• Summary
• Reading List
• Problem Set
• Answers

Week 3 - The Structure of Neutron Stars [2] [4]

Summary

Neutron stars are truly relativistic objects. You cannot understand their structure without general relativity and nuclear physics. The key observational quantities that probe this physics are the mass and radius of neutron stars. You can explore some more and less realistic equations of state using the Mass-Radius Relation Applet and the Neutron-Star Structure Applet.

• Monday 18 January,
  Thursday 21 January: Neutron-Star Structure

Reading List

• ``On Massive Neutron Cores''
  [ ADS, APS, PDF ]
  REF: Oppenheimer, J., Volkoff, G. 1939, Physical Review, 55, 374-381 .

• ``Neutron Star Structure and the Equation of State''
  [ ADS, PDF ]
  REF: Lattimer, J. M., Prakash, M. 2001, Astrophys. J., 550, 426-442 .

• ``Neutron Star Structure''
  [ ADS, PDF ]
  REF: Pethick, C. J., Ravenhall, D. G. 1998, .

Problem Set

Problem 1 - Newtonian Polytropes

1. Consider a star of mass, M, and radius, R. Construct by dimensional analysis a characteristic pressure and a characteristic density from these quantities and Newton's constant, G.
2. A polytropic equation of state is a power-law relationship between pressure and density, P = K ρ^α. Substitute the characteristic pressure and density into the polytropic equation of state to derive a mass-radius relation.
3. Which values of α have special properties? What are they?

Problem 2 - Central Pressures

We will calculate the central pressure of a incompressible star in Newtonian physics and general relativity.

1. Use the General Relativistic equations of hydrostatic equilibrium to determine the central pressure of a star of mass M and radius R. The material is incompressible, i.e. its density is constant. After integrating (9) from the center where the enclosed mass, u, vanishes, it is easiest to integrate (10) from the surface where the pressure vanishes. Write your answer as a function of M and R by eliminating the constant density from the result.
2. By dimensional analysis, figure out where the factors of G and c appear in the General Relativistic equations of hydrostatic equilibrium.
3. Derive the Newtonian equations of hydrostatic equilibrium by taking the limit of c → ∞.
4. Redo the pressure calculation in Newtonian physics.
5. By dimensional analysis, figure out where the factors of G and c appear in your answer to (1).
6. Check your answers by taking the limit of c → ∞ for your answer to (5) and comparing it with the answer to (4).
7. What is the minimal radius for a constant-density star of a given mass? What is the maximal mass for a star of a particular density? What is the maximal mass for a star at nuclear density, 10¹⁵ g cm^-3?

Problem 3 - Neutron Star Masses

Calculate from dimensional analysis the typical mass of a neutron star.

1. Use the characteristic density and pressure of a star that you derived in Problem 1. Neutron stars have relativistic neutrons so the pressure is about the density times c². Use this to derive a relationship between the mass and radius of the star.
2. A relativistic degenerate gas has a density of one particle in a cube a Compton wavelength on a side. Combine this with the result from Part 1 to solve for the mass of the star.