Standard Model for Formation of the Solar System ADACHI Toshitaka Department of Earth Sciences, Undergraduate school of Science, Hokkaido University P
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1 Standard Model for Formation of the Solar System ADACHI Toshitaka Department of Earth Sciences, Undergraduate school of Science, Hokkaido University Planetary Physics Laboratory
2 , Hayashi et al. (1985) Formation of the Solar System. Hayashi et al. (1985),. Hayashi et al. (1985),,.,,. Hayashi et al. (1985),. Hayashi et al. (1985). Hayashi et al. (1985),. Hayashi et al. (1985) VIII. Hayashi et al. (1985) 20,,,.
3 i
4 ii Hill A Hayashi et al. (1985) 30
5 iii I II III IV V a b VI a b c d e f VII a b c
6 iv d VIII a b c IX a b X a b XI
7 ,. 10, 300.,.,,., , T-. 50, 60.,,,.,,,,., 80,., Hayashi et al. (1985)..,., Hayashi et al. (1985),.,,,. 1.2, Hayashi et al. (1985) (2 ).,,,,
8 1 2. 3, Hayashi et al. (1985). 4. A, Hayashi et al. (1985).
9 2 3 2,.,,., Hayashi et al. (1985),. 2.1,, H 2,..,.,.,.,.,,., H K., ( ).,, H 2 He,.,,., 99 H He 1 (O, C, Si, Fe, N, Mg ).,, 0.01M.
10 2 4,.,. H 2 O. 170K, 2.7AU, H 2 O,. 2.7AU,,.,,., 1/10,. 2.2., z.,, z 0., 1µm,.,,..,,. m,,.,.,.,,.,,. km,. cm, km, m. km.
11 ,.,,,.,.,,.,.,..,,,., H 2 O,,., 10M (M ).,. 10M, , , ,,.,,. 2.4,,.,,
12 2 6,.,., 10M.,. 300M, 1M.,.,,,.,..,,,. 2.5,.,.,,.,.,.,.,. 0.2M,,,.,,., ( ), -.
13 3 7 3, Hayashi et al. (1985),. 3.1,..,, ( ).,, , v = v(x, t), ρ = ρ(x, t), p = p(x, t). v, ρ, p. v = 0, ρ = ρ 0, p = p 0, v = v 1 (x, t), ρ = ρ 1 (x, t), p = p 1 (x, t),.,,. ρ t + (ρv) = 0 (3-1) ρ v = p t (3-2) pv = nkt (3-3) V, n, k, T,,,. (3-3) µ m H, p = kt µm H ρ (3-4)
14 3 8. (3-1), (3-2), v = v 0 + v 1, ρ = ρ 0 + ρ 1, p = p 0 + p 1, ρ 1 t + ρ v 1 0 x = 0 (3-5) ρ 0 v 1 t = p 1 x (3-6)., 2. (3-5) (3-6) t, x,, (3-4), 2 ρ 1 t 2 + ρ 2 v 1 0 t x = 0 (3-7) ρ 0 2 v 1 x t = 2 p 1 x 2 (3-8) 2 ρ 1 t 2 = 2 p 1 x 2 2 ρ 1 t 2 = kt µm H 2 ρ 1 x 2 (3-9). (3-9)., c s., 2 ρ t 2 = c s 2 2 ρ x 2 ( ) 1/2 kt c s = (3-10) µm H., T =const., p = c s 2 ρ ,. v = v(x, t), ρ = ρ(x, t), p = p(x, t).,
15 3 9,.,, Poisson 3. ρ t + (ρv) = 0 ρ v t = c s 2 ρ c Φ (3-11) 2 Φ = 4πρG (3-12) Φ, G. v = v 1, ρ = ρ 0 + ρ 1, Φ = Φ 0 + Φ 1, (3-7), ρ 0 v 1 t 2 = c ρ 1 s x Φ x (3-13), 2 v 1 ρ 0 x t = c s 2 2 ρ 1 x 2 ρ 2 Φ 1 0 x 2 (3-14), Poisson 2 Φ 1 x 2 = 4πGρ 1 (3-15). (3-7), (3-14), (3-15), 2 ρ 1 t 2 = c s 2 2 ρ 1 x 2 + 4πGρ 0ρ 1 (3-16). ρ 1 x k, ω,, (3-16), ρ 1 = A exp i(kx ωt) (3-17) ω = c s 2 k 2 4πGρ 0 (3-18). ω 2 < 0 ω, ρ 1 t.,,. ω 2 > 0 ρ 1,. ω = 0 k k J, c s 2 k J 2 = 4πGρ 0 (3-19)
16 3 10, ( ) 1/2 4πGρ0 k J = (3-20) c s 2. k J Jeans. Jeans,. Jeans 2π Jeans λ J. λ J = 2π k J = ( ) 2 1/2 πcs (3-21), Jeans,. Gρ 0,., λ J, 1M ( ). ρ 0 = M /λ J 3 (3-21), λ J = πc s 2 GM (3-22), AU. 3.2,, *1,,.,, H 2 O..,,., H 2 O,.,,. *1 (r, θ, z) rθ
17 ,., ( ). 4πr 2 σ SB T 4 = πr 2 L 4πa 2 (3-23). r ( ), σ SB Stefan-Boltzmann, T, L, a.,, ( ) 1/4 ( ) 1/2 L a T = 280 1AU K (3-24) L., (3-10) T = 280K, ( ) T c s = K. cm s 1 (3-25) z.,, z. z,, z., 2 c s ρ gas ρ gas z = z GM R (3-26). c s, ρ gas, M, R. p = c s 2 ρ. R = (a 2 + z 2 ) 1/2, c s 2 dρ gas ρ gas = GM a 3 z dz (3-27)
18 3 12 ρ gas z = 0 ρ *2 g0, z : 0 z, ρ gas : ρ g0 ρ gas, ( ) c 2 ρgas s ln = GM z 2 a 3 (3-28) 2,., ρ g0 ( ρ gas = ρ g0 exp GM ) 2a 3 c 2 z2 s [ ( ) ] 2 z ρ gas = ρ g0 exp ( 2a 3 ) 2 1/2 c s z 0 = = GM z 0 (3-29) (3-30) ( 2kT a 3 ) 1/2 (3-31) µm H GM, z 0,. a = 1AU, ( a ) 5/4 z 0 = AU (3-32) 1AU., ( 1/20), (A-4) (A-5). σ gas, ρ gas, σ gas =. (3-33) (3-30), σ gas = ρ g0 ρ gas dz (3-33) [ ( ) ] 2 z exp dz (3-34) z 0 = ρ g0 πz0 (3-35) *2 (A-6) ρ gas.
19 3 13.,. ρ g0 = σ ( gas = a ) πz0 1AU (3-36)., ( )., M disk = 36AU 0.35AU 2πaσ gas da (3-37) < a < 36 ( 0.39AU, 30AU). (A-5), M disk g 10 2 M (3-38).,. 3.3,,.,.,.,,,.,, v z., dv z dt = ρ gas ρ mat v th r v z GM a 3 z (3-39)
20 3 14. ρ mat, v th, r z.,. (3-39),. dv z dt v z = ρ mat ρ gas = 0 (3-40) r v th GM a 3 z (3-41) , dm dz = p sπr 2 ρ dust (3-42). m, p s, ρ dust. ρ dust z, ρ dust = σ dust /2z 0, ρ mat m = 4πr 3 ρ mat /3, (3-42), 4πr 2 ρ mat dr = p s πr 2 σ dust dz 2z 0 σ dust dr = p s dz (3-43) 8z 0 ρ mat., r 0 z 0, r 0 r, z 0 z, r = r 0 + p sσ dust 8ρ mat (1 zz0 ) (3-44)., p s. r m = p s σ dust /8ρ mat, z. r = r 0 + (1 zz0 ) r m (3-45)
21 ,. dt = dz/v z. (3-41), dt = ρ gas v th a 3 dz ρ mat r GM z. (3-45), dt = A [ dz 1 r m 1 + r 0 /r m z z 0 (1 + r 0 /r m ) 1 ] z. z : z 0 z, t sed = A ( ) 1 z0 r ln r m 1 + r 0 /r m zr 0. (3-48), (t K Kepler ), A 8 = t m π (3-46) (3-47) A = ρ gasv th a 3 ρ mat GM (3-48) A r m (3-49) r m = p sσ dust (3-50) 8ρ mat a 3 ( ) 2 tk = (3-51) GM 2π ( ) 1/2 8 kt 8 v th = = π µm H π c s (3-52) z 0 = c st K 2π (3-53) 2ρgas t K p s πρ dust = 4t K π 3/2 p s ρ gas rho dust (3-54). ρ gas /ρ dust = ζ (3-49), z t sed = 4 ( ) t K 1 z0 r ln π 3/2 p s ζ 1 + r 0 /r m zr 0., ζ. (3-55)
22 z = 0.,,, Poisson., 2 Φ = 1 r σ dust + 1 t r r (rσ dustv r ) + 1 r v r t + v v r r r = c 2 s σ dust σ dust r v θ t + v r ( r r Φ ) r r θ (σv θ) = 0 (3-56) + v θ 2 r Φ r (3-57) r (rv θ) = 0 (3-58) + 1 r 2 2 Φ θ Φ z 2 = 4πGσ dustδ(z) (3-59) σ dust, r, v r, v θ v r, θ, Φ, δ(z) Dirac. (3-57) 2 z, ρ dust dz = σ dust.,,.,. σ = σ 0, v r = v r0 = 0, v θ = v θ0 = rω K (Ω K Kepler ), Φ = Φ 0, σ = σ 1, v r = v r1, v θ = v θ1, Φ = Φ 1. σ = σ 0 + σ 1, v r = v r0 + v r1, v θ = v θ0 + v θ1, Φ = Φ 0 + Φ 1 (3-56) (3-58), 1 r v r1 t σ 1 t + σ 0 r = 2Ω K v θ1 c s 2 v θ1 ( r Φ 1 r r. r, r (rv r1) = 0 (3-60) σ 1 σ 0 r Φ 1 r (3-61) + v r1ω K = 0 (3-62) t ) Φ 1 z 2 = 4πGσ 1δ(z) (3-63) A exp i(kr ωt) (3-64)
23 3 17. k, ω. (3-60), (3-61), (3-62), iωσ 1 = σ 0 r (1 + ikr)v r1 ikσ 0 v r1 (3-65) iωv r1 = 2Ω K v θ1 c s 2 ikσ 1 + 2πiGσ σ 0 (3-66) iωv θ1 = v r1ω 2 (3-67). (3-65), ( ik/r k 2 ). (3-63), ɛ < z < ɛ, 1 ɛ ( r Φ ) ɛ 1 2 Φ ɛ 1 dz + r ɛ r r ɛ z 2 dz = 4πGσ 1 δ(z) dz (3-68) ɛ. 1, 1 ( r Φ ) 1 r r r z=0, [ 1 z ( r Φ ) ] ɛ [ ] ɛ 1 Φ1 + r r r z=0 z (z) = 4πGσ 1 (3-69) ɛ ɛ 1 2. Φ 1 (z), z, 2 Φ 1 z (ɛ) = 4πGσ 1 (3-70) Φ 1 z (ɛ) = 2πGσ 1 (3-71). z > ɛ Laplace. ( 1 r Φ ) Φ 1 r r r z 2 = 0 (3-72) Φ 1. (3-72), ( ) ik Z r k2 Φ 1 = Z(z) exp i(kr ωt) (3-73) + 2 Z z 2 = 0 (3-74)
24 3 18,. Z 2 Z z 2 = k2 Z (3-75) Z(z) = C[exp(kz) + exp( kz)] (3-76) ( ). Φ 1 0, z > 0 Z = C exp( kz), z < 0 Z = C exp(kz) ( k > 0). z > 0, z, z = ɛ, (3-71), Φ 1 = C exp( kz) exp i(kr ωt) (3-77) Φ 1 z (ɛ) = kc exp( kɛ) exp i(kr ωt) = kφ 1(ɛ) (3-78) Φ 1 (ɛ) = Φ 1 = 2πGσ 1 k (3-79). z < 0. (3-65), (3-66), (3-67), (3-79), ω 2 = Ω 2 K 2πGσ 0 k + c s 2 k 2 (3-80). ω 2 k 2, D,., D 4 = (πgσ 0) 2 (c s Ω K ) 2 (3-81) c s < πgσ 0 Ω K (3-82), ω 2,. (3-80) k, dω 2 dk = 2c2 sk 2πGσ 0 (3-83)
25 3 19 ω 2, 0, k k, k = πgσ 0 c 2 s (3-84) c 2 s = (πgσ 0 ) 2 /Ω 2 K, k = Ω2 K πgσ 0 (3-85), (3-82), k = k. 3.4,., Kepler,., v. m, m, r, r, M = m + m, µ = mm /(m + m ), b.,, v imp.. (3-86), µbv = µ(r + r )v imp (3-86) 1 2 µv2 = 1 2 µv imp 2 GMµ r + r (3-87) v imp = bv r + r (3-88), (3-87), 2 [ b 2 = (r + r ) G(m + ] m ) v 2 (r + r ) (3-89)
26 3 20. σ col, [ σ col = πb 2 = π(r + r ) G(m + ] m ) v 2 (r + r ) (3-90). G(m + m )/v 2 (r + r ) = θ, σ col = π(r + r ) 2 (1 + 2θ) (3-91). θ.,.., f, ( σ col = π(r + r ) 2 f θ ) f (3-92). f 1.8, 2.8., (r, θ) 2. m, 2m = M, m 2 /2m = µ. 2 b, v, 90., [ d 2 r µ dt 2 r µ ( ) ] 2 dθ = GMµ dt ) ( 2 dr dθ dt dt + r d2 θ dt 2 r 2 (3-93) = 0 (3-94). (3-94), ( d r 2 dθ ) = 0 (3-95) dt dt, r 2 dθ dt = const. (3-96)
27 3 21.., r 2 dθ dt = bv (3-97),. (3-94) (3-93), dθ dt = bv r 2 (3-98), d 2 r dt 2 b2 v 2 r 3 = GM r 2 (3-99) d 2 r dt 2 = d dt dr dt = dr dθ dθ dt = bv dr ( ) r 2 dθ dr = b2 v 2 d dt r 2 dθ ( 1 dr r 2 dθ ) (3-100) (3-101), (3-99), ( ) d 1 dr dθ r 2 = 1 dθ r GM b 2 v 2 (3-102). 1/r = u, d dθ [ u 2 d dθ ( )] 1 = u GM u b 2 v 2 (3-103), d 2 ( u dθ 2 = u GM ) b 2 v 2 (3-104), (3-104) w,. d 2 w = w (3-105) dθ2 w = A cos θ + B sin θ (3-106) 1 r GM b 2 = A cos θ + B sin θ (3-107) v2
28 3 22 A B. r θ=0 =,, r θ=3π/2 =,., A = GM b 2 v 2 (3-108) B = GM b 2 v 2 (3-109) 1 r = GM b 2 (sin θ cos θ + 1) (3-110) v2 dr (0) = v, dt dr dt = GM bv v = GM bv (cos θ + sin θ) (3-111) (3-112), 90 b = GM v 2., 90 σ exc, = 2Gm v 2 (3-113) ( ) 2 2Gm σ exc = πb 2 = π (3-114)., 90, 90 n 1/3 Λ,. Λ = v2 2Gm v 2 ( ) 1/3 1 (3-115) v ( ) 2 2Gm σ exc = π ln Λ (3-116) v 2
29 , Kepler. v. Hayashi et al. (1985), v/v K e 2i e, i,, v K Kepler., n, σ col, v, t grow = 1 nσ col v (3-117). t grow,., t grow,, t grow,. a n, n = 1 m σ dust 2 2ai (3-118). m 1 ( ), σ dust, 2 2ai. 2i = v/v K = v/aω K, (3-118), n = σ dustω K 2mv (3-119).,, (3-119) (3-92) (3-117), t grow = ρ matr 3πσ dust t K f 2 (1 + 2θ/f) (3-120). ρ mat, t K, Kepler, m = 4πr 3 ρ mat, Ω K = 2π/t K. r = (3m/4πρ mat ) 1/3, a = 1AU, m = g, (A-26), ( ) ( ) 1/3 m ( a ) 3 t grow = 0.22 f 2 (1 + 2θ/f) g 1AU (3-121)
30 3 24. σ dust (A-4), ρ mat 2g cm 3, 1g cm 3 *3. m = 1M, m = 10M,,,, , , ,., ( , ).,. 3.5,, ,,.,, Mp = 10M,,....,,,. Kepler, ( 10 7 ),,. *3 H 2 O,.
31 Hill, Hill, Hill. Hill,. Hill. Hill,,., - a, Hill r H, M p, Kepler Ω K = GM /a 3,, GM (a r H ) 2 = GM p r 2 H + (a r H ) GM a 3 (3-122).,. Kepler. Taylor, 1,,, GM a 2 ( 1 + 2r ) H = GM p a rh 2 + (a r H ) GM a 3 (3-123) ( ) 1/3 Mp r H = a (3-124) 3M.,., Hill,. M p = 3M a 3 r3 H (3-125)
32 ,. a, 4r H, M gas, M gas = 2πa 4r H σ gas (3-126). σ gas (A-5). M gas = M total (M total ), r H (3-124), ( M total 8πa 2 ) 3/2 σ gas = 1 (3-127) M 3 1/3 M 1000.,. A 13,.,,, ,...,, 1 2 mv2 imp = mc T (3-128). m, v imp, c ( SiO 2 c J/kg K), T., T = v2 imp 2c (3-129)
33 3 27. v imp v esc, M p r p, v imp = 2GM p r p (3-130). M p = 0.1M, v imp m s 1. T (3-128), K. SiO C, m.,,.
34 4 28 4, Hayashi et al. (1985),.,, AU,.,,,.,., H 2 O, H 2 O,.,,., 20 1,.,.,,,.,.,, 10 7,,,,., 10M,,., 10 3 M,.,
35 4 29,.,.,,,.
36 A Hayashi et al. (1985) 30 A Hayashi et al. (1985),., H 2 He,,,. ( ).,. I 1960,. 1960, ( T- ),,,., 1970,, (,, ),,.,., 3. Cameron (1978a), Safronov (1969),. Cameron 2.,., Safronov,. 2,. 2, Safronov,,,,.
37 A Hayashi et al. (1985) 31 1 ( )., (, ).,,,.,,. ( (1981a) ),. I ( )..,.,.,,. 10,,,.,,..,,,.,.,. 2,.., VI,,.,,,,, , T,. 3,,.,,,
38 A Hayashi et al. (1985) 32.,, 10 3,,,. X 1. II.,,
39 A Hayashi et al. (1985) 33. I,,.,.,,,.,.,,,.,,.,.,,,.,,..,.,,,.,,..,,.,,, VI.,, I,., 1960,.
40 A Hayashi et al. (1985) 34 III,.,..,. ( 10K) ( g cm 3 )., ( cm s 1 ). ( kt c s = µm H ) 1/2 = ( T 10K µ(= 2.34) H 2 He. ) 1/2 ( ) 1/ (A-1) µ 1M,, (,, ).. a = GM /2c 2 s = (10K/T ) (AU) (A-2), 10 2 AU ( g cm 3 ),, (T 10K)., H 2 (T 1600K), 7/5 (P ρ 7/5 )., 10 4 K.,, (Bodenheimer[1981] ).,.,,,.,, Miyama et al.(1984) Narita et al.(1984)
41 A Hayashi et al. (1985) 35,.,. Miyama et al.(1984)., 2, α = E th / E grav, β = E rot / E grav,,. β > 0.1, α β, αβ., Case > αβ > Case > αβ 3,.,, (Goldreich and LyndenBell 1965a, b). Case2,. Case1,, J( αβ ) g cm 2 s 1 J = g cm 2 s 1.,,. (αβ << 1). Tscharnuter(1981) 3M, α = 1, β = ( 3M J = g cm 2 s 1 ). 0.1c s z µ. z., M, ρ = g cm 3, T = K, a = AU. Tscharnuter ,,., Narita et al.(1984), ( β = 0). 4 (
42 A Hayashi et al. (1985) 36 )., α 0.2,,. α = 0.4,.,,,., Cameron (Cameron 1978a; DeCampli and Cameron 1979).,, , ( 2M ), L 1551 IRS 5 Kaifu et al.(1984). 0.1pc, (A-2) km s 1., (, ).,,,, ( ).,. Hayashi(1981b),,.,,, (T > 10 3 K, ),.,., ( ),,.,.,
43 A Hayashi et al. (1985) 37.,,. IV, 2., 0.02M, 1M., Cameron (Cameron and Pine 1973; Cameron 1978a; DeCampli and Cameron 1979; chapter by Cameron)..,,.,,.,,,.,,.,,..,,., III,,.., (, Safronov 1969; Kusaka et al. 1970; Weidenschilling 1977b; Wetherill 1980b; Hayashi 1981a, b ). 0.01M,, H 2 He., 0.01M.,,.,,.,
44 A Hayashi et al. (1985) 38.,.,. (Fig.1 ).. Cameron,,., ( V ).,,., a T (K), ( ) 1/4 ( ) 1/2 L a T = 280 (A-3) L 1AU. L, L., T L, L = 1L,, T Hayashi(1981b)., σ dust, ( ).,., ( a ) 3/2 7.1 g cm 2 for 0.35AU < a < 2.7AU 1AU σ dust = ( a ) 3/2 30 g cm 2 for 2.7AU < a < 36AU 1AU (A-4)
45 A Hayashi et al. (1985) 39.,, T < 170K(H 2 O ). (A-3) a > 2.7AU. σ gas ( ). ( σ gas = a ) 3/2 g cm 2 for 0.35AU < a < 36AU (A-5) 1AU σ dust, σ gas M. Hayashi(1981b), z, ρ gas, z 0., ( ρ gas = a ) 11/4 g cm 3 (A-6) 1AU, z 0 = ( 2kT a 3 ) 1/2 ( a ) 5/4 = AU (A-7) µm H GM 1AU. (A-3) (A-7),,,.,,. V,,.,.,, (ter Haar 1950). (Lin and Papaloizou 1980; Lin 1981)., z,.,.,,.
46 A Hayashi et al. (1985) Me, V, E, Ma, J, S, U, N (Hayashi 1981b). a,., z ( Nakagawa et al ). v z = ρ mat ρ gas r v th GM a 3 z (A-8) ρ mat r,, ρ gas, v th ( (8kT/πµm H ) 1/2 )., v z r = 1µm, a = 1AU 10 2 cm s 1.,,.,,.
47 A Hayashi et al. (1985) 41, dm dz = p sπr 2 ρ dust (A-9). m(= 4πρ mat r 3 /3), p s, ρ dust (= σ dust /2z 0 ),,,. r 0 < 1µm, z 0. z 0 (A-7),. p s (A-9), z r, r = r 0 + (1 z/z 0 )r m (A-10) r m = p sσ dust 8ρ mat (A-11). σ dust (A-4). r m r 0, (Safronov 1969). p s = 1,,, r m = 4.4, 3.2, 0.2. ρ mat = 2 g cm 3, 2 1 g cm 3.,., v z r.,,.,. (7), (8), (A-10) dt =dz/v z, z 0 z t sed = 4 ( ) t K 1 z0 r ln π 3/2 p s ζ 1 + r 0 /r m zr 0 (A-12). t K (= 2π(a 3 /GM ) 1/2 ), ζ ( IV a < 2.7AU ζ = 1/240 a > 2.7AU ζ = 1/56 )., z = z 0 /10, p s = 1, r 0 = 1µm,,, t sed = , , yr. t sed t K 10 3, t sed.,
48 A Hayashi et al. (1985) 42.., (A-8), (A-12).,. t sed t K ( ) Nakagawa et al. (1981) Weidenschilling (1980),,,..,,., Nakagawa et al. (1981),, cm, ( 3 ). 3, z. 1 ρ dust (z), z. v z,, ( ), v r
49 A Hayashi et al. (1985) 43., v r v z, v z., z 0.1z 0., (A-11).. Nakagawa et al. (1985),, 1,,,, 18cm, 5cm, 0.5cm.,., Weidenschilling(1980),, 1,, 1 cm. b,. 3.5M /a 3 (Jeans 1929),,. 1AU, 1/10 5. Safronov(1969), Hayashi(1972), Goldreich and Ward(1973),.,,,.,. Ω K [= (GM /a 3 ) 1/2 ], e i(ωt+kr), ω 2 = Ω K 2 2πGσk + c s 2 k 2 (A-13). σ, c s (Toomre 1964; Goldreich and Lynden-Bell 1965a, b). ω 2 < 0,, ω 2 > 0. ω 2 k 2, c s c s = πgσ/ω K,, σ σ = Ω K c s /πg., (A-5) σ gas σ,.,,
50 A Hayashi et al. (1985) 44 Cameron σ > σ (Cameron and Pine 1973; Cameron 1978a).,,.,.,,.,, c 2 s = γ(p gas + p dust )/(ρ gas + ρ dust ). γ 1, p gas, p dust, ρ gas, ρ dust,,., c s. c 2 s γp gas /ρ dust, p gas ρ dust. c s c s, ω2 k = Ω 2 K /πgσ ( 4 ).,,,.., k,, ( ) 2 ( 2π m = πσ dust k = (2π) 2 σdust πa 2 ) 3 M (A-14).,, , , g. (A-14).. M c s = c s, 1AU 1 m. ρ dust AU 0.1 g cm 3, g cm 3., 1,., Hayashi (1977) 2.,.,. Coradini et al.(1981) 2,,., Sekiya (1983), z., 1,, (A-14).
51 A Hayashi et al. (1985) c s c s, k k.,,, (A-14).,., Goldreich and Ward (1973) Weidenschilling (1980),,.,. Weidenschilling (1980), 1AU 1cm,,.,.,. VI V b.,,,..,,
52 A Hayashi et al. (1985) 46 ( Safronov (1969); R. Greenberg et al. (1978b); Wetherill (1980a, b) ).,,,, ( VII c. )..,,.,.,,,,. a ( ), σ col = π(r + r ) 2 (1 + 2θ) (A-15) θ = G(m + m )/v 2 (r + r ) (A-16). r, r, m m, v, θ Safronov (Wetherill (1980b) ),., (A-15),, θ < 1,. Nishida (1983),, 3,.,, σ col v, 2. Nishida,.,.,,. Nakazawa and Hayashi (1985) 2,
53 A Hayashi et al. (1985) 47, R p, N (N = 560).,, ( ρ mat = 3.34 g cm 3 ) 1.8R p, 2.8R p (ρ mat = 1.67 g cm 3 ),., Hill, ( 5).,,., f(r + r )., f = , (A-15), σ col = π(r + r ) 2 f 2 (1 + 2θ/f) (A-17)., σ col, 2θ/f < 1, f 2. (A-17),,. 5. (a): fr P (f R P ),,. (b): fr P,,.
54 A Hayashi et al. (1985) 48 b.,.,.,,.,..,., t exc Chandrasekhar (Chandrasekhar 1942, 48-79)., t exc = 1 nσ exc v (A-18) ( ) 2 2Gm σ exc = π ln Λ (A-19) v 2 Λ = v2 2Gm ( ) 1/3 1 (A-20) n n, v, m. (A-20), Λ 90 2Gm/v 2 n 1/3. (A-19) 90 ln Λ. (A-18) 2.,.,. F D F D = 1 2 C Dπr 2 ρ gas ( u) 2 (A-21)
55 A Hayashi et al. (1985) 49. C D, Re Ma, u. C D = 1., Adachi et al. (1976) Weidenschilling (1977a),, C D ( VI e. )., t dis, t dis = mv F D 2m πr 2 1 ρ gas v (A-22)., u v. t exc = t dis, v, e i, v v K = e = 2i = [( aσdust r 2 ρ gas ) ] 1/5 ( ) 2/5 m ln Λ (A-23) M. a, v K (= (GM/a) 1/2 ) (, Hayashi et al. (1977) Nakagawa (1978) ). e i. σ dust ρ gas (A-4) (A-6). (A-23),.,, e, m 6., m = g e , e.,,,., v, 2θ t exc. 6, (A-17) 2θ/f. 2θ/f 1,, (A-23). t exc (= t dis m 1/15 ) m,,,, 10 4, 10 6, VI c.-f..
56 A Hayashi et al. (1985) 50 6 e 2θ/f..,,, E, J, N. E f = 1.8, J, N 2.8 (Nakagawa et al. 1983). c, VI b.,.,,., (A-4) ( 2 VI e. )., n. n = σ dustω K 2mv (A-24) Ω K (= 2π/t K ) v/ω K ( 2ai) ( )., t grow, (A-17) σ col, (A-23) v., t grow = 1 nσ col v = ρ matr 3πσ dust t K f 2 (1 + 2θ/f) (A-25)
57 A Hayashi et al. (1985) 51. ρ mat (= 3m/4πr 3 ). σ dust (A-4), t grow = ( ) ( ) 1/3 m ( a ) f 2 (1 + 2θ/f) yr (A-26) g 1AU , a < 2.7AU a > 2.7AU. t grow, Safronov, v., 6 2θ/f < 1, t grow ra 3.,, t grow a 3. t K 1/σ dust a 3/2., (A-26), (m=1m ), (m=10 ), ,.,,. d,,., (Nakagawa (1978), Nakagawa et al. (1983)).,, (Adachi et al. (1976) Weidenshilling (1977a))., σ dust,.,. e t exc, D r D r = e 2 a 2 /t exc. D r, t dif, a/10., t dif = (a/10) 2 /D r = t exc /100e 2 (A-27). t dif t grow., g t dif t grow..,, (A-25)., Nakagawa
58 A Hayashi et al. (1985) 52 et al. (1983), ( VI e. ).,. v r ( CD πr 2 v r = m ) ρ gas a(e + i + η)ηv K (A-28). η 1/2, η = 1 ( ) / ( ) dpgas GM ρ ( gas 2 da a 2 = a ) 1/2 (A-29) 1AU (Adachi et al. (1976) Weidenshilling (1977a)). t flow, a/10, t flow = a/10 v r = t dis (1 + η/e) 10η (A-30). v r, t flow t grow.,.,,.,.,. e 2. 1 N, 1.,,., N, ,,. Nakagawa et al. (1983),, a. VI d.,,.
59 A Hayashi et al. (1985) 53, σ(m, a) = (inward flow) + (diffusion) + (coalescence) t (A-31) ( Nakagawa et al. (1983) ). σ(m, a)dm, m m+dm, a. (A-31),,.,, VI a. f., (A-31). ( ),. Fujiwara and Tsukamoto (1980),,.,. Nakagawa et al. (1983) (A-31),,,, 4,,,, g, , , , ( II)., g,., a, (A-26), ( ), a.,.,.,, 2θ/f 1, (A-26) ( 7 ). 8. mσ(m, a) m. Nakagawa et al. (1983), g.,, σ(m, a),, , (A-26) t grow m 1/3 a 3. Safronov (1969) Wetherill (1980a, b) ( ) g
60 A Hayashi et al. (1985) 54 2: (yr) 7, ( )., (A-26) (Nakagawa et al. 1983).,,, 10M ( VII )., , M ( 2).,, g., VI f.,.,,.,,,., 2
61 A Hayashi et al. (1985) 55 8 m/m mσ(m, a). m.. f, (A-25), σ dust, Safronov θ.,,,.,,. θ,. Nakagawa et al. (1983), g, ( VII a. ).,., Takeda et al. (1985), (A-21) C D, (θ >> 1) 1
62 A Hayashi et al. (1985) C D Mach M Gm/rv 2. v, (A-21) u. Gm/rv 2 Safronov. C D Gm/rv 2 Mach., Takeda et al., 2 1/2 1/3.,., VII d., ( ),.,,,, (,, ).
63 A Hayashi et al. (1985) 57 9 Gm/rv 2 M C D , (Takeda et al. (1985)). M, C D Gm/rv 2. mathcalm 1.0, 0.7, 0.5, Gm/rv 2 C D,,., Gm/rv 2,,,.
64 A Hayashi et al. (1985) 58 VII,, g,..,,, ( VII a. ).,,,. Perri and Cameron (1974).. Mizuno et al. (1978) ( ),. Mizuno (1980),.,,,.,.. a,, Hayashi et al. (1979).,.,., L,., L = GM p Ṁ p /R p (A-32) M p R p, Ṁ p. κ( )
65 A Hayashi et al. (1985) 59 κ mol + κ dust. κ mol, κ dust. κ mol Mizuno (1980), κ dust.,.,., VI, Ṁ p = 10M /10 7 yr., M p, R p, κ dust,,. 10, Mizuno (1980) M total M p (M total ). κ dust = 1 cm 2 g 1. M p, Hill, M p M total., M total >> M p,., M p,., Jeans. Mp κ dust. 11, Mp κ dust. :,,. κ dust >> 1 cm 2 g 1, Mp 70M, Perri and Cameron (1974). κ dust 1 cm 2 g 1 (Mizuno (1980); Mizuno and Wetherill (1984))., κ dust, Mp., Mp, 10M. [,.,,., M p /, Mp. ]., Mp,,. M total 2Mp, 20M..,,
66 A Hayashi et al. (1985) M p M total., a = 5.2AU( ), Ṁ p = 10M /10 7, κ dust = 1cm 2 g 1., M p (Mizuno (1980)). 11 κ dust M p
67 A Hayashi et al. (1985) 61 b, Hill., Hill. Hill? Sekiya et al. (1984)., smoothed particle 3 (4000 ).,,,., , 4r H (r H (A-33) Hill ), Hill ( 13 )., 1M /100 yr. HelmholtzKelvin ( ), , HelmholtzKelvin.,. VII c., 10 7.,., M total, Hill r H ( 4r H ) r H = a ( Mtotal 3M ) 1/3 (A-33). a. Hill Hill.,.,,. 4r H, Hill (A-33) r H (M total ) 1/3.
68 A Hayashi et al. (1985) /10., x y,.,, M = 2πa 2(2r H + ea)σ gas (A-34) ( 13 ). a e., σ gas, (A-33) (A-34) M = M total,., (A-5) σ gas,,,,, M, M, M, M.,.,,,., ( VII d. ).
69 A Hayashi et al. (1985) , P, a e. c,, ;. Elmegreen (1978) Horedt (1978), T.,.,. Sekiya et al. (1980a),. III, E b, GM E b = 2r σ gas2πr dr = erg (A-35). T,
70 A Hayashi et al. (1985) 64, L av = Ω 4π (ζ swl sw + ζ uv L uv ) (A-36). ζ L, sw uv. ζ sw ζ uv 0.1 (Sekiya et al. (1981))., Ω, π., t es = E b L av = L ζ sw L sw + ζ uv L uv π Ω yr (A-37). T L sw L uv. T, 10 3., L uv = 10 2 L (Gahm et al. (1979))., , L sw + L uv L,,. L sw + L uv = L,.,, Ezer and Cameron (1965) T., d 4,,,,.,, H 2 He. Slattery (1977) Hubbard and MacFarlane (1980),.,, ( 15M ).,. VII B,.,,.,
71 A Hayashi et al. (1985) ,.,., VI, 10 8,.,,.,, (A-5) σ gas.,., 1/3.5, 100M., 1/3.5..,, T., 2,.,., ( 1M )., 1M., : 1M 0.1M ( VIII A, )., VI D,,.,,. VIII VI, ,.,,. ( Safronov (1969) ).,
72 A Hayashi et al. (1985) 66..,,.,. a, ( ).,.,., T f,. 4πR 2 p σt 4 f. σ Stefan-Boltzmann, R p. (A-32)., T f 4 = L 4πR p 2 σ (A-38).,.,., τ(= κρ dr), dt 4 dτ = 3 L 4σ 4πr 2 (A-39)., T b, T b 2 = Lτ b 4πR p 2 σ = τ bt f 4 (A-40) τ b. τ b 1, T b T f τ b 1/4.. Hayashi et al. (1979) Nakagawa et al. (1985b)..
73 A Hayashi et al. (1985) 67. VII a, Ṁp, M p, κ dust. 3, VI f Ṁp = 1M /10 6, κ dust = 1cm 2 g 1 (Mizuno (1980); Mizuno and Wetherill (1984)). M p. 14, T b M p., (A-38) T f., M p > 0.1M, M p.,,, τ b., M p > 0.1M, T b. ;, ( VIII b ),,., Safronov (1978) Kaula (1979b).,.,,,. ;,.,,,. b,, 0.2M.,., 100km, 10 3 (Sasaki et al. 1983).,,., 0.2M 15 3,,,.
74 A Hayashi et al. (1985) M p, T b (Nakagawa et al. (1985b). T f. Stevenson (1981)., Sasaki and Nakazawa (1985).,.,. 2,., 10 7., ( )., Sasaki and Nakazawa (1985),. 14 T b., 0.2M, *4 ( 16 )..,,.,,, *4,
75 A Hayashi et al. (1985) 69.,.,.,. 1,, FeO FeS,.,, :,,. 15,.,. 3,.,,,.,,, :, (Stevenson 1981).,, ( )..
76 A Hayashi et al. (1985) M, 0.77M, 1.0M ( a, b, c).,. c g, 50atm (Nakazawa et al. 1985b). ;, g ( Ne , Xe ).,. (Brown 1949) (Hamano and Ozima 1978). 2. 1, ( VII c., 10 7 ). 2, H 2 He, Xe Kr. 2 Sekiya et al. (1980a,b), Sekiya et al. (1981).,.
77 A Hayashi et al. (1985) b, p, m, s.,.,,. 17., ρu dh dr = Γ (A-41). Γ, u, H,. Ṁatm = r m H = 1 2 u2 + r m γ P γ 1 ρ GM r Ṁ atm = 4πρur 2 = constant, (A-41) rs / [ 4πr 2 GM Γ dr + 5 3γ GM + γ + 1 ( ) Γr 4(γ 1) r s 4 ρu s ] (A-42) (A-43)., (Sekiya et al. 1980a )., c 2 s = u 2 s = GM + γ 1 2r s 2 ( ) Γr ρu s (A-44)., u 2 m P m/ρ m (GM /r m )., m s ( 17 ).
78 A Hayashi et al. (1985) 72, Ṁatm,., Γ Γ = F uv ζ uv κ uv ρ (A-45). F uv, ζ uv, κ uv,,,., (A-43). r m 1, Ṁatm = 4πr3 mf uv ζ uv GM (A-46). r m = 5R, ζ uv = 0.4( ), F uv = erg cm 2 s 1 ( L uv = L, VII c. ), g 1., g AU Ṁatm (Sekiya et al. 1981). Sekiya et al. (1980a,1981),.,,,,
79 A Hayashi et al. (1985) 73., 1AU 18., Ṁatm F uv.,,, (A-43) GM /r m.,. H 2 O, 0.4 ζ uv., T, 1AU 10 4 erg cm 2 s 1 (Gahm et al. 1979). F uv = erg cm 2 s 1, Ṁatm g yr 1.,., , Hill.,. Sekiya et al. (1980b), Xe Kr. 2.,, H 2,.,. IX Darwin 1879,., ( Ringwood and Kesson 1977; Binder 1977; O Keefe and Sullivan 1978 ),.,. ; Hill,..,.,.,.
80 A Hayashi et al. (1985) 74 a, Hill., Hill.,.,,,.,., Jacobi E 1 2 (ẋ2 ẏ 2 ż 2 ) Ω 2 (x2 y 2 ) GM r 1 GM r 2 const. (A-47). 1 4,,. E. E, E 0.,, E, Hill., : (E < 17GM /a) (E > 17GM /a). a, a 1AU., 19a,, Hill., E ;,, E (Nakazawa et al. 1983).,, (Hayashi et al. 1977; Heppenheimer and Porco 1977).,, ( ).,,,
81 A Hayashi et al. (1985) 75 19b Hill (Nishida 1983)...,,. 19 ( P ). (a) (E = 0.33GM /a), (b) (E = 104GM /a)..,., :, (Mignard 1980,1981), Hill (Ruskol 1972).,,,., Hill.,,.,, Hill., Hill 10 2.,,.
82 A Hayashi et al. (1985) 76.,, Hill., Hill,,,.,, Hill., Nakazawa et al.(1983),, Hill. Nakazawa et al.(1983), t r t, ρ(r, t) = ρ 0 (r) f(t), ( f(t) = exp t t atm ) (A-48). ρ 0 (r), t atm, ( VIII c. )., Hill., E,. E., E ρ(r, t) A. A, A = C DπR 2 M 2M M = cm 2 g 1 (A-49). R M M M, C D ( C D = 1/2 )., Nakazawa et al. (1983) E, E = 2πAGM ρ 0 (r p )f(t)f (A-50). r p, F 0.5.,., E. ρ 0 = g cm 3 ( r = 6R ), (A-50), E = 0.17f(t)GM /a. 0.33GM /a, f(t)GM /a., f(t) > 1/25, Hill.
83 A Hayashi et al. (1985) 77,,., ρ 0 (r) r, r p ( 19a )., Nakazawa et al. (1983) Monte Carlo., f(t) > 1/50, 1/50, Hill,.,.,,, Roche. Nakazawa et al. (1985a),,.,,, f(t) < 1/5 Hill,. (f(t) > 1/50)., 1/5 < f(t) < 1/50, Hill,.,. b, 1 ; 30.,.,.,, ( ).,,., Pollack et al. (1979), Lunine and Stevenson (1982). Hunten (1979)
84 A Hayashi et al. (1985) 78.,,.,.,,.,,,..,.,. VII d.,,..,,,.,,,,.,,., ;,,.,., ( 5 ).,,.,., Roche,,.,., Roche,,,.,, ( 20 ).
85 A Hayashi et al. (1985) 79 X,. 2, σ dust.,. (A-25) 8, g, g, 10 3.,,., VI, VII,.,,,...,,,.,. 20. Roche.,..
86 A Hayashi et al. (1985) 80 a, Nakagawa and Hayashi (1985),. 8 Runge-Kutta,., ( ), dv dt = GM r r 3 GM r r J J r r J 3 (A-51) dv J dt = G(M + M J ) r J r J 3 (A-52). r, r J, v, v J. M J., , 0, , 3/1, 2/1, 3/2,, , 0.3,. ( 21, 22 )., ( 21, 22 )., 3/2, Hill.,,, ( 23 )..,,., (Chapman et al ) g,.,.
87 A Hayashi et al. (1985) e ϖ. 2, 2/1, e (a) (b), (A) (B) ( e cos ϖ, e sin ϖ). ((b) (B)), e, ϖ.,, 2.2AU 3.2AU,.. (A-4) σ dust, 1M, 10 3., g, 10 3.,.,,,., (
88 A Hayashi et al. (1985) e max, a init ( ) e J = 0 e J = 0.048( ). b. ).,, ; e = 0.3, ev K km s 1,,.,,. σ dust,., ;,, 3/2, 10 4,.,,,,.,,,., ,
89 A Hayashi et al. (1985) P, J. S AU ,.,,. b,.,, (McCord and Gaffey 1974; Chapman et al. 1978)., Pribram Lost City,,.,,,.,, (, ),..,
90 A Hayashi et al. (1985) 84,,.,., (A-25) f = 1, θ = 0, t col., t col = 1 nσ col v = ( ρmat r 3πσ dust ) t K (A-53). r g 700km, (A-4), a = 2.7AU σ dust = 2g cm 2, (A-53), ( ) 1 t col = σdust ( r ) 2 g cm 2 700km yr (A-54)., σ dust, km s 1,,,. (A-54),, 10 7., ( 129 I 129 Xe ) ;, t col,,,.,,.,, ( ) 3 : 1.
91 A Hayashi et al. (1985) , 1,. km s 1., ( 0.1km s 1 ) 10km s 1,.,,.,. (, Tsuchiyama and Nagahara [1981] 1 K hr 1 ),. :,,,,.,,,,.,,,,.,., H 2 O,., VI,.,,.,, ( )..,,,,.
92 A Hayashi et al. (1985) 86,, (, ). ( 10 9 ),,. (, ).,.,,, (, ).,,.,.,,,,,. XI,..,,. 24, -,.,. 24,.,.,. ( ),.,.,.,,.,,..
93 A Hayashi et al. (1985) 87,,.,.,, ;,...,,. (, ).,,,., ( T-Tauri ),,.,.,,. ;.,.,.,,.,,,.
94 A Hayashi et al. (1985)
95 A Hayashi et al. (1985) 89,.,,.,,,..,.,..,,,,,,.
96 A Hayashi et al. (1985) 90 1) Adachi, I., Nakazawa, K., Hayashi, C. 1976: The gas drag effect on the elliptical motion of a solid body in the primordial solar nebula. Prog. Theor. Phys., 56, ) Hayashi, C. 1981: Structure of the Solar Nebula, Growth and Decay of Magnetic Fields and Effects of Magnetic and Turbulent Viscosities on the Nebula. Progress of Theoretical Physics Supplement, 70, ) Hayashi, C., Nakagawa, K. and Nakazawa, Y., 1985: Formation of the solar system. In Protostars and Planets II, eds. D. C.Black and M. S. Matthews. Univ. of Arizona Press, ) Nakagawa, Y., Hayashi, C. Nakazawa, K. 1981: Growth and Sedimentation of Dust Grains in the Primordial Solar Nebula. ICARUS 45, ) Nakagawa, Y., Nakazawa, K. Hayashi, C. 1983: Accumulation of Planetesimals in the Solar Nebula, ICARUS 54, ), 2008: 21., 1064pp. 7), 2007:., 204pp. 8),, 1996:., 192pp. 9), 1999:., 173pp. 10) de Pater, I., Lissauer, J. J., 2001: Planetary Sciences. Cambridge University Press, 544pp. 11), 2004: ), 2008:..
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Gmech08.dvi
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13 13.1 O r F R = m d 2 r dt 2 m r m = F = m r M M d2 R dt 2 = m d 2 r dt 2 = F = F (13.1) F O L = r p = m r ṙ dl dt = m ṙ ṙ + m r r = r (m r ) = r F N. (13.2) N N = R F 13.2 O ˆn ω L O r u u = ω r 1 1:
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No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
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II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j
量子力学 問題
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2.6 2.6.1 ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.121) Z ω ω j γ j f j
5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1
4 1 1.1 ( ) 5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1 da n i n da n i n + 3 A ni n n=1 3 n=1
50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () m/s : : a) b) kg/m kg/m k
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chap03.dvi
99 3 (Coriolis) cm m (free surface wave) 3.1 Φ 2.5 (2.25) Φ 100 3 r =(x, y, z) x y z F (x, y, z, t) =0 ( DF ) Dt = t + Φ F =0 onf =0. (3.1) n = F/ F (3.1) F n Φ = Φ n = 1 F F t Vn on F = 0 (3.2) Φ (3.1)
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d M d t = γ M H + α M d M d t M γ [ 1/ ( Oe sec) ] α γ γ = gµ B h g g µ B h / π γ g = γ = 1.76 10 [ 7 1/ ( Oe sec) ] α α = λ γ λ λ λ α γ α α H α = γ H ω ω H α α H K K H K / M 1 1 > 0 α 1 M > 0 γ α γ =
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Formation of hot jupiters by slingshot model Naoya Okazawa Department of Earth Sciences, Undergraduate school of Science, Hokkaido University
Formation of hot jupiters by slingshot model Naoya Okazawa 22070283 Department of Earth Sciences, Undergraduate school of Science, Hokkaido University Planetary and Space Group 2011 4 18 1,.,,.,.,.,..,.,.,,.,.,,,.,.,.,.,,,..
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構造と連続体の力学基礎
II 37 Wabash Avenue Bridge, Illinois 州 Winnipeg にある歩道橋 Esplanade Riel 橋6 6 斜張橋である必要は多分無いと思われる すぐ横に道路用桁橋有り しかも塔基部のレストランは 8 年には営業していなかった 9 9. 9.. () 97 [3] [5] k 9. m w(t) f (t) = f (t) + mg k w(t) Newton
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4 Maxwell- Boltzmann N 1 4.1 T R R 5 R (Heat Reservor) S E R 20 E 4.2 E E R E t = E + E R E R Ω R (E R ) S R (E R ) Ω R (E R ) = exp[s R (E R )/k] E, E E, E E t E E t E exps R (E t E) exp S R (E t E )
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IA 31 4 11 1 1 4 1.1 Planck.............................. 4 1. Bohr.................................... 5 1.3..................................... 6 8.1................................... 8....................................
4.6 (E i = ε, ε + ) T Z F Z = e βε + e β(ε+ ) = e βε (1 + e β ) F = kt log Z = kt log[e βε (1 + e β )] = ε kt ln(1 + e β ) (4.18) F (T ) S = T = k = k
![4.6 (E i = ε, ε + ) T Z F Z = e βε + e β(ε+ ) = e βε (1 + e β ) F = kt log Z = kt log[e βε (1 + e β )] = ε kt ln(1 + e β ) (4.18) F (T ) S = T = k = k](/thumbs/94/119235351.jpg "4.6 (E i = ε, ε + ) T Z F Z = e βε + e β(ε+ ) = e βε (1 + e β ) F = kt log Z = kt log[e βε (1 + e β )] = ε kt ln(1 + e β ) (4.18) F (T ) S = T = k = k")
4.6 (E i = ε, ε + ) T Z F Z = e ε + e (ε+ ) = e ε ( + e ) F = kt log Z = kt loge ε ( + e ) = ε kt ln( + e ) (4.8) F (T ) S = T = k = k ln( + e ) + kt e + e kt 2 + e ln( + e ) + kt (4.20) /kt T 0 = /k (4.20)
5 5.1 E 1, E 2 N 1, N 2 E tot N tot E tot = E 1 + E 2, N tot = N 1 + N 2 S 1 (E 1, N 1 ), S 2 (E 2, N 2 ) E 1, E 2 S tot = S 1 + S 2 2 S 1 E 1 = S 2 E
5 5.1 E 1, E 2 N 1, N 2 E tot N tot E tot = E 1 + E 2, N tot = N 1 + N 2 S 1 (E 1, N 1 ), S 2 (E 2, N 2 ) E 1, E 2 S tot = S 1 + S 2 2 S 1 E 1 = S 2 E 2, S 1 N 1 = S 2 N 2 2 (chemical potential) µ S N
液晶の物理１：連続体理論（弾性，粘性）
The Physics of Liquid Crystals P. G. de Gennes and J. Prost (Oxford University Press, 1993) Liquid crystals are beautiful and mysterious; I am fond of them for both reasons. My hope is that some readers
KENZOU
KENZOU 2008 8 2 3 2 3 2 2 4 2 4............................................... 2 4.2............................... 3 4.2........................................... 4 4.3..............................
2009 2 26 1 3 1.1.................................................. 3 1.2..................................................... 3 1.3...................................................... 3 1.4.....................................................
m(ẍ + γẋ + ω 0 x) = ee (2.118) e iωt P(ω) = χ(ω)e = ex = e2 E(ω) m ω0 2 ω2 iωγ (2.119) Z N ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.120)
2.6 2.6.1 mẍ + γẋ + ω 0 x) = ee 2.118) e iωt Pω) = χω)e = ex = e2 Eω) m ω0 2 ω2 iωγ 2.119) Z N ϵω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j 2.120) Z ω ω j γ j f j f j f j sum j f j = Z 2.120 ω ω j, γ ϵω) ϵ
1 9 v.0.1 c (2016/10/07) Minoru Suzuki T µ 1 (7.108) f(e ) = 1 e β(e µ) 1 E 1 f(e ) (Bose-Einstein distribution function) *1 (8.1) (9.1)
1 9 v..1 c (216/1/7) Minoru Suzuki 1 1 9.1 9.1.1 T µ 1 (7.18) f(e ) = 1 e β(e µ) 1 E 1 f(e ) (Bose-Einstein distribution function) *1 (8.1) (9.1) E E µ = E f(e ) E µ (9.1) µ (9.2) µ 1 e β(e µ) 1 f(e )
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2017 1 2 1.1...................................... 2 1.2......................................... 4 1.3........................................... 10 1.4................................. 14 1.5..........................................
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..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A
![..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A](/thumbs/93/112175219.jpg "..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A")
.. Laplace ). A... i),. ω i i ). {ω,..., ω } Ω,. ii) Ω. Ω. A ) r, A P A) P A) r... ).. Ω {,, 3, 4, 5, 6}. i i 6). A {, 4, 6} P A) P A) 3 6. ).. i, j i, j) ) Ω {i, j) i 6, j 6}., 36. A. A {i, j) i j }.
(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
ii p ϕ x, t = C ϕ xe i ħ E t +C ϕ xe i ħ E t ψ x,t ψ x,t p79 やは時間変化しないことに注意 振動 粒子はだいたい このあたりにいる 粒子はだいたい このあたりにいる p35 D.3 Aψ Cϕdx = aψ ψ C Aϕ dx
i B5 7.8. p89 4. ψ x, tψx, t = ψ R x, t iψ I x, t ψ R x, t + iψ I x, t = ψ R x, t + ψ I x, t p 5.8 π π π F e ix + F e ix + F 3 e 3ix F e ix + F e ix + F 3 e 3ix dx πψ x πψx p39 7. AX = X A [ a b c d x
p = mv p x > h/4π λ = h p m v Ψ 2 Ψ
II p = mv p x > h/4π λ = h p m v Ψ 2 Ψ Ψ Ψ 2 0 x P'(x) m d 2 x = mω 2 x = kx = F(x) dt 2 x = cos(ωt + φ) mω 2 = k ω = m k v = dx = -ωsin(ωt + φ) dt = d 2 x dt 2 0 y v θ P(x,y) θ = ωt + φ ν = ω [Hz] 2π
80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t = 0 i r 0 t(> 0) j r 0 + r < δ(r 0 x i (0))δ(r 0 + r x j (t)) > (4.2) r r 0 G i j (r, t) dr 0
79 4 4.1 4.1.1 x i (t) x j (t) O O r 0 + r r r 0 x i (0) r 0 x i (0) 4.1 L. van. Hove 1954 space-time correlation function V N 4.1 ρ 0 = N/V i t 80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t
x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin
2 2.1 F (t) 2.1.1 mẍ + kx = F (t). m ẍ + ω 2 x = F (t)/m ω = k/m. 1 : (ẋ, x) x = A sin ωt, ẋ = Aω cos ωt 1 2-1 x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ
m d2 x = kx αẋ α > 0 (3.5 dt2 ( de dt = d dt ( 1 2 mẋ kx2 = mẍẋ + kxẋ = (mẍ + kxẋ = αẋẋ = αẋ 2 < 0 (3.6 Joule Joule 1843 Joule ( A B (> A ( 3-2
3 3.1 ( 1 m d2 x(t dt 2 = kx(t k = (3.1 d 2 x dt 2 = ω2 x, ω = x(t = 0, ẋ(0 = v 0 k m (3.2 x = v 0 ω sin ωt (ẋ = v 0 cos ωt (3.3 E = 1 2 mẋ2 + 1 2 kx2 = 1 2 mv2 0 cos 2 ωt + 1 2 k v2 0 ω 2 sin2 ωt = 1
() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y
![() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y](/thumbs/85/92979039.jpg "() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y")
5. [. ] z = f(, y) () z = 3 4 y + y + 3y () z = y (3) z = sin( y) (4) z = cos y (5) z = 4y (6) z = tan y (7) z = log( + y ) (8) z = tan y + + y ( ) () z = 3 8y + y z y = 4 + + 6y () z = y z y = (3) z =
I 1
I 1 1 1.1 1. 3 m = 3 1 7 µm. cm = 1 4 km 3. 1 m = 1 1 5 cm 4. 5 cm 3 = 5 1 15 km 3 5. 1 = 36 6. 1 = 8.64 1 4 7. 1 = 3.15 1 7 1 =3 1 7 1 3 π 1. 1. 1 m + 1 cm = 1.1 m. 1 hr + 64 sec = 1 4 sec 3. 3. 1 5 kg
最新耐震構造解析 ( 第 3 版 ) サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 第 3 版 1 刷発行時のものです.
最新耐震構造解析 ( 第 3 版 ) サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/052093 このサンプルページの内容は, 第 3 版 1 刷発行時のものです. i 3 10 3 2000 2007 26 8 2 SI SI 20 1996 2000 SI 15 3 ii 1 56 6
振動と波動
Report JS0.5 J Simplicity February 4, 2012 1 J Simplicity HOME http://www.jsimplicity.com/ Preface 2 Report 2 Contents I 5 1 6 1.1..................................... 6 1.2 1 1:................ 7 1.3
8 300 mm 2.50 m/s L/s ( ) 1.13 kg/m MPa 240 C 5.00mm 120 kpa ( ) kg/s c p = 1.02kJ/kgK, R = 287J/kgK kPa, 17.0 C 118 C 870m 3 R = 287J
26 1 22 10 1 2 3 4 5 6 30.0 cm 1.59 kg 110kPa, 42.1 C, 18.0m/s 107kPa c p =1.02kJ/kgK 278J/kgK 30.0 C, 250kPa (c p = 1.02kJ/kgK, R = 287J/kgK) 18.0 C m/s 16.9 C 320kPa 270 m/s C c p = 1.02kJ/kgK, R = 292J/kgK
2.1: n = N/V ( ) k F = ( 3π 2 N ) 1/3 = ( 3π 2 n ) 1/3 V (2.5) [ ] a = h2 2m k2 F h2 2ma (1 27 ) (1 8 ) erg, (2.6) /k B 1 11 / K
![2.1: n = N/V ( ) k F = ( 3π 2 N ) 1/3 = ( 3π 2 n ) 1/3 V (2.5) [ ] a = h2 2m k2 F h2 2ma (1 27 ) (1 8 ) erg, (2.6) /k B 1 11 / K](/thumbs/86/94275608.jpg "2.1: n = N/V ( ) k F = ( 3π 2 N ) 1/3 = ( 3π 2 n ) 1/3 V (2.5) [ ] a = h2 2m k2 F h2 2ma (1 27 ) (1 8 ) erg, (2.6) /k B 1 11 / K")
2 2.1? [ ] L 1 ε(p) = 1 ( p 2 2m x + p 2 y + pz) 2 = h2 ( k 2 2m x + ky 2 + kz) 2 n x, n y, n z (2.1) (2.2) p = hk = h 2π L (n x, n y, n z ) (2.3) n k p 1 i (ε i ε i+1 )1 1 g = 2S + 1 2 1/2 g = 2 ( p F
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)