Convection 21 September 2011

Goals • Continue thinking about convection in different kinds of stars

Convection

1. In lecture we saw that the criterion for convection to occur can be expressed as 1 P L L /L γ − 1 r > , 4 Prad LEdd Mr/M γ where the Eddington luminosity is 4πGMc L = . Edd κ Now consider the atmosphere of a massive star with M = 100M , 6 L = 10 L . Approximate the entire luminosity of the star as being generated in the central 1% (by mass) of the star.

(a) What form of opacity dominates?

Solution: High mass stars have generally higher internal tem- peratures (as stated). This means Thomson scattering is more 2 −1 important. Recall for ionized hydrogen that κT = 0.4 cm g .

(b) What type of pressure dominates? Why?

Solution: As we’ve shown several times in the course so far, at these high masses, radiation pressure begins to become very

important, so we can approximate P/Prad ≈ 1.

Solution: Since γ = 4/3 for radiation, We need the ratio P L L /L r > 1. Prad LEdd Mr/M We have said that the pressure is from radiation, which implies the luminosity is at (or very near) the Eddington limit. The luminosity is generated very centrally, so Lr ≈ L for all parts of the star. The inequality simpliﬁes then to M > 1. Mr This is true everywhere, and implies the entire star is convec-

tive. (In reality, the P = Prad approximation begins to break down a bit at large radii.) convection 2

2. In the outer envelope of a white dwarf, the gas is classical and ideal. If we were to use an appropriate opacity and then solve the hydrostatic equilibrium and radiative diffusion equations, we would ﬁnd that If you’re curious as to where this comes P ∝ T17/4 . from, see Phillips §3.4. Is the outer envelope of the white dwarf convective? (That is to say, is the proﬁle we found assuming radiative transport unstable to convection?) If the answer is yes, then the assump- tion of radiative diffusion that we used Solution: We can assume an ideal classical gas with γ = 5/3. Thus to get that relation is violated. our criterion for convection to occur becomes

d ln T ? γ − 1 > d ln P γ 4 ? > 0.4 17 which is clearly not true, so the envelopes of white dwarfs are not convective.

3. We’ve learned that low-mass stars are fully convective, the Sun has a convective envelope, and high-mass stars have convective cores. In words, what are the physical reasons for convection occur- ring where it does in these stars? Solution: The best way to think about this is to look at the convection criterion 1 P L κ γ − 1 r > . 4 Prad 4πMrGc γ In the outer parts of the Sun, the temperature is lower, and thus

the free-free opacity κff is much higher. It becomes increasingly difficult for photons to transport the energy out, and eventually the above criterion is met and convection sets in. In lower mass stars, the opacity is so high everywhere that the star is fully convective. Thus, physically, opacity is an important reason for convection. In high mass stars, however, opacity is not the answer. Thomson scattering tends to dominate, and the opacity κT is independent of both temperature and density. The main quantity that can change dramatically in the above criterion is the luminosity. High-mass stars produce all of their energy in a small region, leading to Lr/L = 1 while Mr/M 1. Essentially, the flux is so large that photons cannot fully transport all that energy, and so convection must set in. This occurs in the core, though the convection zone can grow to a significant fraction of the star in very high-mass stars as you saw in the preceding problem.