Question

Problem 3 - Mean free path (8 marks) Let's assume the opacity inside of a star is constant. The optical depth is T = not where o is the collisional cross-section of an ion. If the optical depth is T = 1, then the photon can't go very far before it's scattered. We write the mean free path as L which is the average distance a photon can go before being scattered. a. If the conditions in the Sun have photons scattering off of densely packed ions, the number density is n= 1.78 x 102 cm-3 and o = 1.06 x 10-27 cm?, find the mean free path of a photon in the center of the Sun. b. A particle undergoing multiple scatterings moves a distance d = Nl where N is the number of scatterings. Find the number of scatterings N a typical photon will go through making its way from the core to the surface by taking d to be the radius of the Sun. c. As the photon goes from the core to the surface it meanders around like a drunk person walking down the street. The photon undergoes N scatterings, and the distance between each is typically l. If the mean free path remained constant for the photon's journey to the surface, a typical photon actually travels a distance D = NE. Calculate this distance using your answer for N from part b and l from part a. Since a photon moves at the speed of light, find the average time it would take for the photon to travel this path. This is the length of time it takes for a photon to escape from the Sun. Express your answer in seconds and years. This problem shows that even radiative transport may be inefficient!

Answer 1

mean free path i ne number density. 6 Collisional cross-section of an son. na 1. 78 810 29 cm3 6 = 1.06X60 27 cm2 cm l = 1.78×10 9 x 1.06410-27 = 5. 2999 xco com .. ~ 5.3 xco 3 cm. This ta is the mean free path of a in photon the centre of the sum.

. d=Nt. et mean free path. N Number of scater. d from the core too the surface. distance. radius of the sum= 69.63 4 X 101 in = a : No (2) 2 Number of scattering t) = (676 calento 16968108 ) - 1:724x100 5.34103

Photon's traves a distance Da Ne = 11784x126 x 5.3 403 = 9:137x10 cm. velocity of Photoni = C= 3x 103 x 10 cm/s. • Average time takes to travel o distana 23 9.137X10 sce = 3.048 x 10 see - 9650774 years 3x10