Assignments 8
Transcription
Assignments 8
Assignments 8 April 23, 2015 Exercise 1: In lecture 5-2, we derived expressions for the central pressure in terms of πΊπ! /π ! (slide 10). i. Show that π! = 4π ! ! π΅! πΊπ! ! π!! ! (hint, use the relation between the central and mean density). ii. Derive an expression for π΅! and calculate its value for π = 0, 1, 3/2, 2, 3, 4 Exercise 2: Here, you will calculate the minimum stellar mass for stars to reach the main sequence. A central temperature of 3x106 K is required for the first step in the protonproton chain. Assume a fully convective star, which you can approximate with a polytrope. i. Calculate the central density for a star that crosses the degeneracy border at 3x106 K. Together with the expression for the central density in terms of the mean density for polytropes, this will yield the first relation between the mass and radius of the star. ii. The second relation between stellar mass and radius follows from the expression for the central temperatures of polytropes. iii. Compare your results to the figure in the lecture notes. Exercise 3: The neutrino opacity for interaction with matter is about 10β20 cm2/g. What is the mean-free path (i.e. the distance between the scatterings) of the neutrino in the solar center? Compare this to the solar radius. Exercise 4: Estimate the density at which neutrons become degenerate.