Guest Lecturer: Dr W J Chaplin Somak Raychaudhury www.sr.bham.ac.uk/~somak/Y2SiU/ Lecture 11 Guest Lecturer: Dr W J Chaplin Nuclear energy generation in stars Revision: Central temperature and pressure in a star Can fusion occur at the centre of stars: overcoming Coulomb repulsion Gamow’s solution The p-p chain

Revision: Density, temperature, pressure within a star The mass density profile of the Sun looks like this A typical model would be Likewise temperature and pressure would peak at the core of the star

Revision We derived the equation of hydrostatic equilibrium, balancing gravity and thermal pressure Central pressure of a star like the Sun Central temperature of the Sun

Atomic nuclei Usually we express the mass of nuclei in atomic mass units u, defined to be 1/12 the mass of 12C (the 12 is the mass number A. 12C has 6 protons, 6 neutrons and 6 electrons) 1u=1.66054x10-27 kg = 931.494 MeV/c2 The mass of 1 proton + 1 electron is 1.0078285u Note 6 p + 6 n + 6 e- = 12.099.

Binding energy The mass of an atom (protons+neutrons+electrons) is not equal to the mass of the individual particles. There is also a binding energy associated with the nucleons themselves. The mass of the Carbon-12 atom: 6 p + 6 n + 6 e- = 12.099. The mass difference is 0.099u, equivalent to 92.22 MeV! This is the binding energy of the C-12 atom

Einstein’s relation: E = mc2 Energy is released in fusion reaction if the sum of masses of initial nuclei is larger that the mass of the final nucleus hydrogen mp + mp hydrogen Positron (antielectron) Deuterium MD + me < 2 mp M = 2 mp- MD - me Energy released E = M c2 neutrino Einstein’s relation: E = mc2 Deuterium has larger binding energy than protons (more tightly bound)

Energy and nuclear reactions Proton rest energy When one proton and one neutron fuse to form a Deuteron nucleus, the final mass is less than the sum of the mass of the four particles. The deficit is the “binding energy”, amounting to 2.2 MeV Here 0.1% of the mass of the particles is being converted to energy

|Ub| There are no heavy elements in the stars Figure 7.16: The red line in this graph shows the binding energy (the energy that holds an atomic nucleus together) for all the different atoms plotted by atomic mass (the number of protons and neutrons in their nucleus). Both fission and fusion nuclear reactions move downward in the diagram (arrows) toward more tightly bound nuclei. Iron has the most tightly bound nucleus, so no nuclear reactions can begin with iron and release energy. There are no heavy elements in the stars

Nuclear time-scale What if you could convert the entire mass of the Sun into energy? At the current luminosity of the Sun, this would be spent in If 0.1% of the mass is converted to energy, the Sun could still last for 1010 yr if powered by nuclear fusion energy. The Sun is currently 4.5 x 109 yr old.

Nuclear energy: fusion Nuclear energy is sufficient to sustain the Sun’s luminosity. But can it actually occur naturally in the Sun?

Coulomb repulsion The repulsive force between like-charged particles results in a potential barrier that gets stronger as the particles get closer: The strong nuclear force becomes dominant on very small scales, 10-15 m What temperature is required to overcome the Coulomb barrier?

Protons should be hot!

Statistical mechanics If the gas is in thermal equilibrium with temperature T, the atoms have a range of velocities described by the Maxwell-Boltzmann distribution function. The number density of gas particles with speed between v and v+dv is: The most probable velocity: The average kinetic energy:

Overcoming the Coulomb barrier Fusion is possible if the average particle kinetic energy (3/2 kT) is equal to or greater than the Coulomb potential energy: For two protons of z1=z2=1 separated by a typical distance of r=10-15 m This is much larger than the central temperature of the Sun

Maxwell-Boltzmann doesn’t help The central temperature of the Sun is The KE of a proton at this temperature is ≈ 2 keV The electrostatic PE of two protons 10-15 m apart is 1 MeV Could the protons at the tail end of the Maxwell-Boltzmann distribution of energies have sufficient kinetic energy to overcome the Coulomb barrier? Energy The relative fraction of protons with thermal energy of 1 MeV is only

Quantum mechanics to the rescue The answer lies in quantum physics. The uncertainty principle states that momentum and position are not precisely defined: The uncertainty in the position means that if two protons can get close enough to each other, there is some probability that they will be found within the Coulomb barrier. This is known as tunneling. The effectiveness of this process depends on the momentum of the particle

Quantum mechanics to the rescue What temperature is required for two protons to come within one de Broglie wavelength of each other?

Quantum tunneling Approximately: tunneling is possible if the protons come within 1 de Broglie wavelength of each other: For two protons, at T~107 K So the protons don’t need to get anywhere near 10-15m before they can begin to tunnel past the barrier Without this quantum effect, fusion would not be possible in the Sun and such high luminosities could never be achieved.

Nuclear reactions So – what are the specific reactions are we talking about?? The probability that four H atoms will collide at once to form a single He atom is exceedingly small. Even this simple fusion reaction must occur via a number of steps.

Proton-proton cycle Figure 7.17: The proton–proton chain combines four protons (at left) to produce one helium nucleus (at right). Energy appears as gamma rays and as positrons, which combine with electrons to convert their mass into energy. Neutrinos escape, carrying away about 2 percent of the energy.

Proton-proton chain (PPI) The net reaction is: But each of the above reactions occurs at its own rate. The first step is the slowest because it requires a proton to change into a neutron: This occurs via the weak force. The rate of this reaction determines the rate of Helium production

Proton-proton chain (PPII and PPIII) Alternatively, helium-3 can react with helium-4 directly: In the Sun, this reaction occurs 31% of the time; PPI occurs 69% of the time. Yet another route is via the collision between a proton and the beryllium-7 nucleus This reaction only occurs 0.3% of the time in the Sun.

The triple-alpha process The burning of helium occurs via the triple alpha process: The intermediate product 8-beryllium is very unstable, and will decay if not immediately struck by another Helium. Thus, this is almost a 3-body interaction Note the very strong temperature dependence. A 10% increase in T increases the energy generation by a factor 50.

Nucleosynthesis At the temperatures conducive to helium burning, other reactions can take place by the capturing of a-particles (He atoms).

Nucleosynthesis The binding energy per nucleon describes the stability of a nucleus. It is easier to break up a nucleus with a low binding energy.

The solar neutrino problem

Neutrino have zero or very small mass and almost do not interact with matter 10,000 years

Neutrino image of the Sun

400,000 liters of perchlorethylene buried 1 mile deep in a gold mine The Davis experiment 400,000 liters of perchlorethylene buried 1 mile deep in a gold mine About 1 Chlorine atom per day is converted into Argon as a result of interaction with solar neutrino Figure 7.18: (a) The Davis solar neutrino experiment used cleaning fluid and could detect only one of the three flavors of neutrinos. (Brookhaven National Laboratory) There are 1032 Cl atoms in a tank! Much more difficult than finding a needle in a haystack!!

Sudbury neutrino observatory: 1000 tons of heavy water D2O

32,000 ton of ultra-pure water 13,000 detectors

Observed neutrino flux is 2 times lower than the theoretical prediction!

Neutrinos should have mass Particle physics models should be modified The problem has been finally solved just recently: Neutrinos “oscillate”! They are converted into other flavors: mu and tau neutrinos Neutrinos should have mass Particle physics models should be modified