ASTR 304 - 2009W [Week 4]

Navigation BarHome PageSyllabusBibliographyProblem SetsNewsContact Info

By week:

Week 4 - Neutron-Star Cooling and Heating [3] [5]

Summary

Neutron stars are born in the fiery explosion of a supernova. Although they are cold in the sense that the Fermi temperature is much greater than the thermodynamic temperature except in the outermost layers, neutron stars radiate like any hot bodies. In fact their interiors and crusts radiate neutrinos and their surfaces radiate soft x-rays. These soft x-rays are some of the few direct data that we get from neutron stars.

Reading List

  • ``Surface X-Ray Emission from Neutron Stars''
    [ ADS, APS, PDF ]
    REF: Chiu, H., Salpeter, E. E. 1964, Physical Review Letters, 12, 413-415 . Top

  • ``Analytical Models of Neutron Star Envelopes''
    [ ADS, PDF ]
    REF: Hernquist, L., Applegate, J. H. 1984, ApJ, 287, 244 . Top

  • ``Almost Analtyic Models of Ultramagnetized Neutron Star Envelopes''
    [ ADS, astro-ph/9805175, PostScript, PDF , Java ]
    REF: Heyl, J. S., Hernquist, L. 1998, Mon. Not. Royal Astr. Soc., 300, 599 . Top

Problem Set

Problem 1 - Thermodynamics and General Relativity Top

In general relativity if two bodies are in thermodynamic equilibrium,
T1
1 + z1 		= T2
1 + z2.
We can exploit this relationship along with Kirchoff's law to derive some interesting facts about how light travels from a neutron star to our telescopes. You can experiment with neutron-star lensing with this Java Applet.

  1. Because everything is in thermodynamic equilibrium, we can safely assume that the neutron star of mass M and radius R emits as a blackbody at a temperature T. Calculate the total power emitted from the neutron star surface in the frame of the neutron star surface.
  2. Calculate the total power received at infinity. Let the redshift of the surface be z.
  3. Let the space surrounding the star be filled with blackbody radiation in thermal equilibrium with the surface of the neutron star. You can imagine that the neutron star is in a gigantic thermos bottle. Let T∞ be the temperature of this blackbody radiation measured at infinity (i.e. z=0). What is T∞?

    4. Now here comes Kirchoff's law: in thermodynamic equilibrium a body emits as much as it receives.

  4. How much power does the neutron star absorb from the blackbody at infinity? This is the product of the surface area of the neutron star with the flux per unit area of the blackbody radiation.

    5. A conundrum: compare the answer to (2) with the answer to (4). They differ. Does the neutron star cool down because (2) is greater than (4)?

    The neutron star can't cool down because it is already in equilibrium, so one of our assumptions must be wrong.

  5. It turns out that the most innocuous sounding assumption is incorrect. The power that the neutron star absorbs is the product of its apparent surface area with the flux per unit area of the blackbody radiation. Let the apparent radius be R∞ and recalculate the answer to (4).
  6. Equate (2) and (5) and solve for R∞.

    7. R∞ ≠ R because in the vicinity of a neutron star light does not travel in a straight line. One can also derive the value of R∞ by solving for a null geodesic that is tangent to the surface of the neutron star.

  7. What is the minimum value of R∞ for a constant value of M? You will need to know that
    1+z = 1
    (1 - 2 G M/R c2)1/2
    What is the value of R? Call this radius Rγ.
  8. Prove the size of the image of the neutron star must decrease or remain the same as the radius of the neutron star decreases. Use the fact that rays that we ultimately see remain outgoing throughout their journey to us (otherwise by symmetry they would hit the surface a second time).
  9. What happens to the size of the image of the star if the radius of the star is less that Rγ?
  10. The calculation of the apparent radius of the star from thermodynamics hinges on the assumption that the outgoing flux from the surface reaches infinity. For R < Rγ, the size of the image no longer increases while thermodynamics says it should, so we must conclude that for radii less than Rγ initially outgoing photons can become incoming photons.

    From these arguments and spherical symmetry speculate what might happen to a photon emitted precisely at Rγ tangentially, i.e. neither ingoing or outgoing.

  11. Calculate how much radiation a star whose radius is less than Rγ will absorb.

    12. The answer to (11) falls short of (2) again. We know that the star can't heat up, so an assumption must be wrong. Within Rγ not every photon emitted can escape to infinity, many photons return and hit the surface.

  12. Using the answers to (2) and (11), calculate the fraction of the outgoing photon flux emitted from the surface that manages to escape.
  13. Use the fact that a blackbody emits isotropically to determine the opening angle of the cone into which the escaping photons are emitted. This region is symmetric around the radial direction.
  14. Pat yourself on the back. You have derived many of the quirky things about the Schwarschild metric (the metric that surrounds a spherically symmetric mass distribution). List the key assumptions that you have made to make this derivation work.

Problem 2 - Light-Envelope Neutron-Star Cooling Top

The presence of hydrogen in the atmosphere of a neutron star strongly affects the emission of the surface of the star and possibly how the star cools.

  1. Calculate the maximum possible density of pure degenerate ionized hydrogen gas (non-degenerate protons and degenerate electrons). What happens above this density?
  2. Assume that the electrons dominate the pressure, what is the pressure at this critical density?
  3. Assume that the gravitational acceleration is constant in this thin layer and is given by gs,141014 cm s-2. What is the column density of the layer?
  4. Assume that the stellar radius is R6 106 cm. How much hydrogen can you pile onto a neutron star?
  5. Assume that the cross section to x-rays per electron-proton pair is given by the Thomson cross-section. What is the column density of an optically thick layer of hydrogen (τ=1)?
  6. The conductivity of the surface layers of a neutron star is inversely proportional to the atomic number of the nuclei. How does the surface luminosity of neutron star change if you pour hydrogen onto its surface?

Last modified: Thursday, 08 April 2010 14:15:29