Today in Astronomy 142
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1 Today in Astronomy 142! Elementary particles and their interactions, nuclei, and energy generation in stars.! Nuclear fusion reactions in stars TT Cygni: Carbon Star Credit: H. Olofsson (Stockholm Obs.) et al. Explanation: TT Cygni is a cool red giant star with a wind. This picture was made using an array of millimeter wavelength radio telescopes and shows emission from carbon monoxide (CO) molecules. The central emission is from material blown off the red giant over a few hundred years while the thin ring, with a radius of about 1/4 light-year, actually represents a shell of gas expanding outward for 6,000 years. Carbon stars are named for their abundance of carbon containing molecules. The carbon is the dredged-up ashes of nuclear helium burning in the stellar interior. Carbon stars lose a significant fraction of their total mass in the form of a stellar wind which ultimately enriches the interstellar gas - the source of material for future generations of stars. TT Cyg is about 500 pc away. Astronomy 142 1
2 Energy and the Sun! Hydrostatic equilibrium and ideal-gas behavior ensure that the center of the Sun is very hot, and energy (in the form of light) is radiated from the center.! The high opacity of the Sun to light determines the rate at which the energy leaks out. As we have seen, it takes a long time for photons to diffuse from center to surface.! This cannot go on forever, without the Sun cooling down, or replacement for the energy that leaks away.! We know that the solar system is about 4.5x10 9 years old (from many radioisotope abundance measurements on meteorites), and that life has existed here for at least 3x10 9 years. Thus the Sun must have had close to its present luminosity for billions of years. Astronomy 142 2
3 Energy sources for the Sun! Cooling time is shorter than age of system, therefor an energy source is needed! Chemical energy doesn t work.! Gravitational energy is not enough.! An energy source of order a percent of is required. Nuclear fusion. burning Hydrogen into Helium. Astronomy 142 3
4 The four fundamental forces are gravity, electromagnetism, the strong nuclear force, and the weak nuclear force.! Gravity ˆr Always attractive, weak, and can t be shielded, so is very long ranged. Holds stars, solar systems, galaxies, and clusters together.! Electromagnetism Long ranged. Much F = GmM r 2 F = qq r 2 ˆr F = q E + v c B stronger than gravity, but is generally shielded over very large distances because it can be either attractive or repulsive (charge can have either algebraic sign). On large scales plasmas tend to be exactly neutral canceling out the force. Holds atoms and molecules together. Light is a form of electromagnetic energy. Astronomy 142 4
5 The four fundamental forces (continued) These two forces are insufficient to explain nature; there must be other attractive forces.! Example: protons in the helium nucleus repel each other electrostatically, and gravity is insufficient to overcome the repulsion. Strong nuclear interaction (no neat formula)! Short ranged: only acts over nuclear dimensions; that s why we don t notice it in everyday life.! Involves a different sort of charge than electromagnetism, that characterizes the strength of interparticle forces. The term for this sort of charge is color.! Always attractive, much stronger than EM, gravity. Astronomy 142 5
6 The four fundamental forces (continued) None of these three forces explains the slow transmutation of some nuclei: the phenomenon of radioactivity. Astronomy 142 6
7 The four fundamental forces (continued) Weak nuclear interaction (also no neat formula)! Even shorter range than strong interaction.! Much stronger than EM and gravity but weaker than the strong interaction by quite a bit.! Causes radioactivity. Along with the forces we distinguish two basic types of matter:! leptons, which take part in gravity, electromagnetic and weak interactions but not strong interactions;! quarks, which take part in all of the interactions.! All of these particles have spin of 1/2, in units of Astronomy 142 7
8 Elementary particles Leptons occur in pairs, the members of which are related by their weak interactions but differ in their mass and electric charge: electrons, muons, tauons, and their neutrinos have so far been observed. Quarks:! Six flavors, called up, down, strangeness, charm, bottom and top (u, d, s, c, b, t).! Three different kinds of the strong-interaction analogue of charge, or colors (usually called red, white and blue).! Individual quarks are never observed ( confinement ).! Nuclear particles are made up of two or three quarks or antiquarks (mesons and baryons, respectively). Astronomy 142 8
9 Elementary particles (continued) It has been shown experimentally and theoretically that the electromagnetic, weak and strong interactions can be thought of as different aspects of the same interaction, that manifest themselves in different energy regimes; this notion is called the standard model of elementary particles and their interactions. In the standard model there are related groups of leptons and quarks, known as generations: Normal atoms and nuclei only contain members of the first generation of leptons and quarks. Astronomy 142 9
10 Quantities conserved in the four interactions! Energy, momentum, angular momentum, etc.! Electric charge! Lepton number, separately for each generation of leptons; = +1 for particle and corresponding neutrino.! Baryon number Astronomy
11 Particles and antiparticles To each of the particles mentioned above corresponds an antiparticle.! Antiparticles have exactly the same mass and spin as the corresponding particle.! They have electric charge, lepton number, and baryon number opposite that of the corresponding particle. Notation: if antiparticle Examples: Astronomy
12 Upshot for stars: energy from nuclear fusion Binding of nuclear particles into nuclei represents a very large negative potential energy due to the strong interaction.! Therefore if free nuclear particles become bound through the strong interaction, energy/momentum conservation demands that the mass of the bound object is less than the sum of the masses of the free objects, and the deficit appears in other forms of energy, such as the kinetic energies of the products of the reaction: Large negative potential energy Conservation of lepton number, baryon number, momentum, etc. Kinetic energy of products Astronomy
13 Nuclear fusion of Hydrogen! Nuclear reactions which burn Hydrogen PP chains CNO cycle (higher temperature)! The rate of each reaction depends on The ambient temperature The composition Astronomy
14 Fusion of two protons Figure: Chaisson and McMillan, Astronomy Today. Astronomy
15 Fusion and E = mc 2 Energy is a form of mass; mass is a form of energy. Even when it s at rest and far from attracting or repelling bodies, a body has a total rest energy Two protons fuse to make a deuteron and two lightweight particles. The deuteron has a proton and a neutron (close to the same mass as the proton), but it also has the large negative potential energy from the strong interaction, so its rest energy is less than the sum of the proton rest energies, or equivalently its rest mass is less than that of the protons. This suggests a convenient accounting method for energy released in fusion processes. Astronomy
16 The proton-proton chains! Several different sequences of reactions that start with the fusion of two protons are the most important such reactions in main sequence stars; collectively they are called the proton-proton (pp) chains.! Here is pp chain I for example (70% of pp chain reactions):! Rest mass of products less than reactants, so the products have more kinetic energy than the reactants (heat!). Astronomy
17 Proton-proton chain I (PPI) Chaisson and McMillan, Astronomy Today. Astronomy
18 The proton-proton chains (continued) Application of all the conservation laws: Mass and baryon number in deuteron; extra charge (+) must be carried off by a non-baryon (i.e. an anti-lepton); neutrino added to conserve lepton number. Energy and momentum cannot both be conserved unless there is more than one particle in the final state. A neutrino-antineutrino pair would also work here, but would happen much less frequently. Astronomy
19 The proton-proton chains (continued) How much kinetic energy do the products have? The difference in rest mass between products and reactants gives the difference in binding (potential) energy: Compare to the average kinetic energy of particles in an ideal gas at 15.7x10 6 K: Astronomy
20 Nuclear catalysis: the CNO bi-cycle One branch, the CN cycle: is a catalyst: it s not used up in the reactions. -- same rest mass difference, and therefore kinetic energy of products, as pp chains. (Requires higher T, though.) Astronomy
21 Thermonuclear reactions The dominant thermonuclear reaction in a star changes with temperature. Collisions of nuclei at higher temperatures can overcome the repulsive force of heavier nuclei (or nuclei with more positive charge) MINIMUM TEMPERATURE 8x10 6 K 20x10 6 K 600x10 6 K REACTION Proton proton chain CNO cycle Carbon helium fusion 10 9 K Carbon burning Astronomy
22 Fusion by tunneling Astronomy
23 Temperature dependence of proton fusion rate A version of a calculation first done by George Gamow in the late 1930s. Consider in one dimension the fusion of two particles with masses m 1 and m 2, charges q 1 and q 2, speeds v 1 and v 2, and separation r. Their reduced mass is m = m 1 m 2 /( m 1 +m 2 ), and their relative speed is v = v 1 - v 2. Classically they can t get any closer together than r min, where But the quantum tunneling probability is equal to, where A is a constant and λ is the de Broglie wavelength: Astronomy
24 Temperature dependence of proton fusion rate (continued) Tunneling through a potential-energy barrier in quantum mechanics. In one dimension, with an infinite barrier of height U: Energy (x <0) = 2k 1 k 1 + k 2 e ik 1x U x 24
25 Temperature dependence of proton fusion rate (continued) The probability density p, integrated over the well on the other side of the barrier, gives the probability that an incident particle with energy E will be found there, on the other side. Note that = h mv where λ is the de Broglie wavelength of the incident particle. Astronomy
26 Temperature dependence of proton fusion rate (continued) So the probability of tunneling past a barrier of width r min can be written as = h mv r min = 2q 1q 2 mv 2 where B is a constant. The probability that the two particles have relative speed v is given by the Maxwell-Boltzmann distribution: where C is another constant. 5 February 2013 Astronomy 142, Spring
27 Temperature dependence of proton fusion rate (continued) so the probability of having tunneling and speed v is the product of these two probabilities: The fusion rate is proportional to this probability P. For what v is the rate largest? Find by setting derivative equal to zero: 5 February 2013 Astronomy 142, Spring
28 or Temperature dependence of proton fusion rate (continued) for fastest rate. Take parameters for proton-proton fusion: and we get, for T = 15.7 MK, 5 February 2013 Astronomy 142, Spring
29 Thus Temperature dependence of proton fusion rate (continued) 5 February 2013 Astronomy 142, Spring
30 Temperature dependence of proton fusion rate (continued) where Thus the fusion rate is very sensitive to temperature. Some numbers: Where T (K) P max /D Earth Sun s center Center of massive star 15.7M M February 2013 Astronomy 142, Spring
31 Implications! Fusion is still fairly slow in the centers of stars. (Good!)! Fusion rates are very sensitive to temperature; they are much higher at higher temperatures.! Fusion rates are much lower for larger values of nuclear charge. This is why the CNO cycles require higher temperatures than the pp chains in order to be significant energy sources.! Nucleosynthesis: fusion in stellar cores produces heavier elements out of hydrogen, in amounts that should tend to decrease with increasing atomic number and nuclear weight. 5 February 2013 Astronomy 142, Spring
32 Abundances of the elements, in the Sun O Mg Si Fe Data from the Clemson University Nuclear Astrophysics Group 5 February 2013 Astronomy 142, Spring
33 Hotter fusion and heavier elements Could stars in principle live forever simply by contracting gravitationally and increasing their temperature to ignite the next heavier source of nuclear fuel whenever they run out?! No. The strong interaction s range is smaller than the diameters of all but the smaller nuclei, but the range of the Coulomb interaction still covers the whole nucleus.! If nuclei get large enough the increase in electrostatic repulsion of protons becomes greater than the increase in binding energy from the strong interaction.! Thus there is a peak in the binding-energy-per-baryon vs. atomic mass number relationship, which turns out to lie at iron (Fe). 5 February 2013 Astronomy 142, Spring
34 Hotter fusion and heavier elements (continued) Binding energy per baryon Figure: Shu, The Physical Universe Atomic mass number Implication: once a star s core is composed completely of iron, it can no longer replenish its energy losses (from luminosity) by fusion. Stars therefore must die, eventually. 5 February 2013 Astronomy 142, Spring
35 Nucleosynthesis- Triple alpha reaction How are elements heavier than Helium produced? The Triple Alpha reaction (3 He s are involved) Carbon is formed in an excited state, originally predicted before it was known that this could happen. Requires temperatures of order 10 8 K. Astronomy
36 Nucleosynthesis (continued) The triple alpha reaction makes Carbon. Add a helium to Carbon and you get an Oxygen. Two carbons can make a Magnesium. To burn heavier elements generally require higher temperatures. Energy is released all the way up to the formation of Iron. Elements are burned at higher and higher temperatures in the core of a massive star until an Iron core forms. If the star doesn t reach high enough temperatures in its core then it can stop at Helium burning (lower mass stars). Eventually stars cannot burn anything more. So how are very heavy elements made in the universe? Astronomy
37 R and S processes Neutrons since they are neutral have an easier time overcoming the Coulomb barrier. The probability of getting within 1 Fermi is higher. Normally neutrons aren t free. What is an example of a reaction that produces free neutrons? By adding neutrons to nuclei, heavier and heavier elements can be produced. Astronomy
38 R and S processes By adding neutrons to nuclei, heavier and heavier elements can be produced. Astronomy
39 R and S-Processes (continued) R-process occurs when heavy bombardment by neutrons makes neutron rich nuclei which beta decay so that the nuclei are stable. decay time Final element ratios are dependent upon the beta decay times compared to the neutron capture times. Neutron capture figure: R=rapid, S=slow processes have different resulting element distributions. Astronomy
40 R, S-processes (continued) S process takes place in the centers of very hot stars. Free neutrons make heavier elements, but slowly. R-process takes place during a supernova and there is a huge flux of neutrons. The solar abundance pattern requires a mix of elements from both types of processes. Astronomy
41 Summary! Standard model particle physics grounding of models of stellar structure and nucleosynthesis.! Ways to burn hydrogen in stars: proton chains and CNO cycle.! Notation on nuclear reactions that help you check conservation laws! Quantum Tunneling estimate for temperature dependence of fusion reactions! Nucleosyntheis Astronomy
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