(Stellar pulsation) 1 ( ) (radial pulsation) (radial oscillation) (δ Cep) 15 km/s ( ) Luminosity Teff 4 R2 ( nonradial pulsation nonradial oscillation
|
|
|
- こうた さくいし
- 5 years ago
- Views:
Transcription
1 (Stellar pulsation) 1 ( ) (radial pulsation) (radial oscillation) (δ Cep) 15 km/s ( ) Luminosity Teff 4 R2 ( nonradial pulsation nonradial oscillation) 1
2 (Gautschy 2008) HR 100 Mira, Semi-regular RCB stars (R Coroni Borealis type stars) ( ) α Cygni variables Cepheids RR Lyrae Core-Helium-Burning stage β Cephei stars, SPB (slowly pulsating B) stars, δ Sct, γ Dor, roap (rapidly oscillating Ap) stars sdbv (subdwarf B variables) Helium-Burning stage sdbv (1990 ) GW Vir DBV DAV GW Vir DBV DB 2
3 DAV DA 5 (Helioseismology ) CoRoT Kepler (Solar-like oscillations) Type T eff (K) Period range Amplitude(mag) Population Mira, Semi-regular d 10 I, II RCB stars d? α Cyg stars d 0.1 I, II Cepheids d I, II RR Lyrae d II β Cephei stars hr I SPB stars d I δ Sct hr I γ Dor d 0.1 I roap stars min 0.02 I sdbv (short) s 0.01 II sdbv (long) hr 0.01 II DAV, DBV 10 4, I, II GW Vir stars s 0.1 I, II Sun min I radial pulsations nonradial pulsations nonradial oscillations radial pulsations nonradial pulsations radial pulsations nonradial pulsations stellar pulsations 3
4 2 Radial pulsations ( ) ( Radial pulsations) coupling time-dependent 2.1 Basic equations Radial pulsations Lagrange r M r (M r Lagrange ) t M r r(m r, t) [ r(m r, t)/ t] Mr [ 2 r(m r, t)/ t 2 ] Mr Basic equations ( ) 1 2 r = P GM r (2.1.1) 4πr 2 t 2 M r 4πr 4 M r r M r = 1 4πr 2 ρ (2.1.2) T = L r 3κ (2.1.3) M r (4πr 2 ) 2 4acT 3 L r = ɛ n T S M r t (2.1.4) (2.1.1) (2.1.2) (2.1.3) L r r (2.1.4) (S entropy ) computer ( ) M r r(m r, t) equilibrium value r 0 (M r ) ξ(m r, t) r(m r, t) = r 0 (M r ) + ξ(m r, t) (2.1.5) f(m r, t) equilibrium value f 0 (M r ) f(m r, t) = f 0 (M r ) + f(m r, t) (2.1.6) f f M r fix (Lagrangian variation) ( ) f(t, Mr ) M r = ( f(t, M r) + f 0 (M r )) M r df 0(M r ) dm r = f(t, M r) M r 4
5 (2.1.5) (2.1.6) (2.1.1) (2.1.2) ξ /r 1 f /f ξ = P + 4 GM r ξ 4πr 2 t 2 M r 4πr 4 r ξ = 1 ( 2 ξ M r 4πr 2 ρ r + ρ ) ρ T = T M r M r ( Lr L r L r = ɛ n T S M r t 4 ξ r + κ κ 3 T T ) (2.1.7) ξ 2.2 Adiabatic radial pulsations timescale (thermal timescale) ( ) P P = Γ ρ ln P 1 ρ ; Γ 1 ln ρ S (2.2.8) (2.2.7) ξ = P + 4 ξ 4πr 2 M r r ξ 1 = M r 4πr 2 ρ GM r 4πr 4 ( 2 ξ r 1 Γ 1 P P r 0 M r ) (2.2.9) dm r = 4πr0ρ 2 0 dr 0 M r r 0 ( 0 ) explicit time-dependence ξ(t, r) = e iσt ξ(r); f(t, r) = e iσt f(r) (2.2.10) real parts σ real part imaginary part 5
6 ( σ ) (2.2.9) 1 d P ρ dr dξ dr = σ 2 ξ + 4 ξ GM r r r 2 = 2 ξ r 1 P Γ 1 P (2.2.11) adiabatic radial pulsations σ 2 linear purturbation analyses Boundary conditions ξ = 0 at r = 0 (2.2.12) P 0 (2.2.11) 1 P d( P/P ) + P ) 1 dp (σ ρ dr P ρ dr = 2 r 3 GMr + 4 ξ GM r r 3 d( P/P ) dr = d ln P dr [ )] P (σ P + 2 r 3 ξ + 4 GM r r d ln P/dr = ρgm/p r 2 pressure scale height H p H p P/P (r = R) (2.2.13) P P ) (σ 2 R3 ξ GM + 4 R at r = R (2.2.13) Linear Adiabatic Wave Equation (2.2.11) P ξ [ ] ( 1 d Γ1 P d(r 2 ξ) + σ 2 + 4GM ) r ξ = 0 (2.2.14) ρ dr r 2 dr r 3 adiabatic radial pulsations linear adiabatic wave equation Operator L L r2 ρ ( d Γ1 P dr r 2 6 ) d 4GM r (2.2.15) dr r 3
7 u(r) (2.2.14) u r 2 ξ (2.2.16) Lu = σ 2 u (2.2.17) Sturm-Liouville weight ρ/r 2 Hermitian R 0 u Lu ρ R r dr = ulu ρ 2 r dr 2 σ 2 j k u j u k R 0 u ju k ρ r 2 dr = 1 4π M 0 0 ξ j ξ k dm r = 0 if j k (dm r = 4πρr 2 dr) {u j } complete set linear combination σ 2 > 0 σ r = R ( e iσt ξ(r) ) = ξ(r) cos(σt) (2.2.18) standing waves σ σ 0 (fundamental pulsation) ξ 0 node( ) σ (overtone pulsations) nodes σ 2 < 0 σ σ 2 < Some examples ξ/r =constant fundamental mode σ 2 0 (2.2.11) P/P = 3Γ 1ξ/r (2.2.11) Γ 1 3Γ ( 1 dp ξ ρ dr r = σ GM ) r ξ r 3 dp/dr = ρgm r /r 2 [ σ0 2 (3Γ 1 4) GM ] r ξ = 0 r 3 w M r r 3 = w M R 3 (2.2.19) 7
8 w σ 2 0 (3Γ 1 4) wgm R 3 (2.2.20) Γ 1 < 4/3 σ 0 < 0 dynamical instability Virial total energy Γ 1 < 4/3 total energy σ (2.2.20) Π 0 Π 0 = 2π σ 0 2π/ (3Γ 1 4)wGM/R 3 1 ρ (2.2.21) free-fall time R 3 /GM internal energy E i GM 2 R 2E i C v T M c 2 sm R 3 /GM R c s (2.2.22) c s (2.2.21) Period-mean density pulsation constant Q Π = Q ρ / ρ (2.2.23) Q 0.04 days Q pulsation constant days Q (polytrope) ξ/r ξ/r Fundamental mode ξ/r first overtone mode ξ/r = 0 (node) n = 3 4 polytrope mode σ 0 /(GM/R 3 ) n = 3 n = 4 polytrope σ (polytropic index n ) σ 2 /(GM/R 3 ) Mode n = 3 n = 4 Fundamental st Overtone nd Overtone
9 (stochastic excitation) ( ) self-excitation 2.3 Hermitian σ 2 σ 2 > 0 (2.3.7) iσ Hermitian σ σ = σ r + iσ i e iσt = e σit e iσrt σ timescale (driving) (damping) entropy 9
10 P dv > 0 (2.3.24) (2.3.24) ; i.e., M 0 [ P dv ] dm r > 0 driving damping entropy local thermal timescale P-V enntropy P dv 0 local thermal timescale thermal balance P-V entropy Pulsation period Local thermal timescale (2.3.24) (2.3.24) epsilon-mechanism kappa-mechanism opacity-mechanism opacity Kappa-mechanism block ( F ) < 0 F = ρdl r /dm r T M r = dt ( Lr dm r L r 4 ξ r + κ ) κ 3 T T kappa-mechanism ( κ T + d dr κ ρ Γ ) > 0
11 Γ 3 1 = (d ln T/d ln ρ) ad kappamechanism drivinig opacity peak opacity (radiative damping) Opacity peaks T 10 4 K T K He T K driving zone local thermal timescale local thermal timescale opacity peak He + opacity peak + H He peak HR δ Scuti RR Lyrae (T eff < K) T K β Cephei SPB Epsilon-mechanism equilibrium internal energy equilibrium (2.3.24) driving epsilon-mechanism epsilon-mechanism epsilon-mechanism epsilon-mechanism 10 2 M 11
12 Strange-mode instability: Thermal time M/L L/M 10 4 thermal time P rad P gas strangemode instability Strange modes (β P gas /P 1) T eff Strange mode strange modes Strange modes ; L/M > 10 4 P gas + P rad P rad = at 4 /3 dp gas dr = GM rρ r 2 L 3 rad 4πr = 4acT dt 2 3κρ dr = c dp rad κρ dr ( 1 κl ) rad 4πcGM r ( = gρ 1 ) κ L rad /L M/M L/M > 10 4 dp gas /dr > 0 density inversion strange mode 12
13 Thermal time (2.1.7) entropy perturbation ( S/ t) Mr 0 (ɛ n = 0) L r /L r 0 P rad T 4 P rad /P rad = 4 T/T L r /L r 0 (2.1.7) L L = κ T 4 P rad P rad ρ κ ρ ρ c d P rad κρl dr 0 β 1 P P rad d P dr ρ ( P ρ) P ρ Strange-mode instability Strange-mode instability L/M HR Instability boundary Stochastic excitation: excitation mechanism Stochastic excitation ( ) characteristic timescale radial nonradial pulsations Stochastic excitation ( 1m/s) 5 stochastic excitation CoRoT,Kepler stochastic excitation : opacity peak kappa-mechanism Strange-mode instability luminosity stochastic excitation 13
14 2.4 Nonlinear pulsations (2.4.1) (2.4.4) thermal timescale Thermal timescale dynamical timescale ( L/M > 10 4 L /M ) Luminosity helium star L/M thermal time sine-curve dynamical timescale luminosity ( ) 14
15 Mira Mira ( sine-curve self-consistent ) shock waves (Höfner & Dorfi 1997, A&A, 319, 648) ( ) Shock wave dust formation ( ) (Willson 2000, Ann.Rev.A.Ap, 38, 573) 15
16 (P-L relation) Π - Π <ρ> 1/2 Π = C R 3 /M (2.5.1) R M C Cepheids core helium burning stagen - (mass-luminosity) L = f L (M) (2.5.2) Cepheids HR (Cepheid instability strip) (T eff ) L = f T (T eff ) (2.5.3) L = 4πR 2 σt 4 eff (2.5.3) L = f R (R) (2.5.4) (2.5.2) (2.5.4) (2.5.1) Π L bolometric correction Period-luminosity relation HR 5M 20M He-burning He-burning 16
17 - Turner et al (2011) M V = ± (2.786 ± 0.075) log P M B = ± (2.386 ± 0.098) log P Sandage et al. (2004) line ( ) - - (PLC) - HR Period changes of Cepheids M 1/2 R 3/2 O C ( ) P n+const. (P n ) E(n) E(n) + const. n 0 ( P 0 + P dt ) dφ φ dφ = P 0 + n2 2 P 0P = P 0 + t2 P 2 P 17
18 O C (φ 0 φ 1 dt/dφ = P ) ( ) 1st crossing He- 2nd 3rd crossings core He burning HR 2nd crossing ( ) P < 0 3rd crossing P > 0 1st crossing He burning 18
19 2.6 Period Luminosity relations of Red Variables Microlensing project OGLE LMC SMC Red variables Period - Luminosity (Kmagnitute) relations (Ita et al 2003). F G Fundamental mode First overtone mode Cepheid P L relations C C Mira variables P L relations P L relations semi-regular variables Mira variables Period - Luminosity relation Mira variables Period-luminosity relation Cepheids Mira variables AGB luminosity 19
20 Mira variables Mira variable luminosity Mira variable mass luminosity mass radius Mira variables Period-luminosity relation 3 Nonradial Pulsations 3.1 Basic equations ρ t ( t + u ψ Poisson + (ρu) = 0 (3.1.1) ) u = 1 P ψ (3.1.2) ρ 2 ψ = 4πGρ (3.1.3) Nonradial pulsations nonradial pulsations 20
21 linear perturbations equilibrium Eulerian perturbations ξ(t, r 0 ) r(t, r 0 ) r 0 f f(t, r) f 0 (r) = f(t, r) f 0 (r 0 ) + f 0 (r 0 ) f 0 (r) = f ξ f 0 (3.1.4) f Eulerian perturbation f Lagrangian perturbation Eulerian perturbation equilibrium ξ Lagrangian perturbation (u 0 = 0) u = u = dξ dt = ξ t (3.1.1) (3.1.3) Eulerian perturbation ( ) ρ t + ξ ρ 0 = 0 (3.1.5) t 2 ξ t = 1 P + ρ dp 0 e 2 ρ 0 ρ 2 r 0 dr ψ (3.1.6) 2 ψ = 4πGρ (3.1.7) Eulerian perturbations ψ nonradial pulsations Cowling approximation Radial pulsations ξ(t, r) = e iσt ξ(r); f (t, r) = e iσt f (r) (3.1.8) displacement vector ξ(r) radial component horizontal component ξ(r) = ξ r (r)e r + ξ h ; ξ h = ξ θ e θ + ξ φ e φ (3.1.9) ξ θ ξ φ θ ξ φ components (3.1.5) (3.1.6) ρ + 1 r 2 r (r2 ρξ r ) + ρ r h ξ h = 0 (3.1.10) σ 2 ρξ r = P r gρ (3.1.11) ξ h = 1 σ 2 ρr hp (3.1.12) 21
22 g GM r /r 2 horizontal differential operator h h = e θ θ + e φ sin θ φ h ξ h = 1 sin θ θ (sin θξ θ) + 1 ξ φ sin θ φ (3.1.13) (3.1.12) (3.1.11) radial pulsation nonradial pulsations p-modes (pressure modes) g-modes (gravity modes) (3.1.10) (3.1.12) Lagrangian perturbations (3.1.6) Eulerian perturbations P P + ξ d ln P r dr = Γ 1 ( ρ ρ + ξ r ) d ln ρ dr Brunt-Väisälä frequency N ( 1 N 2 d ln P g d ln ρ ) Γ 1 dr dr ( = g g c 2 s d ln ρ ) dr (3.1.14) ρ ρ = P Γ 1 P + N 2 (3.1.14) c s adiabatic sound speed c s ( ln P/ ln ρ) s = Γ 1 P/ρ (3.1.12) (3.1.10) ξ h g ξ r (3.1.15) ρ dρ r 2 r (r2 ξ r ) + ξ r dr + ρ + 1 σ 2 r 2 2 hp = 0 (3.1.16) 2 h = 1 sin θ ( sin θ ) + 1 θ θ sin 2 θ (3.1.11) (3.1.16) ρ ( 1 (r 2 ξ r ) h r 2 r σ 2 r 2 = g ξ c 2 r 1 s ρ 22 c 2 s 2 φ 2 (3.1.17) ) P (3.1.18)
23 P r = (σ2 N 2 )ρξ r g P c 2 s (3.1.19) ξ r P (r, θ, φ) explicit 2 h Y m l (θ, φ) N lm P m l (cos θ)e imφ 2 hyl m (θ, φ) = l(l + 1)Yl m (θ, φ) (3.1.20) ξ r = ξ nl (r)y m l (θ, φ), P = η nl (r)y m l (θ, φ) (3.1.21) (3.1.18) (3.1.19) 1 d(r 2 ξ nl ) r 2 dr = g c 2 s ξ nl + ( ) l(l + 1)c 2 s ηnl 1 σ 2 r 2 c 2 sρ (3.1.22) dη nl dr = (σ2 N 2 )ρξ nl g η nl c 2 s (3.1.23) (3.1.21) (3.1.12) displacement (ξ θ, ξ φ ) (ξ θ, ξ φ ) = η ( ) nl Y m l σ 2 ρr θ, 1 Y sin θ φ l ( ) m ( l) l m l = m sectoral modes Radial pulsations (3.1.22) (3.1.23) Hermitian J.P. Cox 1980, 23
24 Theory of Stellar Pulsation adiabatic nonradial pulsations σ 2 m explicit σ 2 m frequency 2l + 1 (3.1.22) (3.1.23) ξ r 2 ξ nl exp ( r 0 g c 2 s ) dr ; η η ( r ) nl ρ exp N 2 0 g dr d ξ dr = h(r)r2 c 2 s [ r ( N 2 h(r) exp 0 g g c 2 s Lamb frequency L l (3.1.24) ( ) L 2 l σ 1 η (3.1.25) 2 d η dr = 1 h(r)r 2 (σ2 N 2 ) ξ (3.1.26) ) ] dr L 2 l l(l + 1)c2 s r 2 (3.1.27) 3.2 Local analysis (3.2.25) (3.2.26) local analysis ξ, η exp(ik r r) (3.2.28) ξ η factor k 2 r = 1 c 2 sσ 2 (σ2 N 2 )(σ 2 L 2 l) (3.2.29) k r radial wave number ξ exp[i(σt + k r r)]y m l k 2 r > 0 radial propagate local propagate σ 2 > N 2 and σ 2 > L 2 l = p modes σ 2 < N 2 and σ 2 < L 2 l = g modes p modes high-frequency oscillations radial pulsations (l = 0) g modes low-frequency oscillations 24
25 wave number k h total wavenumber k k 2 h = l(l + 1) r 2 = L 2 l/c 2 s (3.2.30) (3.2.29) k 2 = k 2 r + k 2 h σ 4 σ 2 (N 2 + k 2 c 2 s) + N 2 k 2 hc 2 s = 0 ( ) k h = 0 radial pulations (l = 0) ( ) σ 2 = c 2 sk 2 r + N 2 g modes σ 2 = 0 k h L 2 l N 2 p modes σ 2 > L 2 l (3.2.29) σ 2 = c 2 s(kr 2 + kh 2 ) p modes wave number frequency g modes σ 2 < N 2 L 2 l (3.2.29) σ 2 N 2 k 2 h/(k 2 r + k 2 h) (3.2.30) g mode oscillations frequency k h k r Brunt Väisälä frequency N k r frequency σ k h 25
26 3.3 Global oscillations r2 r 1 k r dr = π(n + α) (3.3.31) n α r 1 r 2 propagation zone (kr 2 > 0) frequencies frequencies Nonradial pulsations p modes g modes l radial pulsations l units k r radial fix l g modes p modes frequency l fix radial p modes frequencies g modes p modes σ 2 > N 2, L 2 l k r real ( ) nodes g modes σ 2 < N 2, L 2 l k r real g modes nodes 26
27 3.3.1 High-order p modes High-order p-modes (3.3.31) ν nl = σ nl 2π (Tassoul 1980) ν ν = ( R 2 0 (n + l α ) ν (3.3.32) ) 1 dr GM (3.3.33) c s R 3 c 2 s = Γ 1 p/ρ p 3 ρ dm GMr r = dm r r ν global parameters M R (3.3.32) l high-order p modes radial order n ν n 0 ν n 1 2, ν n 1 ν n 1 3 small separation ν δν nl ν nl ν n 1 l+2 (4l + 6) 4π 2 ν nl R 0 dc s dr dr r (3.3.34) ν small separation δν nl 27
28 Aerts et al. (2010) (a,b) l 3 (l = 0 l = 2 l = 1 l = 3 ) l large separation ν ν = 135µHz Echellediagram ( ν ν ) l (3.3.32) 28
29 p- mode large separation ν small separation δ 02 ( X c ) small separation δ 02 ν δ 02 (X c ) High-order g modes High radial order g-modes k r N l(l + 1)/(rσ) (3.3.31) Π = 2π σ 2π2 n l(l + 1) ( r2 r 1 σ l(l + 1) nπ r2 r 1 ) 1 N r dr Π N dr (3.3.35) r 2π 2 l(l + 1) ( r2 r 1 ) 1 N r dr (3.3.36) g-modes ( ) 29
30 ( 1.5M ) Brunt-Väisälä frequency N X c g-modes ( X c ) X c Brunt-Väisälä frequency N (X c ) Kepler γ Dor KIC ( KIC ) 3.4 Solar-like Oscillations (l 3) 30
31 power-spectra amplitude (ν max ) (large separation ν) ν = ( ± 0.04)µHz MR 3 (or ν max = (3120 ± 5)µHz MR 2 (T eff /5777) 1/2 (or ) ν M/M = ν (R/R ) 3 ν max ν max = M/M (R/R ) 2 T eff /T eff T eff ν ν max Kepler (subgiants) R luminosity, L = 4πR 2 σt 4 eff HR plot timescale large separation 3.5 CoRoT Kepler p-mode g-mode (mixed modes) ) 31
32 Brunt-Väisäla frequencyn ( ) radial modes p g-mode gmodes p-mode p-modes g-modes mixed modes high order (n 1) p-modes g-modes ν p (n + l2 ) l(l + 1) + ɛ ν l(l + 1)D, ν g N n ν p-modes large-separation ( 0.5 c s /R; soundtravel time ) N Brunt-Väisälä frequency (g-modes P g N ) ν p ν g N ν g mixed modes ν p ( ) mode inertia I I = 4π R 0 [ξ 2 r + l(l + 1)ξ 2 h]ρr 2 dr (σ 2 I/2 ) ν g inertia ν p ν g ν p inertia 32
33 ( ) Stochastic excitation inertia I ν p ν g Kepler power spectra ν p ν g l ν p large separation ν M/R 3 8µHz HR ν p ν g g-mode P g 1/ N 53s 96s Kepler ν P g P g H-burning shell He Red giant branch P g He-flash core-he Red 33
34 Clump He- P g red giant branch HR ( ) 4 (Rotational splittings) σ Ω ( ) ( ) O(Ω 2 ) linearized momentum equation (σ 0 + σ 1 ) 2 ξ 2m(σ 0 + σ 1 )Ωξ + 2i(σ 0 + σ 1 )Ω ξ = 1 p + ρ dp 0 e ρ 0 ρ 2 r 0 dr ψ (4.0.37) σ 0 σ 1 ( ) ( ) ( ) σ 2 0ξ σ 1 σ 1 ξ = mωξ + iω ξ (4.0.38) ξ σ 1 ξ ξρdv = m Ωξ ξρdv + i ξ (Ω ξ)ρdv (4.0.39) ξ (Ω ξ) = Ω sin θξ rξ φ Ω cos θξ θξ φ + Ω cos θξ φξ θ Ω sin θξ φξ r = 2imΩξ r (r)ξ h (r)yl m Yl m imωξ h (r) 2 cos θ sin θ ( Y m l θ Y l m + Y m l Y m l ) θ (4.0.40) m Ω m Ω r (4.0.39) σ 1 [ξr 2 + l(l + 1)ξh]ρr 2 2 dr = m Ω(r)[ξr 2 + l(l + 1)ξh 2 2ξ r ξ h ξh]ρr 2 2 dr (4.0.41) K(r) K(r) [ξ2 r + l(l + 1)ξh 2 2ξ rξ h ξ h 2]ρr2 [ξ 2 r + l(l + 1)ξh 2]ρr2 dr (4.0.42) 34
35 σ 1 σ 1 = m Ω(r)K(r)dr (4.0.43) σ m ( ) m m l (n, l) m 2l + 1 rotational splitting K(r) (n, l) rotational splittings p-modes g-modes p-modes rotational splittings g-modes splittings l = 1 (dipole) g-modes rotational splittings ( ) (KIC ) (triplets) p-mode g-mode rotational kernel K(r) g-mode p-mode weight (uniform rotation) C nl (2ξr ξ h + ξ 2 h ) ρr2 dr [ξ 2 r + l(l + 1)ξ 2 h ]ρr2 dr (4.0.44) σ 1 = mω (1 C nl ) (4.0.45) C nl Ledoux mωc nl co-rotating High-order g-modes ξ h ξ r C nl 1/[l(l + 1)] high-order p-modes ξ h ξ r C nl red-giants Kepler CoRoT l = 1 rotational splittings core envelope 35
36 KIC dipole (l = 1) rotational splitting large separation ν splittings ( ) rotational splittings splittings maximum dipole modes (l = 1) rotational splitting = (1 C)Ω/(2π) C ( dipole high-order g modes C 0.5) KIC (Goupil et al 2013) (Core-He burning stage) He-burning 36
37 (Aerts 2015) ( log g > 4 log g ) (core) (envelope) 5 20 ( ) (r 2 Ω) = 4π (ρr 4 ΩU r ) + t M P 5 M P M P [ (4πρr 3 ) 2 D Ω ] M P 37
38 1µHz 38
総研大恒星進化概要.dvi
The Structure and Evolution of Stars I. Basic Equations. M r r =4πr2 ρ () P r = GM rρ. r 2 (2) r: M r : P and ρ: G: M r Lagrange r = M r 4πr 2 rho ( ) P = GM r M r 4πr. 4 (2 ) s(ρ, P ) s(ρ, P ) r L r T
1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
Contents 1 Jeans (
Contents 1 Jeans 2 1.1....................................... 2 1.2................................. 2 1.3............................... 3 2 3 2.1 ( )................................ 4 2.2 WKB........................
vol5-honma (LSR: Local Standard of Rest) 2.1 LSR R 0 LSR Θ 0 (Galactic Constant) 1985 (IAU: International Astronomical Union) R 0 =8.5
2.2 1 2.2 2.2.1 (LSR: Local Standard of Rest) 2.1 LSR R 0 LSR Θ 0 (Galactic Constant) 1985 (IAU: International Astronomical Union) R 0 =8.5 kpc, Θ 0 = 220 km s 1. (2.1) R 0 7kpc 8kpc Θ 0 180 km s 1 270
Mｏｔｔ散乱によるParity対称性の破れを検証
Mott Parity P2 Mott target Mott Parity Parity Γ = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 t P P ),,, ( 3 2 1 0 1 γ γ γ γ γ γ ν ν µ µ = = Γ 1 : : : Γ P P P P x x P ν ν µ µ vector axial vector ν ν µ µ γ γ Γ ν γ
II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re
![II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re](/thumbs/94/118770263.jpg "II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re")
II 29 7 29-7-27 ( ) (7/31) II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I Euler Navier
磁性物理学 - 遷移金属化合物磁性のスピンゆらぎ理論
email: takahash@sci.u-hyogo.ac.jp May 14, 2009 Outline 1. 2. 3. 4. 5. 6. 2 / 262 Today s Lecture: Mode-mode Coupling Theory 100 / 262 Part I Effects of Non-linear Mode-Mode Coupling Effects of Non-linear
1 ( ) Einstein, Robertson-Walker metric, R µν R 2 g µν + Λg µν = 8πG c 4 T µν, (1) ( ds 2 = c 2 dt 2 + a(t) 2 dr 2 ) + 1 Kr 2 r2 dω 2, (2) (ȧ ) 2 H 2
1 ( ) Einstein, Robertson-Walker metric, R µν R 2 g µν + Λg µν = 8πG c 4 T µν, (1) ( ds 2 = c 2 dt 2 + a(t) 2 dr 2 ) + 1 Kr 2 r2 dω 2, (2) (ȧ ) 2 H 2 = 8πG a 3c 2 ρ Kc2 a 2 + Λc2 3 (3), ä a = 4πG Λc2 (ρ
.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T
NHK 204 2 0 203 2 24 ( ) 7 00 7 50 203 2 25 ( ) 7 00 7 50 203 2 26 ( ) 7 00 7 50 203 2 27 ( ) 7 00 7 50 I. ( ν R n 2 ) m 2 n m, R = e 2 8πε 0 hca B =.09737 0 7 m ( ν = ) λ a B = 4πε 0ħ 2 m e e 2 = 5.2977
4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c
( ) ,
II 2007 4 0. 0 1 0 2 ( ) 0 3 1 2 3 4, - 5 6 7 1 1 1 1 1) 2) 3) 4) ( ) () H 2.79 10 10 He 2.72 10 9 C 1.01 10 7 N 3.13 10 6 O 2.38 10 7 Ne 3.44 10 6 Mg 1.076 10 6 Si 1 10 6 S 5.15 10 5 Ar 1.01 10 5 Fe 9.00
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
4/15 No.
4/15 No. 1 4/15 No. 4/15 No. 3 Particle of mass m moving in a potential V(r) V(r) m i ψ t = m ψ(r,t)+v(r)ψ(r,t) ψ(r,t) = ϕ(r)e iωt ψ(r,t) Wave function steady state m ϕ(r)+v(r)ϕ(r) = εϕ(r) Eigenvalue problem
untitled
MRR Physical Basis( 1.8.4) METEK MRR 1 MRR 1.1 MRR 24GHz FM-CW(frequency module continuous wave) 30 r+ r f+ f 1.2 1 4 MRR 24GHz 1.3 50mW 1 rf- (waveguide) (horn) 60cm ( monostatic radar) (continuous wave)
2007 5 iii 1 1 1.1.................... 1 2 5 2.1 (shear stress) (shear strain)...... 5 2.1.1...................... 6 2.1.2.................... 6 2.2....................... 7 2.2.1........................
Formation process of regular satellites on the circumplanetary disk Hidetaka Okada Department of Earth Sciences, Undergraduate school of Scie
Formation process of regular satellites on the circumplanetary disk Hidetaka Okada 22060172 Department of Earth Sciences, Undergraduate school of Science, Hokkaido University Planetary and Space Group
QMI_09.dvi
25 3 19 Erwin Schrödinger 1925 3.1 3.1.1 3.1.2 σ τ 2 2 ux, t) = ux, t) 3.1) 2 x2 ux, t) σ τ 2 u/ 2 m p E E = p2 3.2) E ν ω E = hν = hω. 3.3) k p k = p h. 3.4) 26 3 hω = E = p2 = h2 k 2 ψkx ωt) ψ 3.5) h
QMI_10.dvi
25 3 19 Erwin Schrödinger 1925 3.1 3.1.1 σ τ x u u x t ux, t) u 3.1 t x P ux, t) Q θ P Q Δx x + Δx Q P ux + Δx, t) Q θ P u+δu x u x σ τ P x) Q x+δx) P Q x 3.1: θ P θ Q P Q equation of motion P τ Q τ σδx
V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H
199 1 1 199 1 1. Vx) m e V cos x π x π Vx) = x < π, x > π V i) x = Vx) V 1 x /)) n n d f dξ ξ d f dξ + n f = H n ξ) ii) H n ξ) = 1) n expξ ) dn dξ n exp ξ )) H n ξ)h m ξ) exp ξ )dξ = π n n!δ n,m x = Vx)
gr09.dvi
.1, θ, ϕ d = A, t dt + B, t dtd + C, t d + D, t dθ +in θdϕ.1.1 t { = f1,t t = f,t { D, t = B, t =.1. t A, tdt e φ,t dt, C, td e λ,t d.1.3,t, t d = e φ,t dt + e λ,t d + dθ +in θdϕ.1.4 { = f1,t t = f,t {
KENZOU Karman) x
KENZO 8 8 31 8 1 3 4 5 6 Karman) 7 3 8 x 8 1 1.1.............................. 3 1............................................. 5 1.3................................... 5 1.4 /.........................
02-量子力学の復習
4/17 No. 1 4/17 No. 2 4/17 No. 3 Particle of mass m moving in a potential V(r) V(r) m i ψ t = 2 2m 2 ψ(r,t)+v(r)ψ(r,t) ψ(r,t) Wave function ψ(r,t) = ϕ(r)e iωt steady state 2 2m 2 ϕ(r)+v(r)ϕ(r) = εϕ(r)
数学の基礎訓練I
I 9 6 13 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 3 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1..............
. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n
003...............................3 Debye................. 3.4................ 3 3 3 3. Larmor Cyclotron... 3 3................ 4 3.3.......... 4 3.3............ 4 3.3...... 4 3.3.3............ 5 3.4.........
I
I 6 4 10 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1..............
http://www.ike-dyn.ritsumei.ac.jp/ hyoo/wave.html 1 1, 5 3 1.1 1..................................... 3 1.2 5.1................................... 4 1.3.......................... 5 1.4 5.2, 5.3....................
基礎数学I
I & II ii ii........... 22................. 25 12............... 28.................. 28.................... 31............. 32.................. 34 3 1 9.................... 1....................... 1............
ver.1 / c /(13)
1 -- 11 1 c 2010 1/(13) 1 -- 11 -- 1 1--1 1--1--1 2009 3 t R x R n 1 ẋ = f(t, x) f = ( f 1,, f n ) f x(t) = ϕ(x 0, t) x(0) = x 0 n f f t 1--1--2 2009 3 q = (q 1,..., q m ), p = (p 1,..., p m ) x = (q,
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
H.Haken Synergetics 2nd (1978)
27 3 27 ) Ising Landau Synergetics Fokker-Planck F-P Landau F-P Gizburg-Landau G-L G-L Bénard/ Hopfield H.Haken Synergetics 2nd (1978) (1) Ising m T T C 1: m h Hamiltonian H = J ij S i S j h i S
Hanbury-Brown Twiss (ver. 2.0) van Cittert - Zernike mutual coherence
Hanbury-Brown Twiss (ver. 2.) 25 4 4 1 2 2 2 2.1 van Cittert - Zernike..................................... 2 2.2 mutual coherence................................. 4 3 Hanbury-Brown Twiss ( ) 5 3.1............................................
Untitled
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
B 1 B.1.......................... 1 B.1.1................. 1 B.1.2................. 2 B.2........................... 5 B.2.1.......................... 5 B.2.2.................. 6 B.2.3..................
(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x
![(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x](/thumbs/104/162595168.jpg "(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x")
Compton Scattering Beaming exp [i k x ωt] k λ k π/λ ω πν k ω/c k x ωt ω k α c, k k x ωt η αβ k α x β diag + ++ x β ct, x O O x O O v k α k α β, γ k γ k βk, k γ k + βk k γ k k, k γ k + βk 3 k k 4 k 3 k
201711grade1ouyou.pdf
2017 11 26 1 2 52 3 12 13 22 23 32 33 42 3 5 3 4 90 5 6 A 1 2 Web Web 3 4 1 2... 5 6 7 7 44 8 9 1 2 3 1 p p >2 2 A 1 2 0.6 0.4 0.52... (a) 0.6 0.4...... B 1 2 0.8-0.2 0.52..... (b) 0.6 0.52.... 1 A B 2
輻射の量子論、選択則、禁制線、許容線
Radiative Processes in Astrophysics 005/8/1 http://wwwxray.ess.sci.osaka- u.ac.jp/~hayasida Semi-Classical Theory of Radiative Transitions r r 1/ 4 H = ( cp ea) m c + + eφ nonrelativistic limit, Coulomb
215 11 13 1 2 1.1....................... 2 1.2.................... 2 1.3..................... 2 1.4...................... 3 1.5............... 3 1.6........................... 4 1.7.................. 4
A 99% MS-Free Presentation
A 99% MS-Free Presentation 2 Galactic Dynamics (Binney & Tremaine 1987, 2008) Dynamics of Galaxies (Bertin 2000) Dynamical Evolution of Globular Clusters (Spitzer 1987) The Gravitational Million-Body Problem
H+He H+He M > 10 M He He Wolf-Rayet Minit = 13 ~ 100 M, Zinit = 0.0001 ~ 0.02 CNO SN 2007bi (Type Ic) progenitor Minit > 100 M, Zinit = 0.004 WO star? Core collapse Rotating star model Saio, Nomoto, and
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
4 19
I / 19 8 1 4 19 : : f(e, J), f(e) Phase mixing Landau Damping, violent relaxation : 2 2 : ( ) http://antwrp.gsfc.nasa.gov/apod/ap950917.html ( ) http://www-astro.physics.ox.ac.uk/~wjs/apm_grey.gif
2000年度『数学展望 I』講義録
2000 I I IV I II 2000 I I IV I-IV. i ii 3.10 (http://www.math.nagoya-u.ac.jp/ kanai/) 2000 A....1 B....4 C....10 D....13 E....17 Brouwer A....21 B....26 C....33 D....39 E. Sperner...45 F....48 A....53
TOP URL 1
TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7
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
(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
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
(5) 75 (a) (b) ( 1 ) v ( 1 ) E E 1 v (a) ( 1 ) x E E (b) (a) (b)
(5) 74 Re, bondar laer (Prandtl) Re z ω z = x (5) 75 (a) (b) ( 1 ) v ( 1 ) E E 1 v (a) ( 1 ) x E E (b) (a) (b) (5) 76 l V x ) 1/ 1 ( 1 1 1 δ δ = x Re x p V x t V l l (1-1) 1/ 1 δ δ δ δ = x Re p V x t V
Note.tex 2008/09/19( )
1 20 9 19 2 1 5 1.1........................ 5 1.2............................. 8 2 9 2.1............................. 9 2.2.............................. 10 3 13 3.1.............................. 13 3.2..................................
: 2005 ( ρ t +dv j =0 r m m r = e E( r +e r B( r T 208 T = d E j 207 ρ t = = = e t δ( r r (t e r r δ( r r (t e r ( r δ( r r (t dv j =
72 Maxwell. Maxwell e r ( =,,N Maxwell rot E + B t = 0 rot H D t = j dv D = ρ dv B = 0 D = ɛ 0 E H = μ 0 B ρ( r = j( r = N e δ( r r = N e r δ( r r = : 2005 ( 2006.8.22 73 207 ρ t +dv j =0 r m m r = e E(
) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8)
4 4 ) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) a b a b = 6i j 4 b c b c 9) a b = 4 a b) c = 7
211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,
( ; ) C. H. Scholz, The Mechanics of Earthquakes and Faulting : - ( ) σ = σ t sin 2π(r a) λ dσ d(r a) =
1 9 8 1 1 1 ; 1 11 16 C. H. Scholz, The Mechanics of Earthquakes and Faulting 1. 1.1 1.1.1 : - σ = σ t sin πr a λ dσ dr a = E a = π λ σ πr a t cos λ 1 r a/λ 1 cos 1 E: σ t = Eλ πa a λ E/π γ : λ/ 3 γ =
0406_total.pdf
59 7 7.1 σ-ω σ-ω σ ω σ = σ(r), ω µ = δ µ,0 ω(r) (6-4) (iγ µ µ m U(r) γ 0 V (r))ψ(x) = 0 (7-1) U(r) = g σ σ(r), V (r) = g ω ω(r) σ(r) ω(r) (6-3) ( 2 + m 2 σ)σ(r) = g σ ψψ (7-2) ( 2 + m 2 ω)ω(r) = g ω ψγ
Part () () Γ Part ,
Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1.
1.1 1. 1.3.1..3.4 3.1 3. 3.3 4.1 4. 4.3 5.1 5. 5.3 6.1 6. 6.3 7.1 7. 7.3 1 1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N
006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................
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
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
global global mass region (matter ) & (I) M3Y semi-microscopic int. Ref.: H. N., P. R. C68, ( 03) N. P. A722, 117c ( 03) Proc. of NENS03 (to be
Gogny hard core spin-isospin property @ RCNP (Mar. 22 24, 2004) Collaborator: M. Sato (Chiba U, ) ( ) global global mass region (matter ) & (I) M3Y semi-microscopic int. Ref.: H. N., P. R. C68, 014316
* 1 1 (i) (ii) Brückner-Hartree-Fock (iii) (HF, BCS, HFB) (iv) (TDHF,TDHFB) (RPA) (QRPA) (v) (vi) *
* 1 1 (i) (ii) Brückner-Hartree-Fock (iii) (HF, BCS, HFB) (iv) (TDHF,TDHFB) (RPA) (QRPA) (v) (vi) *1 2004 1 1 ( ) ( ) 1.1 140 MeV 1.2 ( ) ( ) 1.3 2.6 10 8 s 7.6 10 17 s? Λ 2.5 10 10 s 6 10 24 s 1.4 ( m
25 7 18 1 1 1.1 v.s............................. 1 1.1.1.................................. 1 1.1.2................................. 1 1.1.3.................................. 3 1.2................... 3
t = h x z z = h z = t (x, z) (v x (x, z, t), v z (x, z, t)) ρ v x x + v z z = 0 (1) 2-2. (v x, v z ) φ(x, z, t) v x = φ x, v z
I 1 m 2 l k 2 x = 0 x 1 x 1 2 x 2 g x x 2 x 1 m k m 1-1. L x 1, x 2, ẋ 1, ẋ 2 ẋ 1 x = 0 1-2. 2 Q = x 1 + x 2 2 q = x 2 x 1 l L Q, q, Q, q M = 2m µ = m 2 1-3. Q q 1-4. 2 x 2 = h 1 x 1 t = 0 2 1 t x 1 (t)
6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2
![6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2](/thumbs/92/109118076.jpg "6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2")
1 6 6.1 (??) (P = ρ rad /3) ρ rad T 4 d(ρv ) + PdV = 0 (6.1) dρ rad ρ rad + 4 da a = 0 (6.2) dt T + da a = 0 T 1 a (6.3) ( ) n ρ m = n (m + 12 ) m v2 = n (m + 32 ) T, P = nt (6.4) (6.1) d [(nm + 32 ] )a
1 1 1 1-1 1 1-9 1-3 1-1 13-17 -3 6-4 6 3 3-1 35 3-37 3-3 38 4 4-1 39 4- Fe C TEM 41 4-3 C TEM 44 4-4 Fe TEM 46 4-5 5 4-6 5 5 51 6 5 1 1-1 1991 1,1 multiwall nanotube 1993 singlewall nanotube ( 1,) sp 7.4eV
1. z dr er r sinθ dϕ eϕ r dθ eθ dr θ dr dθ r x 0 ϕ r sinθ dϕ r sinθ dϕ y dr dr er r dθ eθ r sinθ dϕ eϕ 2. (r, θ, φ) 2 dr 1 h r dr 1 e r h θ dθ 1 e θ h
IB IIA 1 1 r, θ, φ 1 (r, θ, φ)., r, θ, φ 0 r
2 Chapter 4 (f4a). 2. (f4cone) ( θ) () g M. 2. (f4b) T M L P a θ (f4eki) ρ H A a g. v ( ) 2. H(t) ( )
http://astr-www.kj.yamagata-u.ac.jp/~shibata f4a f4b 2 f4cone f4eki f4end 4 f5meanfp f6coin () f6a f7a f7b f7d f8a f8b f9a f9b f9c f9kep f0a f0bt version feqmo fvec4 fvec fvec6 fvec2 fvec3 f3a (-D) f3b
19 σ = P/A o σ B Maximum tensile strength σ % 0.2% proof stress σ EL Elastic limit Work hardening coefficient failure necking σ PL Proportional
19 σ = P/A o σ B Maximum tensile strength σ 0. 0.% 0.% proof stress σ EL Elastic limit Work hardening coefficient failure necking σ PL Proportional limit ε p = 0.% ε e = σ 0. /E plastic strain ε = ε e
128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds = 0 (3.4) S 1, S 2 { B( r) n( r)}ds
127 3 II 3.1 3.1.1 Φ(t) ϕ em = dφ dt (3.1) B( r) Φ = { B( r) n( r)}ds (3.2) S S n( r) Φ 128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds
S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d
S I.. http://ayapin.film.s.dendai.ac.jp/~matuda /TeX/lecture.html PDF PS.................................... 3.3.................... 9.4................5.............. 3 5. Laplace................. 5....
量子力学 問題
3 : 203 : 0. H = 0 0 2 6 0 () = 6, 2 = 2, 3 = 3 3 H 6 2 3 ϵ,2,3 (2) ψ = (, 2, 3 ) ψ Hψ H (3) P i = i i P P 2 = P 2 P 3 = P 3 P = O, P 2 i = P i (4) P + P 2 + P 3 = E 3 (5) i ϵ ip i H 0 0 (6) R = 0 0 [H,
医系の統計入門第 2 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 2 版 1 刷発行時のものです.
医系の統計入門第 2 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/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α 2 ), ϕ(t) = B 1 cos(ω 1 t + α 1 ) + B 2 cos(ω 2 t
85 4
85 4 86 Copright c 005 Kumanekosha 4.1 ( ) ( t ) t, t 4.1.1 t Step! (Step 1) (, 0) (Step ) ±V t (, t) I Check! P P V t π 54 t = 0 + V (, t) π θ : = θ : π ) θ = π ± sin ± cos t = 0 (, 0) = sin π V + t +V
2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP
1. 1 213 1 6 1 3 1: ( ) 2: 3: SF 1 2 3 1: 3 2 A m 2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP
0.1 I I : 0.2 I
1, 14 12 4 1 : 1 436 (445-6585), E-mail : sxiida@sci.toyama-u.ac.jp 0.1 I I 1. 2. 3. + 10 11 4. 12 1: 0.2 I + 0.3 2 1 109 1 14 3,4 0.6 ( 10 10, 2 11 10, 12/6( ) 3 12 4, 4 14 4 ) 0.6.1 I 1. 2. 3. 0.4 (1)
H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [
![H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [](/thumbs/99/141438442.jpg "H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [")
3 3. 3.. H H = H + V (t), V (t) = gµ B α B e e iωt i t Ψ(t) = [H + V (t)]ψ(t) Φ(t) Ψ(t) = e iht Φ(t) H e iht Φ(t) + ie iht t Φ(t) = [H + V (t)]e iht Φ(t) Φ(t) i t Φ(t) = V H(t)Φ(t), V H (t) = e iht V (t)e
ssp2_fixed.dvi
13 12 30 2 1 3 1.1... 3 1.2... 4 1.3 Bravais... 4 1.4 Miller... 4 2 X 5 2.1 Bragg... 5 2.2... 5 2.3... 7 3 Brillouin 13 3.1... 13 3.2 Brillouin... 13 3.3 Brillouin... 14 3.4 Bloch... 16 3.5 Bloch... 17
1 filename=mathformula tex 1 ax 2 + bx + c = 0, x = b ± b 2 4ac, (1.1) 2a x 1 + x 2 = b a, x 1x 2 = c a, (1.2) ax 2 + 2b x + c = 0, x = b ± b 2
filename=mathformula58.tex ax + bx + c =, x = b ± b 4ac, (.) a x + x = b a, x x = c a, (.) ax + b x + c =, x = b ± b ac. a (.3). sin(a ± B) = sin A cos B ± cos A sin B, (.) cos(a ± B) = cos A cos B sin
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)
I-2 (100 ) (1) y(x) y dy dx y d2 y dx 2 (a) y + 2y 3y = 9e 2x (b) x 2 y 6y = 5x 4 (2) Bernoulli B n (n = 0, 1, 2,...) x e x 1 = n=0 B 0 B 1 B 2 (3) co
16 I ( ) (1) I-1 I-2 I-3 (2) I-1 ( ) (100 ) 2l x x = 0 y t y(x, t) y(±l, t) = 0 m T g y(x, t) l y(x, t) c = 2 y(x, t) c 2 2 y(x, t) = g (A) t 2 x 2 T/m (1) y 0 (x) y 0 (x) = g c 2 (l2 x 2 ) (B) (2) (1)
) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
![) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4](/thumbs/92/107866707.jpg ") ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4")
1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
Gmech08.dvi
51 5 5.1 5.1.1 P r P z θ P P P z e r e, z ) r, θ, ) 5.1 z r e θ,, z r, θ, = r sin θ cos = r sin θ sin 5.1) e θ e z = r cos θ r, θ, 5.1: 0 r
A
A04-164 2008 2 13 1 4 1.1.......................................... 4 1.2..................................... 4 1.3..................................... 4 1.4..................................... 5 2
9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) Ĥ0 ψ n (r) ω n Schrödinger Ĥ 0 ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ0 + Ĥint (
9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) 2. 2.1 Ĥ ψ n (r) ω n Schrödinger Ĥ ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ + Ĥint (t)] ψ (r, t), (2) Ĥ int (t) = eˆxe cos ωt ˆdE cos ωt, (3)
m dv = mg + kv2 dt m dv dt = mg k v v m dv dt = mg + kv2 α = mg k v = α 1 e rt 1 + e rt m dv dt = mg + kv2 dv mg + kv 2 = dt m dv α 2 + v 2 = k m dt d
m v = mg + kv m v = mg k v v m v = mg + kv α = mg k v = α e rt + e rt m v = mg + kv v mg + kv = m v α + v = k m v (v α (v + α = k m ˆ ( v α ˆ αk v = m v + α ln v α v + α = αk m t + C v α v + α = e αk m
Gravothermal Catastrophe & Quasi-equilibrium Structure in N-body Systems
2004/3/1 3 N 1 Antonov Problem & Quasi-equilibrium State in N-body N Systems A. Taruya (RESCEU, Univ.Tokyo) M. Sakagami (Kyoto Univ.) 2 Antonov problem N-body study of quasi-attractivity M15 T Antonov
,., 5., ,. 2.2,., x z. y,.,,,. du dt + α p x = 0 dw dt + α p z + g = 0 α dp dt + pγ dα dt = 0 α V dα dt = 0 (2.2.1), γ = c p /c
29 2 1 2.1 2.1.1.,., 5.,. 2.1.1,. 2.2,., x z. y,.,,,. du dt + α p x = 0 dw dt + α p z + g = 0 α dp dt + pγ dα dt = 0 α V dα dt = 0 (2.2.1), γ = c p /c v., V = (u, w), = ( / x, / z). 30 2.1.1: 31., U p(z),
D v D F v/d F v D F η v D (3.2) (a) F=0 (b) v=const. D F v Newtonian fluid σ ė σ = ηė (2.2) ė kl σ ij = D ijkl ė kl D ijkl (2.14) ė ij (3.3) µ η visco
post glacial rebound 3.1 Viscosity and Newtonian fluid f i = kx i σ ij e kl ideal fluid (1.9) irreversible process e ij u k strain rate tensor (3.1) v i u i / t e ij v F 23 D v D F v/d F v D F η v D (3.2)
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)
QMII_10.dvi
65 1 1.1 1.1.1 1.1 H H () = E (), (1.1) H ν () = E ν () ν (). (1.) () () = δ, (1.3) μ () ν () = δ(μ ν). (1.4) E E ν () E () H 1.1: H α(t) = c (t) () + dνc ν (t) ν (), (1.5) H () () + dν ν () ν () = 1 (1.6)
63 3.2,.,.,. (2.6.38a), (2.6.38b), V + V V + Φ + fk V = 0 (3.2.1)., Φ = gh, f.,. (2.6.40), Φ + V Φ + Φ V = 0 (3.2.2). T = L/C (3.2.3), C. C V, T = L/V
62 3 3.1,,.,. J. Charney, 1948., Burger(1958) Phillips(1963),.,. L : ( 1/4) T : ( 1/4) V :,. v x u x V L, etc, u V T,,.,., ( )., 2.1.1. 63 3.2,.,.,. (2.6.38a), (2.6.38b), V + V V + Φ + fk V = 0 (3.2.1).,
Z: Q: R: C: sin 6 5 ζ a, b
Z: Q: R: C: 3 3 7 4 sin 6 5 ζ 9 6 6............................... 6............................... 6.3......................... 4 7 6 8 8 9 3 33 a, b a bc c b a a b 5 3 5 3 5 5 3 a a a a p > p p p, 3,
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
30
3 ............................................2 2...........................................2....................................2.2...................................2.3..............................
Microsoft Word - 11問題表紙（選択）.docx
A B A.70g/cm 3 B.74g/cm 3 B C 70at% %A C B at% 80at% %B 350 C γ δ y=00 x-y ρ l S ρ C p k C p ρ C p T ρ l t l S S ξ S t = ( k T ) ξ ( ) S = ( k T) ( ) t y ξ S ξ / t S v T T / t = v T / y 00 x v S dy dx
CMP Technical Report No. 4 Department of Computational Nanomaterials Design ISIR, Osaka University 2 2................................. 2.2......................... 2 3 3 3................................
* 1 2014 7 8 *1 iii 1. Newton 1 1.1 Newton........................... 1 1.2............................. 4 1.3................................. 5 2. 9 2.1......................... 9 2.2........................
( )
7..-8..8.......................................................................... 4.................................... 3...................................... 3..3.................................. 4.3....................................
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
q quark L left-handed lepton. λ Gell-Mann SU(3), a = 8 σ Pauli, i =, 2, 3 U() T a T i 2 Ỹ = 60 traceless tr Ỹ 2 = 2 notation. 2 off-diagonal matrices
Grand Unification M.Dine, Supersymmetry And String Theory: Beyond the Standard Model 6 2009 2 24 by Standard Model Coupling constant θ-parameter 8 Charge quantization. hypercharge charge Gauge group. simple
/ 18 12 4 ( ) ( ) : : f(e, J), f(e) Phase mixing Landau Damping, violent relaxation : 2 2 : ? m i d 2 x i dt 2 = j i f ij (1) x i m i i f ij j i f ij G f ij = Gm i m j x j x i x j x i 3, (2) ( ) 2 (