ASTR 304 - 2003W [Week 7]

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Week 7 - Millisecond Pulsars [6] [8]

Summary

Millisecond pulsars are among the most extreme denizens of the neutron star zoo. They spin up to 600 times per second (faster that a kitchen blender). How do they form? What can they tell us?

Reading List

  • ``A millisecond pulsar''
    [ ADS ]
    REF: Backer, D. C., Kulkarni, S. R., Heiles, C., Davis, M. M., Goss, W. M. 1982, Nature, 300, 615-618 . Top

  • ``Models for the Formation of Binary and Millisecond Pulsars''
    [ ADS, PDF ]
    REF: Bhattacharya, D. 1996, , , 243-256 . Top

Problem Set

Problem 1 - How much mass? Top

How much mass does it take to spin up a neutron star to a period of 1.6 milliseconds?
  1. Let us suppose that the radius of the inner edge of the accretion disk is given by r. What is the orbital frequency (ω) at the inner edge of the disk?
  2. Using Newtonian physics, what is the specific angular momemtum (the angular momemtum per unit mass) of material at the inner edge of the disk?
  3. Let us assume that the moment of inertia of a neutron star is given by I=0.2 M R2 and that R is constant as the mass of the star increases. What is the angular momentum of the star if its rotation frequency is Ω (M) ?
  4. What is the derivative of the angular momentum of the star with respect to mass, assuming that Ω is a function of M?
  5. Set dL/dM in part (3) equal to l/m in part (2) and solve for d&Omega / dM.
  6. This equation actually has an analytic solution (you are welcome to use Maple to find out). However, we don't want to work so hard, so let's estimate the various terms. Ω varies from zero to 4000 Hz. Take M=1.4 and R=r=10 km.
  7. You should find that one term in the numerator is much larger than the other. Neglect the smaller term and replace dΩ with ΔΩ = 4000 Hz and dM with ΔM. Solve for ΔM.
  8. We replaced the differential equation with a difference equation that we could solve algebraically. This won't give a large error if ΔM << M. Is this the case?

Problem 2 - The Birth Line Top

We will calculate where in the P-Pdot diagram neutron stars that have been spun up by accretion should end up. Our simple model is that if a neutron star is spinning faster than the material orbits at the inner edge of the accretion disk (rA) it will spin down. The star is in spin equilibrium if its spin rate is equal to the orbital frequency at the inner edge of the disk.
  1. What is the spin frequency of a neutron star in equilibrium with an accretion disk whose inner radius is rA?
  2. Substitute for rA using the expression from the lectures that gives rA in terms of μ, G, M, R and L.
  3. Now let's pretend that the accretion has stopped so we can use the radio pulsar expression for μ = Bp R3 for dipole spin-down. Substitute this expression for μ into the answer for (2), substitute Ω = 2π/P and solve for P-dot in terms of P.
  4. To make sense of this expression let M=m, L=1.3 (l m) x 1038 erg s-1, R=106 R6 cm, I=1045I45 g cm2 and sinα = 1. Does your birth line agree with the line in the lectures or the Bhattacharya paper?
Last modified: Tuesday, 06 April 2004 07:28:15