Rolf Kudritzki SS 2017 3. Stellar radial pulsation and stability
mV δ Cephei
Teff
spectral type
vrad
R = vdt Z stellar disk Δ from R 1 Rolf Kudritzki SS 2015
1.55
1.50
1.45
1.40 LD Angular diam. (mas)
A 1.35 0.04 0.0 0.5 1.0 0.02 0.00 −0.02
res. (mas) B −0.04 0.0 0.5 1.0
Phase Interferometric observations of δ Cep (Mérand et al. 2005) M. Groenewegen2 MIAPP, 17 June 2014 – p.8/50 Rolf Kudritzki SS 2017 instability strip
Hayashi limit of low mass fully convective stars
3 Rolf Kudritzki SS 2017 the instability strip stars in instability strip have significant layers not too deep, not too superficial, where ↵ stellar opacity increases with T. T , ↵ 0 ⇠ 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
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↵ stars to the left of instability strip T , ↵ 0 are hotter and H and He are ionized ⇠ in the interior.
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 energy transport is dominated by very effective convection and not by radiation transport. The compression heat is, thus, effectively transported away and the opacity behavior is irrelevant. 5 RolfKippenhahn Kudritzki SS, Weigert 2015 , excitation of Weiss, 2012 pulsation
6 Rolf Kudritzki SS 2017 a simple model of radial pulsation
pressure gradient dP0 Mr0 balances gravity = G 2 ⇢0 dr0 r0
r0 mass confinedM =4⇡ ⇢ r2dr within r r0 0 0 0 0 Z0 2 dMr =4⇡⇢0r dr0 mass within dr0 0 0
r0 radial coordinate in equilibrium P0 pressure ρ0 mass density
“Gedanken Experiment”: compress star and then release à oscillation T ↵ , ↵ 0 à compression energy stored ⇠ à no oscillation damping 7 Rolf Kudritzki SS 2017 r pulsation leads to radial shift of mass shells within the stars: = x(t) r0 new – time dependent – coordinate of mass shells: r(t)=r0[1 + x(t)]
x(t) is a time dependent perturbation and we 2 2 ⇢r @r = ⇢0r @r0 assume that the moving shells conserve their mass: 0
3 with @ r =[1+ x ( t )] @ r 0 we then obtain for the density: ⇢ = ⇢0[1 + x(t)] ⇢ P = P0 if the mass shells do not exchange ⇢0 energy, we have an adiabatic situation: ✓ ◆ 3 P = P0[1 + x(t)] because of the perturbation the original hydrostatic equation does not hold anymore. In the coordinate frame co-moving with @Mr @2r we need to add the inertia term ⇢ @t2 2 @P Mr @ r eom = G 2 ⇢ ⇢ 2 @r r @t 8 Rolf Kudritzki SS 2017 small perturbation: x(t) 1 and Mr = Mr | | ⌧ 0 ⇢ = ⇢ [1 3x(t)] 0 P = P0[1 3 x(t)] 1 1 = [1 2x(t)] r2 r2 left hand side of eom: 0 @P @P @r @P 1 3 x @P = 0 = 0 0 (1 3 x)(1 x) @r @r0 @r @r0 1+x ⇡ @r0
Mr0 = G 2 ⇢0(1 3 x)(1 x) r0
Mr0 G 2 ⇢0(1 x(3 + 1)) ⇡ r0 neglecting x2 terms
9 Rolf Kudritzki SS 2017 on right hand side of eom we have the terms
Mr Mr0 Mr0 G 2 ⇢ = G 2 (1 2x)⇢0(1 3x) G 2 ⇢0(1 5x) r r0 ⇡ r0 @2r @2x and ⇢ = ⇢ (1 3x)r @t2 0 0 @t2 left hand eom side equal to right hand side
2 Mr0 Mr0 @ x G 2 ⇢0(1 x(3 1)) = G 2 ⇢0(1 5x)+⇢0(1 3x)r0 2 r0 r0 @t
@2x M (4 3 ) r = G r0 x(t) 0 @t2 r2 (1 3x) 0 10 Rolf Kudritzki SS 2017 at the outer edge of the star r0 = R* we then obtain the differential equation @2x 3G⇢¯ M with ⇢¯ = ⇤ 2 +(3 4) x(t)=0 4⇡ 3 @t 4⇡ 3 R ⇤ mean stellar density
i!t with the usual solution x(t)=x0e 2 3G⇢¯ eigenfrequency ! =(3 4) of star 4⇡ 1 P pulsation period puls ⇠ ⇢¯ of star r
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pulsation period: strong radius dependence absolute magnitudes weak mass dependence
3 1 1 2 Ppuls R M 2 MV,I,J,H,K 5logR ⇠ ⇢¯ ⇠ ⇤ ⇠ ⇤ r
10 predicted MV,I,J,H,K logPpuls period luminosity ⇠ 3 relationship
MV 2.77logPpuls observed (Ngeow, 2009, ApJ 693, 6791) ⇠ MI 2.96logPpuls good agreement despite simplifications ⇠ such as ⇢ MJ 3.11logPpuls adiabatic assumption P = P0 ⇠ ⇢0 MH 3.21logPpuls r ✓ ◆ ⇠ = x ( t ) independent of r M 3.19logP r0 K ⇠ puls 12 Rolf Kudritzki SS 2015 PERIOD-LUMINOSITY RELATIONS
L. Macri, 2014, MIAPP13 WS LMC MACRI, NGEOW, KANBUR+ (IN PREP.) Rolf Kudritzki SS 2015
14 G. Fiorentino Rolf Kudritzki SS 2015
15 G. Fiorentino Rolf Kudritzki SS 2015
16 G. Fiorentino Rolf Kudritzki SS 2017 stellar stability
time dependent radial movement of mass shells r(t)=x(t)r0 described by oscillator equation, eigenfrequency and adiabatic exponent γ 2 3G⇢¯ cp ⇢ @ x 2 2 = P = P + ! x(t)=0 ! =(3 4) 0 ⇢ @t2 4⇡ cv ✓ 0 ◆
4 2 periodic solution only for > ! > 0 3
normal mono-atomic gas non-relativistic Fermi gas =5/3 oscillation 2 ! t however, for 4/3 ! 0 x(t)=x0e±| | exponential collapse or expansion 2 ! =0 x(t)=v0t linear growth or collapse
instability !!!! 17 Rolf Kudritzki SS 2017 examples
• White Dwarf at Chandrasekhar-limit (Chapter 2)
• very massive stars dominated by radiation pressure 1 P P = aT 4 ⇡ rad 3 4 with T = f(ρ(r)) from stellar interior structure à ⇠ 3
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