3. Stellar radial pulsation and stability

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1 3. Stellar radial pulsation and stability m V δ Cephei T eff spectral type v rad R = Z vdt stellar disk from ΔR 1

2 1.55 LD Angular diam. (mas) res. (mas) Phase A B Interferometric observations of δ Cep (Mérand et al. 2005) M. Groenewegen 2 MIAPP, 17 June 2014 p.8/50

3 instability strip Hayashi limit of low mass fully convective stars main sequence 3

4 the instability strip stars in instability strip have significant layers not too deep, not too superficial, where stellar opacity increases with T. This is caused by the excitation and ionization of H and He. Consider a perturbation causing a local compression à T increase à κ increase à perturbation heat not radiated away but stored à expansion beyond equilibrium point à then cooling and T decreasing à gravity pulls gas masses back à cyclically pulsation apple T, 0 4

5 stars to the left of instability strip are hotter and H and He are ionized in the interior. apple T, 0 The opacity decreases with increasing T. For a perturbation causing a local compression we have again à T increase but now à κ decreases à perturbation heat is quickly radiated away à perturbation is damped, no pulsation To the right of the instability strip the domination energy transport is by very effective convection and not by radiation transport. The compression heat is, thus, effectively transported away and the opacity behavior is irrelevant 5

6 Kippenhahn, Weigert, Weiss, 2012 Rolf Kudritzki SS 2015 excitation of pulsation 6

7 a simple model of radial pulsation pressure gradient balances gravity in hydrostatic equilibrium dp 0 dr 0 = G M r 0 r mass confined M r0 =4 within r 0 Z r0 0 0 r 2 0dr 0 mass within dr 0 dm r0 =4 0 r 2 0dr 0 r 0 radial coordinate in equilibrium P 0 pressure mass density ρ 0 Gedanken Experiment : compress star and then release à oscillation à compression energy stored apple T, 0 à no oscillation damping 7

8 pulsation leads to radial shift of mass shells within the stars: r 0 r = x(t) new time dependent coordinate of mass shells: r(t) =r 0 [1 + x(t)] x(t) is a time dependent perturbation and we assume that the moving shells conserve their mass: r = 0 r 2 0@r 0 =[1+x(t)]@r 0 we then obtain for the density: if the mass shells do not exchange energy, we have an adiabatic situation: = 0 [1 + x(t)] P = P 0 0 P = P 0 [1 + x(t)] 3 3 because of the perturbation the original hydrostatic equation does not hold anymore. In the coordinate frame co-moving r we need to add the inertia @r = GM r r 2 eom 8

9 left hand side of eom: small perturbation: x(t) 1 = 0 [1 and 3x(t)] P = P 0 [1 3 x(t)] 1 r 2 = 1 r0 2 [1 2x(t)] M r = @r (1 3 x)(1 0 = G M r 0 r (1 3 x)(1 x) G M r 0 r 2 0 neglecting x 2 terms 0 (1 x(3 + 1)) 9

10 on right hand side of eom we have the terms G M r r 2 = GM r 0 r 2 0 (1 2x) 0 (1 3x) G M r 0 r (1 5x) 2 2 x 0(1 3x)r 2 left hand eom side equal to right hand side G M r 0 r (1 x(3 1)) = G M r 0 r (1 5x)+ 0 (1 3x)r 2 2 r 2 2 = GM r 0 r 2 0 (4 3 ) (1 3x) x(t) 10

11 @ 2 2 +(3 at the outer edge of the star r 0 = R * we then obtain the differential equation 4)3G 4 x(t) =0 with = M 4 3 R3 mean stellar density with the usual solution x(t) =x 0 e i!t! 2 =(3 4) 3G 4 eigenfrequency of star P puls r 1 pulsation period of star 11

12 pulsation period: strong radius dependence weak mass dependence P puls r 1 absolute magnitudes R 3 2 M 1 2 M V,I,J,H,K 5logR M V,I,J,H,K 10 3 logp puls predicted period luminosity relationship M V 2.77logP puls M I 2.96logP puls M J 3.11logP puls M H 3.21logP puls M K 3.19logP puls observed (Ngeow, 2009, ApJ 693, 6791) good agreement despite simplifications such as adiabatic assumption P = P 0 0 r 0 r = x(t) independent of r 12

13 PERIOD-LUMINOSITY RELATIONS LMC L. Macri, 2014, MIAPP WS 13 MACRI, NGEOW, KANBUR+ (IN PREP.)

14 G. Fiorentino 14

15 G. Fiorentino 15

16 G. Fiorentino 16

17 stellar stability r(t) =x(t)r 2 2 +!2 x(t) =0 time dependent radial movement of mass shells described by oscillator equation, eigenfrequency and adiabatic exponent γ! 2 =(3 4) 3G 4 periodic solution only for > > 0 normal mono-atomic gas non-relativistic Fermi gas =5/3 4 3! 2 = c p c v P = P 0 oscillation however, for apple 4/3! 2 apple 0 x(t) =x 0 e ±! t exponential collapse or expansion! 2 =0 x(t) =v 0 t linear growth or collapse 0 instability!!!! 17

18 examples White Dwarf at Chandrasekhar-limit (Chapter 2) very massive stars dominated by radiation pressure P P rad = 1 3 at 4 =0 18

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