.2) . . , . exp(−0.5) . , v 0 ρ 0 r0 2 = aρ(r c)rc 2 .umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 41 A.8 (M = v/a) , r/r 0 r 0 . . . .umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 42A.1.1.4, . R ∗ M ∗ , ρ 0 T c . , ( A.29) v esc = 2a r c . , , .Ṁ = 4πρ c ar 2 c ≈ 4πρ 0 ar 2 0( ) 2 {rcexp − r }c − r 0· r0r 0 H 0 r c(A.31), exp(−0.5) .{Ṁ = 4πρ 0 ar02 vesc (r 0 )2a{= 4πρ 0 ar02 vesc (r 0 )2a} 2exp{− r c − r 0r 0− 1 }H 0 r c 2}} 4exp{− v2 esc(r 0 )2a 2 + 3 2 1 .(A.32) v esc /a . v 2 esc(r 0 )/2a 2 r 0 /H 0 .1 × 10 6 K ρ 0 = 10 −14 gcm −3 1.6 ×10 −14 M ⊙ yr −1 . 2.0 × 10 −14 M ⊙ yr −1 .A.1.1.5, . . v(r c ) = a = v esc (r c )/2 , r 0 1 v 0 . r 0 , Ṁ = 4πρ 0 v 0 r0 2 . , umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 43 1 ρ 0 = 10 −14 g/cm 3 M ∗ (M ⊙ ) R ∗ (R ⊙ ) v esc (km/s) T(K) a(km/s) H 0 (R ∗ ) r c (R ∗ )r c −R ∗H 0Ṁ(M ⊙ /yr)1 1 617.5 1 × 10 5 37.2 7.3 × 10 −3 68.7 9.3 × 10 3 1.2 × 10 −683 × 10 5 64.5 2.2 × 10 −2 22.9 1.0 × 10 3 1.5 × 10 −281 × 10 6 117.7 7.3 × 10 −2 6.9 8.0 × 10 1 1.6 × 10 −143 × 10 6 203.9 2.2 × 10 −1 2.3 5.9 8.2 × 10 −115 × 10 6 263.2 3.6 × 10 −1 1.4 1.1 4.0 × 10 −101 100 61.7 3 × 10 3 6.4 2.2 × 10 −2 22.9 1.0 × 10 3 1.5 × 10 −251 × 10 4 11.8 7.3 × 10 −2 22.9 8.1 × 10 1 1.6 × 10 −113 × 10 4 20.4 2.2 × 10 −1 2.3 5.9 8.2 × 10 −85.10 4 26.3 3.6 × 10 −1 1.4 1.1 4.0 × 10 −710 10 617.5 1 × 10 5 37.2 7.3 × 10 −3 68.7 9.3 × 10 3 1.2 × 10 −663 × 10 5 64.5 2.2 × 10 −2 22.9 1.0 × 10 3 1.5 × 10 −261 × 10 6 117.7 7.3 × 10 −2 6.9 8.0 × 10 1 1.6 × 10 −123 × 10 6 203.9 2.2 × 10 −1 2.3 5.9 8.2 × 10 −95 × 10 6 263.2 3.6 × 10 −1 1.4 1.1 4.0 × 10 −810 1000 61.7 3 × 10 3 6.4 2.2 × 10 −2 22.9 1.0 × 10 3 1.5 × 10 −231M ⊙ /yr = 6.303 25 g/s, µ = 0.601 × 10 4 11.8 7.3 × 10 −2 22.9 8.1 × 10 1 1.6 × 10 −93 × 10 4 20.4 2.2 × 10 −1 2.3 5.9 8.2 × 10 −65 × 10 4 26.3 3.6 × 10 −1 1.4 1.1 4.0 × 10 −5umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 44. , . , . , exp(−0.5) .. , r > r c a .A.1.2r −2 f , . . , .. . T(r) = T , , ., r −2 f . , . F r −2 , g rad = κ F F(r)/c =κ F F(R ∗ )(r/R ∗ ) −2 /c r −2 . , κ F . κ F ., r −2 , . ( A.22) ( A.32) . , . r −2 . umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 45, . , r −2 . . . f = Ar −2 .v dvdr = −1 dpρ dr − GM ∗r 2 + A r 2 (A.33) A , . ( A.1) , ( A.5) , ( A.6) .{1 dv 2a2v dr = − GM ∗r r 2 + A }r 2 /{v 2 − a 2 }(A.34) ( A.8) . , dv/dr . (ρ 0 , T, r 0 ) , v 0 1. A A 0 .A.1.2.1 r −2 f A 0 . *5 . , ( A.34) ( A.8) .M eff = M ∗ − A G = M ∗(1 − Γ)(A.35)*5 umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 46 Γ .Γ = A/GM ∗(A.36)Γ < 1 M eff > 0 , ( A.12) . , v(r c ) = a , (1 − Γ) . 2 ( A.32) . H 0 ( g 0 ), (( A.29) ). r 0 . ρ c /ρ 0 . Ṁ .. Γ r c (Γ) = GM ∗(1 − Γ)2a 2(A.37). Γ , r c r 0 . Γ max .{ } 2Γ max = 1 − 2a2 r 0 2a= 1 −(A.38)GM ∗ v esc (r 0 )2a ≪ v esc (r 0 ) Γ max ≃ 1 . Γ = Γ max , ( A.34) r 0 . Γ > Γ max r 0 v 0 < a , r > r 0 , . , . (dv/dr < 0) . . , , ρ . , . , . . (3.3) 0 < Γ < Γ max Γ = A/GM ∗ . M = v/a , ρ 0 umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 47. a 2 = 0.05GM ∗ /r 0 . Γ . ρ 0 Ṁ v 0 . Γ 0 0.8 , M(x = 1) = v 0 /a . , . Γ v 0 . A.9 M = v/a , f = Γ(GM ∗ /r 2 ) r/r 0 . . Γ , v(r 0 ) Ṁ ∝ v 0 .umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 48A.1.2.2 r −2 f r d , r < r d Γ = 0 r > r d 0 < Γ < 1 r −2 . . r d . , Γ Γ = 0.5 . (3.4) . RT/µ = a 2 =(GM ∗ /r 0 )/2 √ 10 . logr c /R ∗ =0.5 . Γ , r c (Γ) . 5 Ṁ . v c = a , ρ(r 0 ) Ṁ v(r 0 ). Γ = 0 ( A.12) r c (0) . Γ = 0.5 ( A.37) r c (Γ) . .r d > r c (0) , Γ . Γ = 0 . . . 4 . r > r d Γ > 0 ( A.34) , ( A.34) Γ = 0 . Γ , r > r d . Γ 1 .r d < r c (Γ) , r c (Γ) . r c ( A.34) . r 0 < r < r c (Γ) Γ . ( A.34) Γ(r) . Γ r d < r < r c (0) Γ = 0 . r c (Γ) umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 49 ρ , .Γ r c (Γ) r c (0) , Γ(r) . Γ , ( A.34) . ( A.37) .. Γ = 0 r c = r c (0) , r c Γ 0. Γ 0 r c = r c (Γ) , r c (Γ) Γ = 0 . Γ r c (Γ) < r < r c (0) r , . ( A.35) .2a 2r c− GM ∗rc2 {1 − Γ(r c )} = 0 (A.39)r c1 − Γ(r c ) = GM ∗2a 2(A.40) Γ(r) . Γ = 0 . r c (0) v = a , Γ = 0 .A.1.2.3 Γ Γ > 0 , Γ . Γ .0 < Γ < Γ max , , . , Γ , . Ṁ . , , . Ṁ ( A.32) . r −2 v 2 esc → v 2 esc(1 − Γ), r c → r c (1 − Γ) H → H/(1 − Γ) , { }ṀṀ(Γ = 0) = (1 − Γ) · exp Γr 0(A.41)H(Γ = 0)umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 50 A.10 f (r) = Γ(r)GM ∗ /r 2 . Γ , . Γ(r) = 0 . , . Γ(r) > 0 , Ṁ ∝ v(r 0 ) .umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 51. Γ max {Ṁ(Γ max ) 2H(Γ = 0) r 0= expṀ(Γ = 0) r 0 H(Γ = 0) − 3 }(A.42)2, Ṁ(Γ max ) = 4πr 2 0 aρ 0 *6 . Ṁ (3.2) . Ṁ Γ .A.1.2.4 r −2 .1. Γ Γ max ( A.38) . , , Γ > Γ max . . Γ > 1 . , .2. 0 < Γ < Γ max , . , .3. Γ > 0 , .*6 ( A.38) ( A.41) ,,.{ } 2Γ max = 1 − 2a2 r 0 2H(Γ = 0) 2a= 1 − = 1 −GM ∗ r 0v esc (r 0 )Ṁ(Γ max )Ṁ(Γ = 0) = (1 − Γ max) · exp{ }Γmax r 0=H(Γ = 0)Ṁ = 4πρ 0 ar 2 0 exp(−0.5){2H(Γ = 0)expr 0}r 0H(Γ = 0) − 2umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 524. r −2 r c (Γ) < r d < r(0) Γ(r) 0 Γ , ( A.39) .. r 0 r d ( A.29) , *7 .{Ṁ ≃ 4πrd 2 aρ 0 exp − (r }d − r 0 )H(r 0 ) · r0r d(A.43)r 0 ≃ R ∗ ≃ 10 3 R⊙, T wind ≃ T eff ≃ 3 × 10 3 K, M ∗ ≃ 20M⊙, ρ 0 ≃ ρ(τ ≃ 1) ≃ 10 −10 g cm −3 , T 10 3 K r d ≃ 4R ∗ . H 0 = 7 × 10 −3 R ∗ = 5 × 10 11 cm . Ṁ10 −50 M ⊙ /yr . , r d 6 × 10 2 . 8 5 × 10 11 cm 4 × 10 12 cm , 10 −5 M ⊙ /yr, 45 . . 7 .A.1.3v(dv/dr) f , . , . . 8 . 1cm 3 1 F ν (r) · ∆ν/c . , F ν (r) ≃ F ν (R ∗ ) · (R ∗ /r) 2 r *7 ( A.29) ,{ρ d = ρ(r d ) = ρ 0 exp − (r }d − r 0 ) r 0H 0 r d, , ,Ṁ = 4πr 2 d ρ da ≃ 4πr 2 d aρ 0 exp{− (r }d − r 0 )· r0H(r 0 ) r dumemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 53, ∆ν 1cm . , ∆r = 1cm ∆ν = (ν 0 /c)(dv/dr)∆r *8 . . , f rad r −2 (dv/dr) , f rad ρ −1 r −2 (dv/dr)v(dv/dr) . v(dv/dr) .A.1.3.1.v dvdr = −1 dpρ dr − GM ( )∗ dvr 2 + B · vdr(A.44) B . B . ( A.6) , .{1 dv 2a2v dr = − GM }∗r r 2 / { v 2 (1 − B) − a 2} (A.45) ( A.44) , ( A.8) 1 − B . ( A.45) B < 1 v(r) v ′ (r) = v(r)(1 − B) −1/2 , ( A.8) . ( A.8) v v ′*8 ν 0 v ν . ∆ν 1 ν = ν 0c − vc∆ν 1 = ν 0 − ν = ν 0c v ∆r ∆v = ( )dvdr ∆r ∆ν ∆ν = ν 0c( ) dv∆rdrumemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 54, . dv/dr > 0 1 , .r c = GM ∗2a 2 v c ≡ v(r c ) =a(1 − B) 1/2 (A.46) B < 1 . , v(r c ) = v esc (r c )/2 . a (1 − B) −1/2 . . . . dv/dr .A.1.3.2v v(1 − B) 1/2 , , f = Bv(dv/dr) . v ′ 0 = v 0(1 − B) −1/2 ( A.45) . .Ṁ ≃ 4πρ 0ar02 { } 2 vesc (r 0 )√ exp{− v2 esc(r 0 )(1 − B) 2a2a 2 + 3 }2(A.47) v 0 → v 0 (1 − B) −1/2 ( A.16) ( A.23) . vdv/dr (1 − B) −1/2 . v(r 0 ) . (1 − B) −1/2 ( A.24) . (1 − B) −1/2 , , . . B > 1 . , ( A.45) v . r 0 < r < r c , r c , . v 0 , r c . umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 55, 1 . . 8 . , f rad vdv/dr f rad r −2 , . .A.1.3.3vdv/dr , f = Bvdv/dr . , (1− B) −1/2 . (1− B) −1/2 , B = 0 . r c = GM ∗ /2a 2 , a . , a(1 − B) −1/2 . , . d lnv/dr.A.1.4. . f (r,v) + g(r,v) × vdv/dr , r, v dv/dr . , (dv/dr) α , . A.1.4.1.1 dv (v 2 − a 2) − 2a2 + GM ∗v drr r 2 − f (r,v) − g(r,v)v dvdr = 0(A.48)umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 56.{1 dv 2a2v dr = − GM }∗r r 2 + f (r,v) / { v 2 [1 − g(r,v)] − a 2} (A.49)2a 2r c− GM ∗r 2 c+ f (r c ,v c ) = 0 (A.50)v 2 c {1 − g(r c ,v c )} = a 2(A.51). ( A.51) . , . , . v = v c r = r c . (v ≪ a) , (v ≫ a) g(r,v) f (r,v) , .(dv/dr) r0 > 0 (dv/dr) ∞ ≥ 0 , g(r 0 ,v 0 ) > 1 − a2v 2 f (r 0 ,v 0 ) < GM ∗0r02 − 2a2r 0(A.52), r ≫ R ∗ v > a g(∞,v ∞ ) ≤ 1 − a2v 2 ∞≃ 1 f (∞,v ∞ ) ≥ GM ∗r 2− 2a2r= 0 (A.53) ( A.49) . r 0 f , g . 3.2, . f . (A.50) ( A.51) . ( A.49) 2a 2 /r GM ∗ /r 2, . ( A.52) ( A.53) , a 2 /r , . . , a 2 /r .umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 57, 1 . f (r,v) g(r,v) , . , f (r,v) ( A.49) r > r c , g(r,v) r > r c , . 9 .A.1.4.2f (r,v) g(r,v)vdv/dr , r c v(r c ) . , v 0 . r 0 , ρ 0 v 0 , r 0 r c . f g .A.1.5 d lnv/dr , . r c v c r c . r v , . , dv/dr , . r 0 , r 0 r c v(r c ) = v c . r 0 v 0 . r 0 , Ṁ = 4πρ 0 v 0 r0 2 . v c r > r c . .. r > r c , . , ., . v ≤ 0.5a , vdv/dr (1/ρ)dp/dr umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 58. v . , . r 0 , . .A.2 . (a) . . (b) . , . e(r) ., , .e(r) = v2 (r)2 − GM ∗+ γ RTr γ − 1 µ = v2 (r)2 − GM ∗+ 5RTr 2µ(A.54), γ = c p /c v = 5/3, (5/2)RT/µ , (A.54) . . , v(r 0 ) ≪ v esc (r 0 ) RT(r 0 )/µ ≪ v 2 esc(r 0 )/2 ., v 2 (r) > v 2 esc , r . , , ., T(r) = . . , . , , . , , ., , umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 59. () , (A.1.1 ). (A.1.2 ). , .A.2.1 () , () . f = dp/dt . .. . .A.2.1.1 f = f (r) v dvdr + 1 dpρ dr + GM ∗r 2 = f (A.55). f cm s −2 . du = dQ − pdρ −1 ., du , dQ , pdρ −1 . . dudt = dQdt − pdρ−1 dt(A.56)umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 60. u, Q, ρ r , vd/dr . r . du = da 2 /(γ − 1) , .q ≡ dQdr = 1 R dTγ − 1 µ dr + pdρ−1 dr= 1 d(p/ρ)+ p dρ−1γ − 1 dr dr(A.57) q g −1 cm . ( A.57) ( A.55) , ρ −1 dp/dr + pdρ −1 /dr = d(p/ρ)/dr , .(d v2dr2 + γγ − 1RTµ − GM )∗≡ d e(r) = f + q (A.58)r dr. ,e(r) = v2 2 + γ RTγ − 1 µ − GM ∗r∫ r∫ r= e 0 + f (r)dr + q(r)drr 0 r 0= e 0 + W(r) + Q(r) (A.59). , , e(r) r 0 e 0 . Q(r) W(r) .Q(r) =W(r) =∫ r∫ rr 0q(r)dr(A.60)r 0f (r)dr (A.61)Q(r) , r 0 r . r 0∆e = W(∞) + Q(∞)(A.62). , .∆e = v2 ∞2 − v 02 + γ Rγ − 1 µ (T ∞ − T 0 ) + GM ∗r 0(A.63)umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 61v ∞ ≫ v 0 T ∞ ≪ T 0 , ( A.63) 2 .∆e ≃ v2 ∞2 + GM ( )∗ γ RT0−r 0 γ − 1 µ(A.64)v 2 ( )∞2 ≃ γ RT0γ − 1 µ − GM ∗+ ∆e (A.65)r 0r = ∞ r 0 ., (q = 0, f = 0) , . r 0 .A.2.1.2 ( A.55) q(r) . , . , . . . ρ −1 dp/dr p/ρ 1 dpρ dr = d(p/ρ) − p dρ−1dr dr= d(p/ρ)dr+ p ρd lnρdr(A.66). d lnρ = −d lnv − 2d lnr *9 . *9 4πv(r)r 2 ρ(r) = constant ρ(r) = C (C ). , lnρ(r) = lnC −vr2 lnv − 2lnr. , d lnρ = −d lnv − 2d lnr .umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 62 a(r) 2 = p/ρ = RT/µ .1 dpρ dr = da2dr − 2a2 − a 2 d lnvr dr(A.67).v dvdr − a2 dvv dr + da2dr − 2a2 + GM ∗r r 2 = f (A.68){1 dv 2a2v dr = − da2r dr − GM }∗ {vr 2 + f2 − a 2} −1(A.69) da 2 /dr = (R/µ)dT/dr . ( A.58) .da 2dr = R dTµ dr = γ − 1γ{f + q − GM ∗r 2 − v dvdr}(A.70) da 2 /dr ( A.70) , γ . c s = √ γa 2 .{1 dv 2c2v dr = sr− GM }∗{v }r 2 + f − (γ − 1)q2 − c 2 −1s(A.71). q > 0 f (γ − 1)q . , . ( A.69) ( A.71) . v = a , v = c s . ( A.69) . . ( A.70) . vdv/dr v = a v = c s .( A.71) , . , f qumemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 63 dv/dr . dv/dr , ( A.71) , . vdv/dr A.1.3 . v > a , r v = a . v c s . f q r v , dv/dr . v = c s .A.2.1.3 ( A.59) ( A.68) . , . . . ( A.59) ( A.68) , . , c s ( A.68) . . M = v/c s . Holzer and Axford(1970) .M c s M 2 − 1 dM 22M 2 dr= 2 r − M2 − 12c 2 s. c s dc 2 s/dr dc 2 sdr + f − (γ − 1)q − (GM ∗/r 2 )c 2 sc 2 s = γRT {µ = (γ − 1) e(r) + GM }∗− v2r 2. v = Mc s .c 2 s =2(γ − 1){(γ − 1)M 2 e(r) + GM ∗+ 2 r}(A.72)(A.73)(A.74)umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 64,dc 2 sdr = (γ − 1) {f + q − GM ∗r 2 − 1 2dv 2 }drdc 2 {sdr = 2(γ − 1)(γ − 1)M 2 f + g − GM ∗+ 2 r 2 − c2 s2dM 2 }dr(A.75)(A.76). ( A.74) ( A.76) , γ = 5/3 .M 2 − 1 dM 2M 2 dr= 4(3 + M2 )3{ [e(r)/r] + f − (3 + 5M 2 })(q/12)e(r) +GM ∗ /r(A.77) e(r) = e 0 + W(r) + Q(r) . , 2 ( A.77) e(r) +GM ∗ /r .{3(M 2 − 1)/4(3 + M 2 )M 2 }dM 2 /dr ln(M 2 + 3) − (1/4)ln(M 2 ) , γ = 5/3 .{} ⎧ln(M 2 + 3) − ln(M2 ) ⎪⎨−4 ⎪⎩ ln(M2 0 + 3) − ln(M2 0 )⎫⎪⎬4 ⎪⎭ =∫ r{ [e(r)/r] + f − (3 + 5M 2 })(q/12)drr 0e(r) +GM ∗ /r(A.78), M 0 ()., , . r , . ( A.77) M = 1 r c . .e(r c )+ f (r c ) − 2q(r c)= 0 (A.79)r c 3, e(r c )/r c . ( A.77) , , , . (, umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 65) , . , ., , . , (A.77) . dM 2 /dr . ( A.77) f (r) q(r) . , f (r) q(r) M = 1 ( A.79) . , . M(r) e(r) , v 2 = M 2 γRT/µ ( A.73) . γ = 5/3 ,T(r) = 6 5µ{Re 0 + W(r) + Q(r) + GM ∗r}/M 2 (r) + 3 (A.80). M(r) T(r) , ρ 0 , .A.2.1.4 r c M r > r c M > 1 , . . . {d e(r)+ f − (3 + 5M 2 ) q }> 0dr r12r c(A.81). de/dr = f + q ( A.79) e(r c ) = {(2/3)q − f }r c , .2 fr + d fdr + 1 q3 r − 2 dq3 dr − 5 12 qdM2 dr> 0 (A.82), q(r c ) = 0 (dq/dr) rc = 0 , umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 66, .ddr (r2 f ) rc > 0(A.83) r −2 , .. κ , f ∝ κL ∗ /r 2 , . , . ., , ( A.82) .− r1/2 c d{ q}q(r c ) dr r 1/2 = − d {ln(qr −1/2 ) } > 5 { } dM2> 0drr c 8 drr c(A.84) q r 1/2 . , . r c (dM 2 /dr) > 0 . q , . , ., , . . r −2 , , .umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 67A.2.1.5 ( A.77) . . ( ) q , dM 2 /dr . q(r) . ( ) r 0 r e(r) , dM 2 /dr . dv/dr dM/dr .(a) , .(b) dM 2 /dr , dM 2 /dr .(c) dM 2 /dr , dM 2 /dr . (r 0 ) , v(r 0 ) M(r 0 ) . .(a) , dv/dr .(b) , . *10(c) , ., 3.4 . (3.1.3 ) . *10 , .umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 68. , .A.2.2, T(r) ∝ ρ(r) Γ−1 . Γ . , . Γ . Γ = 5/3 , Γ = 1 ( A.18) , , 1 < Γ < 5/3 T(r) . Γ . Γ , Γ ..T = T 0( ρρ 0) Γ−1(A.85)p = RρT ( ) ( ) Γ ρT ρµ = p 0 = p 0 (A.86)ρ 0 T 0 ρ 0 Γ Γ = γ = c p /c v . 5/3 . , p ∝ ρ γ , (Appendix 2 ).A.2.2.1 ( A.1) ( A.55) . ρ −1 dp/dr .1 dpρ dr = p d ln p= ΓRT {− 1 dvρ dr µ v dr − 2 }r(A.87)umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 69 d ln p = Γd lnρ . .v dvdr + ΓRT {− 1 dvµ v dr − 2 }+ GM ∗r r 2 = 0 (A.88){1 dv 2Γa2v dr = − GM }∗r r 2 / { v 2 − Γa 2} (A.89) a r . ( A.89) ( A.8). , r c =GM ∗2Γa 2 (r c )(A.90)v 2 (r c ) = Γa 2 (r c ) = GM ∗= v2 esc(r 0 )2r c 4(A.91). a √ Γa . Γ = γ = c p /c v , √ Γa . Γ γ , . . ( A.8) ( A.89) . a . , . , . .A.2.2.2 ( A.89) p ∝ (p/ρ) Γ/Γ−1 . Γ 1 1 dpρ dr = p d ln p= Γ p d ln(p/ρ)= Γ dρ dr Γ − 1 ρ dr Γ − 1 dr (p/ρ)(A.92)umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 70, .v dvdr +ΓΓ − 1ddr( pρ)+ GM ∗r 2= d dr{ v22 + (ΓΓ − 1) RTµ − GM }∗= 0 (A.93)r Γ 1 .( )e Γ ≡ v2 2 + Γ RTΓ − 1 µ − GM ∗r(A.94) e Γ , , , ΓRT/(Γ − 1)µ . e Γ . e Γ r → ∞ e Γ . , , T(r) . T ∝ ρ Γ−1 ρ(r → ∞) = 0 . r → ∞ .e Γ = e(r → ∞) = v2 ∞2(A.95).e Γ r 0 < r c < ∞ . r c e Γ ( A.94) . ( A.91) . ( ) ( ) ( ) 5 − 3Γe Γ = · Γa2 c 5 − 3ΓΓ − 1 2 = · v2 c 5 − 3ΓΓ − 1 2 = GM∗Γ − 1 4r c(A.96). 1 < Γ < 5/3 . Γ = 5/3 . ( A.96) e Γ . r 0 < r c < ∞ . , .0 < e Γ

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 71. ( A.94) r 0 .v 2 (∞2 = v2 02 +ΓΓ − 1) RTµ − GM ∗r 0(A.98)v ∞ > 0 Γ . , Γ/(Γ − 1) . Γ . v ∞ > 0 . T(r) ∝ ρ(r) Γ−1 Γ . Γ , .e Γ e(r) , .e(r) = v2 2 − GM ∗r+ 5 2RTµ = e Γ − 5 − 3Γ RT(r)2(Γ − 1) µ(A.99)∆e = e(∞) − e(r 0 ) = 5 − 3Γ RT 02(Γ − 1) µ(A.100) Γ 1 e(r) .d edr = − 5 − 3Γ R dT2(Γ − 1) µ dr(A.101)Γ = 5/3 , . p ∝ ρ 5/3 . , e(r 0 ) , , .1 < Γ < 5/3 , , . Γ = 1 , . Γ < 1 , .( A.101) . 1 < Γ < 5/3 Γ , de/dr . , . . ( A.101) . Γ , umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 72. Γ, dT/dr, de/dr . , , ( A.101) . Γ, dT/dr, de/dr . , de/dr . dT/dr Γ ≡ d ln p/d lnρ de/dr .A.2.2.3, ( A.89) , ( A.94). ( A.94) .e Γ = v2 2 +Γ ( ) 2−2Γ ( ) 1−Γ r vΓ − 1 a2 c− v2 esc(r c ) r cr c v c 2 r(A.102), RT/µ = a 2 a 2 = a 2 c(T/T c ) = a 2 c(ρ/ρ c ) Γ−1 . ρ v r . c r c . x = r/r c w = v/v c(A.103), .v 2 c2 w2 + 1Γ − 1 v2 c x 2−2Γ w 1−Γ − 2v2 cx = e Γ (A.104)v c v esc (r c ) a c , ( A.91) . e Γ v c (A.96) , . ( A.104) w Γ−1 , .( 4w Γ+1 − w Γ−1 x + 5 − 3Γ )+ 2Γ − 1 Γ − 1 x2−2Γ = 0(A.105) w x . .umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 73w ≪ 1 , .w ≃{ ( 2Γ − 1 x3−2Γ / 4 + 5 − 3Γ )} 1/(Γ−1)Γ − 1 x(A.106)x → ∞ , w Γ > 1 √ (5 − 3Γ)/(Γ − 1) . ( ) ( ) 5 − 3Γ 5 − 3Γv 2 ∞ = v 2 GM∗c =Γ − 1 Γ − 1 2r c(A.107) ( A.11) Γ = 1 Γ = 5/3 . Γ ( A.105) . Γ = 5/3 , . . . M(r) = v(r)/ √ Γa(r) . . * 11 . T ∝ ρ Γ−1 ∝ v 1−Γ · r 2−2Γ . ,( ) 1−Γ ( ) 2−2ΓT(r) v rT(r c ) = v c r c(A.108). , Γ , . . , . ( A.11) v(r) 1 < Γ < 3/2 , 3/2 < Γ < 5/3 . Γ = 3/2 , . de/dr (5R/2µ)dT/dr , , . Γ = 3/2 , e Γ . v 2 /2 = v 2 ∞/2 = e Γ , {Γ/(Γ − 1)RT/µ} GM ∗ /r . , T r −1 ρ ∝ r 1/(1−Γ) . , T ρ T ∝ ρ Γ−1 . v = , ρ ∝ r −2 . ρ 1/(1 − Γ) = −2, Γ = 3/2 .*11 M(r) =v√ = vΓa(r) v c√ √v c Γa(rc ) T(r c )√ = x · √ = w ·Γa(r) Γa(r) T(r) .umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 74 ( A.11) , Γ , . v(r) , . , r c v c . , , Γ . r 0 v 0 T 0 , Γ , e Γ. Γ 5/3 1 , . v/v c r/r c ( A.11) . v c . r c r 0 [( A.96) ]. r e Γ . v ∞ = √ 2e Γ [( A.95) ]. ( A.12) T(r 0 ) = 4 × 10 6 K, µ = 0.60, v esc (r 0 ) = 600[kms −1 ] Γ = 7/6 , , , . 1g r 0 r 1.5r 0 . Γ < 5/3 . dT/dr , .. .A.2.2.4 r 0 v 0 , r 0 ρ 0 , . , v 0 w 0 = w(x 0 ) . w 0 . (A.91) ΓRT c /µ = GM ∗ /2r c . T ∝ ρ Γ−1 , umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 75M=v/c sv/v crit4.03.02.01.03.0 0.02.01.0Γ =7/6Γ =4/3Γ =3/2log T/T crit1.0 0.00.50.0-0.5-1.0-1.0 -0.5 0.0 0.5 1.0log r/r crit A.11 x = r/r c Γ = 7/6, 4/3, 3/2 , , Γ = 3/2 km s −1 , 3/2 < Γ < 5/3() .umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 76. , T ∝ v 1−Γ r 2−2Γ . .T c = T 0 w Γ−10x 2Γ−20(A.109) x = 1 w = 1 . ( A.91) .ΓRT 0µ· w Γ−10x 2Γ−20= GM ∗2r 0· x 0 (A.110)w Γ−10x 2Γ−30= GM ∗2r 0·µΓRT 0(A.111). w Γ−10· x 2Γ−30. , w x ( A.105) . w 0 x 0 . x 0 . , r c = r 0 /x 0 . ( A.109) , v c = √ ΓRT c /µ . v 0 = w 0 v c √ΓRT 0v 0 = · w (Γ+1)/2µ0x0 Γ−1(A.112), .Ṁ = 4πr 2 0 ρ 0√ΓRT 0· w (Γ+1)/2µ0x0 Γ−1(A.113), v 2 esc(r 0 ) ≫ 4ΓRT 0 /µ , x 0 w 0 w ≪ 1 ( A.106) w(x) . w = w 0 ( A.106) ( A.111) , .{ 8ΓRT0 /µx 0 ≃v 2 esc(r 0 )}− 4Γ + 4 /(5 − 3Γ) (A.114)umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 77.r c ≃(5 − 3Γ)r 0{4 − 4Γ + 8ΓRT0 /µv 2 esc(r 0 ) } (A.115)( A.114) 0 < x 0 < 1 (Γ − 1)/2Γ < RT 0 /µv 2 esc(r 0 ) < (Γ + 1)/8Γ , .w 0 ( A.111) .{ v2} 1/(Γ−1)w 0 = esc (r 0 )x (3−2Γ)/(Γ−1)4ΓRT 0 /µ0(A.116) w 0 x 0 ( A.113) v 2 esc(r 0 ) ≫ 4Γa 2 0 .√ {Ṁ ≃ 4πr0 2 ρ ΓRT 0 v2} (Γ+1)/2(Γ−1) [{ } ]esc (r 0 )(5−3Γ)/2(Γ−1)8ΓRT0 /µ0µ 4ΓRT 0 /µv 2 − 4Γ + 4 /{5 − 3Γ} (A.117)esc(r 0 ) 4.1 . , Γ . R⊙, 600km s −1 , 10 −14 g cm −3 , 4 × 10 6 K . . Γ , , . , . .A.2.2.5, . , . Γ = d ln p/d lnρ . Γ γ = c p /c v = 5/3 , , . Γ 5/3 3/2 , , . , . Γ < 3/2 . Γ < 3/2 (5/2)RT 0 /µ .umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 78 A.12Γ = 7/6, T(r 0 ) = 4 × 10 6 m, v esc (r 0 ) = 600km s −1 . .umemoto˙B.tex2010/03/11()

A Henny J.G.L.M.Lamer and Joseph P.Cassinelli (1999) p.60-111 79 e Γ [( A.94) ]. , ({Γ/(Γ − 1)}RT/µ) . e Γ , v 2 ∞/2 = e Γ .. r 0 , T 0 , ρ 0 , Γ , .umemoto˙B.tex2010/03/11()

B Γ 80 BΓ 4 .B.1 H Escape flux[kg/s]10 2010 1510 1010 5Γ=1Γ=1.05Γ=7/6Γ=4/3Γ=7/510 01.0×10 23 1.0×10 24 1.0×10 25 1.0×10 26 1.0×10 27Mass [kg] B.1T = 300Kumemoto˙B.tex2010/03/11()

B Γ 8110 20Escape flux[kg/s]10 1510 1010 510 01.0×10 23 1.0×10 24 1.0×10 25 1.0×10 26 1.0×10 27Mass [kg] B.2T = 1000K10 20Escape flux[kg/s]10 1510 1010 510 01.0×10 23 1.0×10 24 1.0×10 25 1.0×10 26 1.0×10 27Mass [kg] B.3T = 2000Kumemoto˙B.tex2010/03/11()

B Γ 8210 20Escape flux[kg/s]10 1510 1010 510 01.0×10 23 1.0×10 24 1.0×10 25 1.0×10 26 1.0×10 27Mass [kg] B.4T = 3000Kumemoto˙B.tex2010/03/11()

B Γ 83B.2 H 2 Escape flux[kg/s]10 2010 1510 1010 5Γ=1Γ=1.05Γ=7/6Γ=4/3Γ=7/510 01.0×10 23 1.0×10 24 1.0×10 25 1.0×10 26Mass [kg] B.5T = 300K10 20Escape flux[kg/s]10 1510 1010 510 01.0×10 23 1.0×10 24 1.0×10 25 1.0×10 26Mass [kg] B.6T = 1000Kumemoto˙B.tex2010/03/11()

B Γ 8410 20Escape flux[kg/s]10 1510 1010 510 01.0×10 23 1.0×10 24 1.0×10 25 1.0×10 26Mass [kg] B.7T = 2000K10 20Escape flux[kg/s]10 1510 1010 510 01.0×10 23 1.0×10 24 1.0×10 25 1.0×10 26Mass [kg] B.8T = 3000Kumemoto˙B.tex2010/03/11()

B Γ 85B.3 H 2 O Escape flux[kg/s]10 2010 1510 1010 5Γ=1Γ=1.05Γ=7/6Γ=4/3Γ=7/510 01.0×10 21 1.0×10 22 1.0×10 23 1.0×10 24 1.0×10 25Mass [kg] B.9T = 300K10 20Escape flux[kg/s]10 1510 1010 510 01.0×10 21 1.0×10 22 1.0×10 23 1.0×10 24 1.0×10 25Mass [kg] B.10T = 1000Kumemoto˙B.tex2010/03/11()

B Γ 8610 20Escape flux[kg/s]10 1510 1010 510 01.0×10 21 1.0×10 22 1.0×10 23 1.0×10 24 1.0×10 25Mass [kg] B.11T = 2000K10 20Escape flux[kg/s]10 1510 1010 510 01.0×10 21 1.0×10 22 1.0×10 23 1.0×10 24 1.0×10 25Mass [kg] B.12T = 3000Kumemoto˙B.tex2010/03/11()

87[1] Henny J.G.L.M.Lamer and Joseph P.Cassinelli, 1999: Introduction to stellar wind p.60-113[2] , 1997: 12 p.259-p.265[3] , 2009: 1 , Vol.18,No.4,2009 p.194-p.215[4] D.C.Catling and K.J.Zahnle, 2009: The planetary Air Leak, , SCIENTIFIC AMERICAN May 2009, 8 )[5] Andrew J.Watoson,Thomas M. Donahue, and Jamas C. G.Walker, 1981: The Dynamicsof a Rapidly Escaping Atmosphere: Applications to the Evolution of Earth andVenus, Icarus 48, 150-166[6] , 2009: , , , 2009 3 [7] , 2001: Alfven . .umemoto˙B.tex2010/03/11()