ASTR 304 - 2009W [Answers for Week 4]

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Week 4 - Neutron-Star Cooling and Heating

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.

Answer for Problem 1

  1. Pemiited = 4 π R2 σ T4
  2. Each photon loses energy by a factor of (1+z), and the rate of emission goes down by a factor of (1+z).
    Preceived = 4 π R2 σ T4 ( 1+z )-2
  3. T∞ = T / (1+z).
  4. Pabsorbed = 4 π R2 σ T4 ( 1 + z )-4
  5. Pabsorbed = 4 π R∞2 σ T4 ( 1 + z )-4
  6. R∞ = R (1 + z)
  7. First we calculate 1+z as a function of M and R.
    1 + z =
    		1
    ( 1 - 2 M
    R     		)1/2
    		 so R∞ =

    R
    ( 1 - 2 M
    R     		)1/2
    We take the derivative of R∞ with respect to R. This is (R - 3 M) R1/2 (R - 2 M)-3/2. So Rγ=3M and the minimum value of R∞ is 3 √3 M
  8. If we trace back the photons that we detect, they all pass through a sphere slightly larger than the surface of the star. We could pretend that this larger sphere was the surface of the star because all of the photons are outgoing, but unlike at the real surface of a star they don't necessarily fill the entire outgoing hemisphere. The image produced by this slightly smaller bundle of rays passing through this imaginary surface must be slightly smaller than if the rays filled the entire outgoing hemisphere. Putting this all together tells us at that the image produced by a star of a given radius must be smaller or equal in size to the image produce by a star of a slightly larger radius.
  9. The radius of the image must remain at 3 √3 M.
  10. It keeps going tangentially, i.e. the photon orbits the star. That is why R=3 M is called the photon sphere.
  11. Pabsorbed = 108 π M2 σ T4 ( 1 + z )-4
  12. The energy emitted is given by the answer to (2); the energy absorbed is given by (11), so the fraction of the photon flux that actually makes it to infinity is (11)/(2) or
    fescape = 27 M2
    R2    		 ( 1 - 2 M
    R    		)
  13. Calculating the opening angle is somewhat tricky. You have to remember that the outgoing flux is the integral of product of the intensity times the cosine of the angle with the radial direction over solid angle, so the total outgoing flux within an angle ψ of the radial direction is
    Fout(ψ) = ∫ &psi
    (I cos θ) 2 π sin &theta d θ = π I ( 1 - cos2 ψ ) = π I sin2 ψ
    0
    and the total outgoing flux is
    Fout = π I
    so
    sin2 ψ = fescape = 27 M2
    R2    		 ( 1 - 2 M
    R    		)
  14. You needed to know the value of the gravitational redshift as a function of radius; this is related to g00. You had to assume that thermodynamics holds in general relativity, that the spacetime surrounding a static neutron star is static, spherically symmetric and asymptotically flat.

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 14:15:28