Lecture 27 - Stellar Structure (3/25/96)

Seeds: Chapters 9, 10

1. Other Nuclear Reactions
   • The CNO cycle (for Carbon - Nitrogen - Oxygen) also turns 4 H -> He, but using carbon as a catalyst. This provides about 10% of the energy of the Sun, and requires temperatures of 16 million K.
   • The Triple Alpha reaction turns 3 He -> C ("alpha particles" are another term for the helium nucleus). This reaction needs temperatures above 100 million K, and do not happen in the Sun at this time. Later, however ...
   • Carbon Fusion can occur above about 600 million K, and produces oxygen, neon, magnesium and silicon. At still higher temperatures around 3.5 billion K, silicon can be fused to form heavier elements, including iron.
   • The iron barrier: if you remember our consideration of the binding energy per nucleon of the nucleus as a function of the atomic weight of the nucleus, we showed that you could extract energy from fusion only for elements lighter than iron. Thus, iron is the "end of the line" for fusion as an energy generator.
2. The Pressure-Temperature Thermostat
   • Pressure from energy generation in the core balances the gravitational weight of the layers of the star above it.
   • When the core is undergoing thermonuclear fusion, the pressure from the energy generation balances gravity.
   • Because the rate of nuclear reactions depends very sensitively on the temperature, a slight increase in temperature will cause and increase in the energy generation rate, thus increasing the pressure of the core, thus expanding the core against gravity, thus reducing the temperature (with the decrease in density), thus reducing the energy produced, and the pressure, and radius, and the system comes back to equilibrium.
   • This is an example of negative feedback in an astrophysical system. A postive change in one of the parameters causes changes in the system that feed back into a negative change in the original parameter, bringing the system back into an equilibrium. This is like a "cruise control" for the nuclear reactions in the star.
   • If the core is not generating thermonuclear energy, like in the original collapse of a protostar, then the gravitational energy of contraction generates heat, and thus pressure against gravity. There is still a negative feedback against temperature increases, though the core will slowly contract to generate the heat to maintain the pressure against the weight of the star.
   • Pressure-temperature regulation is only part of the story of stellar structure. We now consider the other factors that go into a stellar model.
3. Stellar Structure
   • There are 4 global physical quantities of a star: L=luminosity, T=temperature, M=mass, and R=radius.
   • A star can be modeled as a spherically symmetric gas that is in equilibrium, that is it doesn't change with time (at least on timescales short compared to the energy generation timescale).
   • We can build a star by calculating the physical properties as a function of radius r within the star.
   • There are 2 quantities that depend upon the shells below directly: L(r) the luminosity at any shell generated by the shells below it, and M(r) the mass within the radius r.
   • The temperature T(r) is a local quantity, the temperature of the gas in the shell at radius r.
   • To help us calculate, we also use 3 other local quantities: the gas mass density rho(r) in kg/m^3 at radius r, the gas pressure p(r), and the energy (luminosity) generation rate E(r).
   • We calculate the star model by adding shells consecutively from the center of the star (r=0) to the surface of the star (r=R). At the surface M(R),T(R),L(R) are just the physical quantities we can observer for the star.
   • We use four laws to tell us how to make a change in the quantities when we add a shell:
     1. Mass - add the mass in the shell to the total
     2. Luminosity - add the luminosity generated in the shell to the total
     3. Pressure - adjust the pressure to balance the weight of the new shell (reduce the pressure, since the shell contributes to the pressure of lower layers only)
     4. Temperature - heat is transported from the hot innner layers to cooler outer layers by convection, radiation or conduction.
   • This prescription is carried out through four equations that give the changes in each quantity with radius.
   • This purely mathematical model describes the structure of the star. Furthermore, you can take into account the changes with the star's energy production with time, and thus calculate the evolution of the star's structure.
   • The mass-luminosity relation (L/Lsun) = (M/Msun)^3.5 is an approximation to the solution of the equations of stellar structure for a star of mass close to that of the Sun. The actual exponent ranges from 2.5 to over 4 over the main sequence, and of course does not obey anything like this off the main sequence.
   • It turns out that the stellar structure that you find depends only upon the mass M of the star! If the energy is produced by fusion as given by E(r)=E(T(r),rho(r)), then the star falls on the main sequence and the surface temperature T, and luminosity L depend on the mass M such that they increase as you increase the mass.
   • Because higher mass stars are much more luminous, they burn their fuel faster, and have shorter lifetimes on the main sequence. Note that the lifetime t is proportional to M/L (fuel/rate of burning) and for stars of mass near the Sun's: t/tsun = (M/Msun)/(L/Lsun) = M/M^3.5 = 1/M^2.5 Thus, a star of twice the mass of the Sun would live only 18% of the life span of the Sun. A star of 0.5 Msun would live 5.6 times longer.
   • The most massive stars live only a few millions of years! (Short by cosmic standards.)
4. The Life Cycles of Stars
   • Stars have "lives" in that they are born out of dust and gas, grow under gravity, start burning nuclear fuel and become full-fledged stars, go through stages as different fuel sources are found, exhaust their energy and die.
   • Stars return much (in fact sometimes most) of their mass back to the interstellar medium, from which new generations of stars are born.
   • The Sun is 4.5 billion years old, while the oldest stars known appear to be 18 billion years old. The Sun is not a first generation star, it was formed from material that had been processed through at least one generation of stars that lived and died previously!
   • Almost all the elements heavier than helium were "created" from fusion in the cores of stars. The material that our bodies are made of was synthesized in the center of stars now long dead!
   • The understanding of the stages of a star's life comes from understanding the Hertzsprung-Russell (H-R) diagram of L vs. T.
   • A star like the Sun spends 90% of its life burning hydrogen on the main sequence. Eventually the hydrogen runs out and the star must change its ways. This is when things get interesting.
5. The Tale of the Core
   • When hydrogen is exhausted in the center (only about the innermost 10% of the stars mass is available for fusion as we will see), the core, which is now made of mostly helium, must contract to derive gravitational energy to keep up the pressure against gravity.
   • This contraction is slow, as the Sun could actually generate its current luminosity by contraction of 40 meters per year! Q: How long can the Sun keep this up before contracting to nothing?
   • As the helium core contracts, it heats up, and this ignites the unfused hydrogen in a shell around the core. This further liberates new energy which flows out into the envelope of the star.
   • The envelope structure was set up to be in equilibrium with the luminosity of the hydrogen burning core. Now it has extra energy pumped in at the bottom, and it is too dense to let all this new heat and radiation pass through. To compensate it will expand in radius, dropping in density, and thus in surface temperature at the outside.
   • The radius will increase greatly, by as much as 25 times! The star moves to the right (becomes cooler) in the H-R diagram.
   • Eventually the envelope will drop in density enough to allow the new luminosity to come out through the surface, and the star will increase in luminosity, moving upward on the H-R diagram.
   • Thus, after hydrogen core burning on the main sequence, a star move upward and to the right (higher L, lower T).
   • The star is much larger in radius, and is called a giant star. It is cooler, and thus redder in color, so it is commonly called a red giant.
   • Some stars are even larger than these giants, with radii almost a thousand times the radius of the Sun. These are called supergiants.
   • Supergiants are high mass stars that have evolved off the main sequence. Betelgeuse in the constellation Orion is a red supergiant with mass 15 Msun and radius of nearly 1000 Rsun (almost 4 AU!).
   • Blue giants and supergiants usually refer to high-mass (and thus high temperature) main sequence stars.

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