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 s t = L r M r + ε N ε ν (3). ε N ε ν radiation conduction convection λ (= λ rad + λ cond + λ conv, λ rad λ cond λ conv L r = λ T r = λ T rad r λ T cond r λ T conv r. (4) L r,rad L r,cond L r,conv κ L r = 4πGcM r κ 4aT 4 /3 P log T log P. (4 )
a c λ = 4πr2 4 ρκ 3 act 3, (5) /κ =/κ rad +/κ cond ++/κ conv (6) 2. eqs. ()-(4) M r P ρ T r L r 5 4 M r =0 r = L r = 0 (7) M r =0 P = ρ = 0( radiative zero boundary condition), or (8) τ = κρdr =2/3 P = ρdr, and T = T eff ( L 4πR 2 σ )/4. (9) R R L luminosity σ (= ac/4) Stephan-Boltzmann T eff effective temperature R L =4πR 2 σt 4 eff (0) II.. P = Kρ +/N (K, N ) () eqs. () (2) ρ c P c ξ = r/r 0, ϕ = M r /M 0, ϖ = P/P c, θ = ρ/ρ c (2) P c R 0 =( ) /2 Pc 3, M 0 =( 4πG ρ c 4πG 3 ρ 4 c ) /2 dϕ dξ = ξ2 θ, dϖ dξ = ϕθ ξ 2, ϖ = θ+/n (3) 2
ξ = ξ/ +N θ = θ /N d dθ ξ 2 (ξ 2 dξ ξ )= θ N (3 ). eq. (3) eq. (3 ) Lane-Emden 2. Lane-Emden Lane-Emden polytropic index, N, 0 N 5 (θ =0) dθ /dξ =0atξ =0: θ = Lane-Emden N =0 5 ϖ = ξ 2 /6 θ = sin(ξ/ 2)/(ξ/ 2) θ =[+ξ 2 /8] 5/4 N =0 N = N =5. Lane-Emden * ξ, ϕ, N =0 N soft ρ c /ρ (= 3ξ 3 /ϕ ) N N =5 ϕ ξ * N ξ ϕ ρ c /ρ 0 2.449 4.899.0 4.442 8.886 3.290.5 5.777 0.73 5.99 3.0 3.79 6.4 54.8 5.0 25.56 Lane-Emden M R ρ c P c Pc 3 M =[ 4πG 3 ρ 4 c ] /2 ϕ, R =[ 4πG P c ] /2 ξ ρ 2, (4) c * N =5 t = log ξ z = 2ξ θ * N =5 stellar dynamics Plummer 3
M R ρ c P c P c =( ξ ) GM ρ c ϕ R, ρ c =( ξ3 ) M (4) ϕ 4πR 3 (d log P/dlog r) (d log ρ/d log r) = N + N ( ) N 3 P = k µm a ρt (5) (µ m a k atomic mass unit Boltzmann ) (4 ) T c = µm a k P c ρ c.2 0 7 K, ρ c 77 g cm M =.99 0 33 g and R =6.95 0 0 cm 4 R T c R ρ c R 3 T 3 c /ρ c =(µm a /k) 3 (P 3 c /ρ 4 c)=(µm a /k) 3 4πG 3 (M/ϕ ) 2 (6) T c ρ /3 c T c M 2/3 ρ c M 2 (6) (4) P = P g + P rad = k ρt + a µm a 3 T 4 (4 ) T c R ρ c R 3 β β = P g /P = kρt/µm a P, β = P rad /P = at 4 /3P (7) (4) β c β 4 c =( µm a k )4 a 3 4πG3 ( M ϕ ) 2. (8) 4
β c M β 4 0 4 M>0M 4 M r M luminosity L L L Edd (κ el /κ)( β) (9) κ el L Edd 4πGcM/κ el (20) Eddington β d log P/dlog ρ ( log P/ log ρ) ad = 32 24β 3β2 3(8 β) 4/3 (2) ( γ =5/3 ) N =3 N =3 β β (8) β β M 2 and hence L M 3 (κ el /κ) (22) M>0M β L L Edd M (23) * III.. Virial u γ u =/(γ ) kt =/(γ ) P/ρ (24) µm a * Lane-Emden 5
E T E G M P (γ )E T = 0 ρ dm r = =[ 4πr3 P ] R 0 3 = 3 M 0 R 0 P 4πr 2 dr ( 4πr3 3 )dp dr dr GM r dm r = r 3 E G (25) Virial E E = E T + E G = (3γ 4)E T = 3γ 4 3γ 3 E G. (26) E E T E G γ =5/3 E = E T + E G = E T = 2 E G (27) 2. 6
timescale free-fall timescale τ ff (= / 4πGρ) 6 T c ρ /3 c R T c 0 6 L L N E/ t = L + L N < 0 E/ t = L+L N > 0 L N = L * ** IV. = * ** 7
. Fermi-Dirac ΨkT n e = P e = 3 0 0 exp(ɛ/kt Ψ) + p ɛ p exp(ɛ/kt Ψ) + 8πp 2 dp, (28) h3 8πp 2 dp, (29) h3 ɛ =(m 2 ec 4 + p 2 c 2 ) /2 mc 2 Ψ > 0 n e = e Ψ 2(2πm e kt/h 2 ) 3/2 (30) ρ > µ e m a 2(2πm e kt/h 2 ) 3/2 =8.µ e T 3/2 6 (3) (µ e T 6 = T/0 6 K), Fermi p F n e = 8π 3 (m ec h )3 ( p F m e c )3 (32) P e = { π 3 ( m ec h π 3 ( m ec h )3 m e c 2 8 5 ( p F m e c )5 )3 m e c 2 2( p F m e c )4 (N.R.) (E.R.) Fermi ɛ F =(m 2 ec 4 +p 2 F c2 ) /2 mc 2 2kT 64π ρ>µ e m a 3 (m ec h )3 =7.8 0 6 µ e gcm 3 (33) { 4 π 2/3 ( h 5 3 m P e = e c )2 m e c 2 ρ ( µ e m a ) 5/3 (N.R.) π 2/3 3 ( h m e c )m (34) ec 2 ρ ( µ e m a ) 4/3 (E.R.) N =.5 N =3 - P e P i 8
E E T E G / E T I II E >0 3 τ exp = c P T/ε N c P τ ff =/ 4πGρ (a) τ exp >τ ff L N = L (b) τ exp <τ ff E = E T /E G (b) Ia 9
(a) T c ρc /3 M 2/3 T c ρ 2/3 c (8) (3) T deg T deg = 2π(8πµ em a ) 2/3 µ 2 m 2 am e G 2 ( M ) 4/3 (35) h 2 k ϕ polytropic index N =.5 M =0.023M (µ 3 µ e ) /2 T 3/4 6 (36) T 2 0 6 K µ = 0.6 µ e =.2 M 0.08M 0.08M Kelvin-Helmholtz τ KH (GM 2 /R)/L 3 0 7 (M/M ) 2 (R/R ) (L/L ) yr (37) (L M 3 4 ) Gyr f τ = MXE Hf L =6.8 0 9 ( M )( X M 0.7 )( L ) E H f ( L 6 0 8 )( )yr (38) erg g 0. X E H 00 L M 3 4 2 3 0
L Edd τ =7.2 0 6 f( L Edd )yr (39) L f 2.5M 8M 8 M 2M Lane-Emden Fermi N =.5 h 2 R =[ 4πGm e 5 ( 3 8π )2/3 µ e m a 3h 6 M =[ 32π 2 G 3 m 3 e 5/3 ] /2 5.777 ρ /6 c 5 3 ( ) 5 ] /2 0.73 ρ /2 c µ e m a (40) RM /3 = 5 ( 3 32π 2 µ e m a ) 2/3 h 2 Gµ e m a m e ξ ϕ /3 =0.040( µ e ) 5/3 R M /3 (4) (µ =2) 0.0R /00 Fermi P ρ 4/3, N =3 3 M = M crit =[ 32π ( hc ) 3 ] /2 µ 2 6.4 =.42M 2 ( 2 ) 2 (42) 4Gm a m a µ e M crit Chandrasekhar hc 3 R =[ ( ) /3 ] /2 ( ) 2/3 3.79 ρ /3 c =0.0( ) 2/3 ρ /3 c,6 R (43) 4πGm a 8πm a µ e µ e
(ρ c,6 = ρ c /0 6 gcm 3 ) ρ c M 2 R M /3 M crit 2. M ρ (3 N)/2N c homologous / N =3 Γ +/N =4/3. N < 3 Γ > 4/3 : M / 2. N =3 Γ =4/3 : M / 3. N>3 Γ < 4/3 : M ρ c γ =( log P/ log ρ ) ad =5/3 > 4/3 ( ) γ<4/3 H p + e / / 0 9 K γ<4/3 M>2M 56 Fe 3 4 He + 4n + 24.4Mev (44) 2
M < 30M < 2M M > 30M II T>0 9 2γ e + e + (45) γ<4/3 0 9 K 3. IV. 3