Assignments 8 April 23, 2015
Exercise 1: In lecture 5-2, we derived expressions for the central pressure in terms of 𝐺𝑀! /𝑅!
(slide 10).
i.
Show that 𝑃! = 4𝜋 ! ! 𝐵! 𝐺𝑀! ! 𝜌!! ! (hint, use the relation between the
central and mean density).
ii.
Derive an expression for 𝐵! and calculate its value for 𝑛 = 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 10−20 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.