Astrophysics 1. ADVANCED and NORMAL (15 marks) (a) From the fact that Saturn’s rings are reflective to ground-based radar transmissions, we know that the chunks of ice making up the rings are comparable to, or larger than, the radar wavelength of about 0. 1 meters. Since the rings cast shadows on the planet, we know they are optically thick at the wavelength of visible light. Plus other evidence suggests that, despite being more than 250, 000 km in diameter, the rings are no more than 10’30 m thick. (i) Assume the ice chunks are spheres of radius 0. 1 m, and that the thickness of the rings is 20 m. If the rings are just optically thick, estimate the number density of ice chunks in the ring, and hence the average distance between chunks.
Explain your reasoning, and comment on your result. (ii) The dense main rings have an inner radius of 68, 000 km and an outer radius of 140, 000 km. If all the ice in these rings was combined into a single solid object with the same density, how big would it be? (b) The first step in the pp chain is the reaction p + p? d + e + +? where d is the deuteron, 2H. The estimated cross-section for this reaction is s = 8 Г— 10-52 m2. Assume we try to measure this cross-section in the laboratory using an intense beam of protons with a circular cross-section that is 2. 0 mm in diameter. If the beam is incident on a dense hydrogen target, with a surface density of 1024 protons per m2, and the rate of incoming protons is 1015 s-1, estimate the average time between reactions. 2. NORMAL ONLY (15 marks) (a) Use the Saha equation to determine the fraction of hydrogen atoms that are ionised, NII/Ntot, at the centre of the Sun.
Here the temperature is 15. 7 Г— 106 K and the number density of electrons is about ne = 6. 1 Г— 1031 m-3. (Use ZI = 2, and ZII = 1, since ionised H has only one state). Comment on your result. Does it agree with the fact that practically all of the Sun’s hydrogen is ionised at the Sun’s centre? (b) Why do sunspots appear dark, even though they are hot? The average temperature of sunspots is 4500 K, while the Sun’s overall effective temperature is 5770 K (derivedin class). How much brighter is a section of the Sun’s surface than a sunspot of the same area? (c) Estimate the hydrogen-burning lifetimes of stars near the lower and upper ends of the main sequence. The lowest-mass stars have M = 0. 07 M? and L = 4Г—10-5 L?, while the most massive stars have M = 100 M? and L = 2 Г— 106 L?. Assume that the 0. 07 M? star is fully convective so that, through convective mixing, all of its hydrogen becomes available for burning, rather than just the inner 10%.
Hint: The lifetimes of stars depend on one of the timescales we talked about in the first stellar astrophysics lecture. 2 4. ADVANCED and NORMAL (15 marks) (a) In order for nuclear fusion to take place, the nuclei of atoms must collide, which means they need enough energy to overcome the Coulomb barrier. We make the assumption that this energy is provided by the thermal energy of the gas, which is at temperature T. By equating the Coulomb potential of two particles with charge q1 and 12 U = 1 4pe0 q1q2 r with the average kinetic energy of a particle (3 2 kT), write down an expression for the temperature required for two protons to approach within a distance r. (b) In classical physics, for the protons to fuse they need to approach within a distance roughly the size of a nucleus, where the strong nuclear force acts to bind nucleons together. If this distance is r = 10-15 m, find the temperature required for fusion to take place.
Compare this with the central temperature of the Sun (Tc 1. 5 Г— 107 K). (c) In quantum mechanics, the position and momentum of the particle cannot both be specified exactly, and particles are able to В«tunnel» through the Coulomb barrier. Assume the protons need to approach within one De Broglie wavelength, ? = h/p in order for this tunnelling to take place. By finding the wavelength at which the kinetic energy of a particle with momentum p equals the barrier height, and using your result from (a), write down an expression for the temperature required for quantum mechanical fusion to take place. (d) Calculate this temperature, and commment.