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Nuclear fusion in stars Paul Hickson October 19, 2016 Kelvin-Helmholtz mechanism In the late 19th century Lord Kelvin and Hermann von Helmholtz proposed a mechanism to explain the source of the energy radiated by the Sun. They reasoned that as the Sun radiated energy, it would cool, lowering the gas pressure (which according to the ideal gas law is proportional to T ). This would result in a slow contraction of the Sun, which would maintain the temperature and pressure needed for hydrostatic equilibrium, allowing the Sun to continue to radiate energy. With this mechanism, the source of the energy is gravitational potential energy. Kelvin-Helmholtz timescale Let’s estimate how long this mechanism could power the Sun. The internal gravitational potential energy of a body is defined as the energy needed to separate all the atoms and move them away to infinity, against their mutual gravitational attraction. It is negative because we have to do work to separate the atoms. For a homogeneous sphere of mass M and radius R, the internal gravitational energy is U “´ 3GM 2 . 5R If we divide this energy by the luminosity (the present rate at which energy is radiated), we get a characteristic time, the Kelvin-Helmholtz time tKH “ |U | GM 2 „ . L RL Powering the Sun Lets evalute the Kelvin-Helmholtz time for Sun, tKH » 6.671 ˆ 10´11 ˆ p1989 ˆ 1030 q2 “ 9.88 ˆ 1014 s “ 31 Myr. 6.955 ˆ 108 ˆ 3.838 ˆ 1026 This is much shorter than the geological age of the Earth, so there must be another source of energy. It is easy to show that chemical energy is insufficient. For example the reaction O ` 2H Ñ H2 O produces 1.26 eV of energy (2.02 ˆ 10´19 J) per oxygen molecule. The number of oxygen atoms in the Sun is about 0.00954Md {16mH “ 7.09 ˆ 1054 , so the total energy available is 1.43 ˆ 1035 J. This would only power the Sun for 1.43 ˆ 1035 J “ 3.7 ˆ 108 s “ 11.8 yr. 3.85 ˆ 1026 J/s Nuclear fusion The problem was solved by the discovery of nuclear fusion in the 20th century. The idea that fusion powers the stars was first proposed by the English astrophysicist Arthur Stanley Eddington in 1920. In 1939, German physicist Hans Bethe discovered the two main stellar fusion reaction sequences, for which he received a Nobel prize. Before examining these, lets review some properties of particles and light nuclei, Properties of particles and light nuclei Particle electron positron neutrino photon proton neutron deuteron tritium helium-3 helium Symbol e e` νe γ p n d, 2 H 3H 3 He 4 He, α Charge{e -1 1 0 0 1 0 1 1 2 2 Mass{mp 0.000544 0.000544 0.000000 0 1.000000 1.001378 1.999552 2.993718 2.993154 3.972600 where mp “ 1.672622 ˆ 10´27 kg is the mass of a proton. Fusion energy The rest-mass energy associated with mass m is given by Einstein’s equation E “ mc2 . From the table, we see that four protons are less massive than a 4 He nucleus (an α particle). Therefore, if we could somehow combine four protons to make a 4 He nucleus, there would be a release of energy equal to ∆E “ p4 ´ 3.972600qmp c2 “ 4.119 ˆ 10´12 J “ 25.71 Mev. This is more than 10 million times greater than chemical energy! We know that a fraction X “ 0.747 of the Sun’s mass is hydrogen. If it were all converted to helium, this could provide enough energy to power the Sun for t“ XMd ∆E E “ “ 2.3 ˆ 1018 s » 74 Gyr. L 4mp L Proton-proton chain The principal reaction that occurs in the Sun and lower-mass stars is the proton-proton chain (pp I). p`pÑ 2 H ` e` ` νe pˆ2q H`pÑ 3 He ` γ pˆ2q He ` 3 He Ñ 4 He ` p ` p 2 3 The net result of this is 4p Ñ 4 He ` 2e` ` 2νe ` 2γ. The first step in this chain is the most critical. When two protons collide they form a “diproton” which is unstable and quickly splits into two protons. Bethe realized that a small fraction of the time, one of the protons could decay into a neutron, positron and neutrino before the diproton can split, thus producing a deuteron. Proton-proton chain Several other branches also occur, for example the pp II branch is dominant at temperatures between 14 and 23 MK. 3 He ` 4 He Ñ 7 7 ´ Be ` e Ñ Li ` p Ñ 4 7 Be ` γ 7 Li ` νe He ` 4 He And the pp III branch dominates at temperatures above 23 MK, 3 He ` 4 He Ñ 7 8 Be ` p Ñ BÑ 8 8 7 Be ` γ 8 B`γ ` Be ` e ` νe Be Ñ 4 He ` 4 He The pp IV (Hep) branch occurs very rarely, 3 He ` p Ñ 4 He ` e` ` νe Proton-proton chain The neutrinos carry about 2% of the released energy. They interact with matter only very rarely, and generally escape from the Sun. The positrons annihilate with electrons, producing two 511 keV gamma-ray photons each e` ` e Ñ 2γ. All the photons have energies in the gamma-ray region of the spectrum. They interact strongly with free electrons by the process of electron scattering. Absorbed photons transfer energy to the gas, which re-emits photons with the Planck energy distribution, according to the local temperature. At the photosphere of the Sun, the temperature has dropped to about 5780 K. Photons emitted here can escape, producing the radiation that we see. CNO cycle Stars having central temperature higher than about 18 MK produce most of their energy by the CNO cycle, 12 13 N`γ NÑ 13 C ` e` ` νe C`pÑ 14 N`γ N`pÑ 15 O`γ OÑ 15 N ` e` ` νe N`pÑ 12 C `4 He C`pÑ 13 13 14 15 15 The net result of this is 4p Ñ 4 He ` 2e` ` 2νe ` 3γ. In this cycle, carbon acts as a catalyst. Nitrogen and oxygen are produced as byproducts (not all of these nuclei are consumed). Triple-alpha reaction The final reaction that we will discuss is the triple-alpha reaction, 4 He `4 He Ñ 8 8 Be `4 He Ñ 12 Be ` γ C`γ The 8 Be nucleus produced in the first reaction is unstable and splits into two 4 He nuclei in only 2 ˆ 10´16 s. However, if the temperature and density are high enough, a third 4 He nucleus may fuse with the 8 Be before it decays. Typically this requires temperatures greater than 100 MK. Other reactions also occur, producing elements with greater atomic mass, up to iron. Iron is the most stable nucleus, having the lowest binding energy, so it is not easily fused to form other elements. Stability of stars If fusion powers the stars, why don’t they explode like hydrogen bombs? Some do, but most are stable. The rate at which the nuclear reactions proceed depends very sensitively on the temperature in the core of the star. If the temperature increases, more energy is generated which makes the star expand. This in turn lowers the temperature, reducing the energy generation rate. So, most stars are self-regulating. The equilibrium is not always perfect. Some stars pulsate, periodically expanding and contracting. Examples are RR-Lyrae variables and Cepheid variables. Davis/Bahcall experiment In the late 1960s Astrophysicists Raymond Davis and John Bahcall set out to detect solar neutrions. Davis did the experimental work and Bahcall did the theoretical calculations Davis installed a 380 m3 tank of CCl2 (perchloroethylene) 1.5 km underground at the Homestake Gold mine in South Dakota. Raymond Davis, Jr. The reaction that they were looking for is νe ` 37 Cl Ñ 37 Ar ` e Davis was able to detect neutrinos, but only at 1{3 the rate predicted by Bahcall. John Bahcall Sudbury Neutrino Observatory In 1984, The American physicist Herb Chen realized that a neutrino detector employing heavy water would be sensitive to all neutrinos, not just νe . Following up on this, Canadian physicist Arthur McDonald led a project to build such a facility, deep in a mine in Sudbury Ontario. The Sudbury Neutrino Observatory, has a 12 m diameter acrylic sphere filled with heavy water, surrounded by 9,600 photomultiplier tubes that detect photons produced by neutrino reactions. SNO showed that the total neutrino flux agreed with Bahcall’s prediction, but that 2{3 of the νe produced in the Sun had changed into νµ and ντ by the time they reached Earth. SNO