Stellar Structure and Evolution Stellar Evolution (2) 10. Red Giants 10 Red Giant Stars 1.INTRODUCTION We should now continue our exploration of the main pathways of stellar evolution, while keeping in mind that we are still neglecting important aspects such as rotation and more importantly, mass loss. Remember that our aim in this course is to understand the main observational features of the Hertzsprung-Russel diagram. The HRD will be continue to be our guide in the following. The main results from this Chapter are: the Schönberg-Chandrasekhar mass limit the motion of stars in the HRD during the transition from H-burning to He-burning the mirror effect: as the core contracts, the envelope expands the Hertzsprung gap, the subgiant branch, and the Red Giant Branch (RGB) the Hayashi line is the location of convective stars in the HRD there are no stars right to this line in the HRD In this Figure, we see the HRD of globular clusters, as obtained with the ESO/Gaia satellite. A few noticeable features are outlined. In this Lecture, we wish to explain, on physical grounds, the origin of these features. In doing so, we will realize that, although some do not depend qualitatively on the mass of the star—the turn- off point, the sub-giant, the Red Giant Branch, the Horizontal Branch—there are some important qualitative differences between low-mass stars (푀 < 2M⊙) on the one hand and intermediate (2 − 9M⊙) and massive stars on the other hand. The HRD of open clusters show a specific feature: the Hertzsprung gap, which is a direct consequence of the first principles we have established throughout this course. 1 Stellar Structure and Evolution Stellar Evolution (2) 10. Red Giants 2.THE SCHÖNBERG-CHANDRASEKHAR MASS Equilibrium condition of the stellar core The Virial Theorem applied to a sphere of gas reads: 3 3 훾 − 1 푈 + 훺 = 4휋푅 푃푠 where we recall that U is the total internal energy, which is related to the kinetic energy of the particles through: 3 3 퐾 = 훾 − 1 푢 = 푢 2 푁푓 This gives the more familiar form of the viral theorem 3 훺 + 2K = 4휋푅 푃푠 As you notice, we have kept the surface pressure terms in the right-hand side of the equation. This is important for the situation we are about to study: the condition of mechanical equilibrium of the burning core. We could neglect the surface-pressure term when studying the equilibrium of stars. However, the pressure acting at the surface of the burning con is not negligible. let us substitute for the internal energy and gravitational energy their expression in term of the physical parameters. The internal energy is related to the pressure and temperature through the equation of state. As long as the core remains an ideal gas: 푀퐶 푀푐 3 3 푃 3 푅푔푇푐 퐾 = 푁푘퐵푇 = dm = dm 2 2 휌 2 µ푐 0 0 when you see that we use quantities relative to the core itself. On the other hand the gravitational energy is: 퐺푀2 훺 =− 훼 푐 푅푐 where again 훼 is a factor of order unity. We then obtain the following expression for the virial theorem: 푀퐶 2 3 푅푔푇푐 퐺푀푐 4휋푅푐 푃푠 = 3 dm − 훼 µ푐 푅푐 0 The surface pressure is thus given by: 2 3 푅푔푇푐푀푐 퐺푀 푃 = − 훼 푐 푠 4휋 3 4 푅푐µ푐 4휋푅푐 This relation expresses the mechanical equilibrium between external pressure, internal pressure, and self- gravity. One sees that the surface pressure presents a maximum: 2 Stellar Structure and Evolution Stellar Evolution (2) 10. Red Giants The surface pressure is zero at a radius; 훼퐺 푀푐µ푐 푅0 = 3푅푐 푇푐 and has a maximum at 4 4 퐺푀푐µ푐 푅1 = 푅0 = 훼 3 9 푅푐푇푐 Now let us consider a stellar burning core of mass Mc and radius Rc such that the core is in mechanical equilibrium against the pressure exerted above by the envelope. We see that the core can find such a mechanical equilibrium provided that the pressure at its boundary is not larger than Pmax, the value of 푃 at R1. We find that 푇4 푃 ∝ 푐 max 2 4 푀푐 휇푐 Whenever the surface pressure overcomes this value, the core can no longer find an equilibrium configuration. Conversely, given a stellar mass, the pressure at the core surface is what it is, but since Pmax decreases with Mc it is clear that there is a maximum value of Mc for which the core can maintain mechanical equilibrium. This upper limit is called the Schönberg-Chandrasekhar limit. Stability of the equilibrium Consider a core of mass Mc, in mechanical equilibrium under the surface pressure exerted by the envelope. From the shape of the pressure-radius curve, we see that there are two equilibrium radii, one smaller than R1 and one that is larger. However, only one corresponds to a stable equilibrium. Let us see why. First, we notice that above the curve, there is no equilibrium because the internal energy is too small compared to the gravitational energy. Hence, if we start initially from an equilibrium configuration with 푅 larger than R1, and we increase the radius at constant external pressure, then the core finds itself in a region where the gravitational potential is larger than the internal energy: the core contracts and goes back to its initial location. Similarly, if we decrease the radius at constant internal pressure, the core would have too large an internal energy and would thus expand, going back to its initial location. We thus conclude that the right branch with 푅 larger than R1 corresponds to stable equilibrium configurations. Conversely, a similar analysis would show that the left branch corresponds to unstable equilibrium configurations. Note that a more conventional analysis led to first order in perturbation amplitude would lead to a more general expression of the stability condition, ∂P/∂R < 0. 3 Stellar Structure and Evolution Stellar Evolution (2) 10. Red Giants Consequences for the stability of the nuclear burning core Now that we have seen that the core can be in mechanical equilibrium provided that the pressure does not exceed Pmax, we will now see that this translates into a condition on its mass. One way to see this, is to realize that the value of Pmax depends on Mc and is such that Pmax decreases when Mc increases. Yet, this would be true if neither Tc nor μc vary in such a way as to compensate for it. However, the key point is that, as mentioned already, in the hydrogen-poor core, there is no nuclear reaction: ∂ℓ = ϵ(m) = 0. ∂m Hence there is no energy flow, hence ℓ = 0 in the core. Since the temperature gradient in a radiative core is proportional to ℓ, this gradient vanishes and the temperature is constant: Inert cores are isothermal. The inert core is however surrounded by a hydrogen-burning shell which makes the separated the core from the inert envelope. As the burning proceeds in the shell, hydrogen is consumed and the burning-shell thus moves outwards, leaving behind an inert core of growing mass. However, as we have just seen, Mc can not grow without limit for it soon will reach the critical mass above which the core will no longer be able to find an equilibrium configuration where the kinetic pressure can oppose both the gravity and the pressure exerted by the envelope. We thus expect the core to contract. In fact, it will, but not the same way for low-mass stars (say, below 2 M⊙) and for intermediate (2 < 푀 < 9M⊙) and massive stars. We now look at what will happen in intermediate stars and in low-mass stars after the inert has started to contract. 3.RED GIANTS Red Giants are stars transiting from hydrogen burning to helium burning. Their name directly reflects their most striking observational properties: they have a low effective temperature, and are located to the right of the HRD, and they are very luminous. Since 2 4 퐿 = 4πR σ T푒푓푓, this means that they are huge. An overview of the evolution of an intermediate mass star is best summarized in the Iben 1991 review, from which the Figure below was extracted. Intermediate mass stars The inert core has grown due to the helium production in the surrounding burning shell. When the core mass has reached the Schönberg-Chandrasekhar mass, MSC, it contracts, for there is no equilibrium configuration available. In doing so, the pressure in the core increases and, since the gas is described by a perfect gas EOS, the temperature also increases throughout the isothermal core. Hence, the temperature at the base of the burning shell also increases thus enhancing the rate of nuclear energy production. In such stars, hydrogen burning in the shell is due to the CNO-cycle and its ν ≈ 20 temperature 4 Stellar Structure and Evolution Stellar Evolution (2) 10. Red Giants exponent. Therefore, the production rate of nuclear ashes, namely helium, is greatly enhanced: the mass of the core further increases. A key point is that the growth of the mass of the core is fast in such stars compared to low-mass stars (see below). The temperature in the core soon reaches the threshold of ≈2(8) K to ignite He burning through the 3α- process. When this is achieved, the temperature stabilizes due the thermostat-effect of nuclear burning, and the core finds a new mechanical equilibrium. The star is now burning helium in the core, and burning hydrogen in a shell above, migrating outwards. We stress an important point: when the core reaches the threshold for He burning, it finds a stable equilibrium, because of the stabilizing effect of nuclear burning in a perfect gas.