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Week 12 - Understanding Cosmological Gamma-Ray Bursts [11] [13]
Summary
The central engines of gamma-ray bursts are completely hidden by the
gamma-ray emission from the bursts themselves, but gamma-ray emission
and other observations give clues to the nature of power behind
GRBs, specifically the collapse of the core of a massive star and the
subsequent hyperaccretion onto the central black hole.
Problem Set
Problem 1 - Burst Rates 
We are going to calculate the fraction of stars that are born that will
end up as binary neutron stars or collapsars. We will use the Salpeter
IMF: dN = A M-2.25 dM for
M>0.1  .
-
Black holes: what fraction of stars has initial masses greater than
20 ?
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Neutron stars: what fraction of stars has initial masses greater than
8 and less
than 20 ?
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In steady state, if a galaxy has a star-formation-rate of
1 /yr, what is
the rate of supernova that produce black holes and neutron stars?
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Binary neutron stars: let's assume that half of all stars are in binaries and
that stars pick their companions randomly,
what fraction of neutron stars will have companions that will become neutron
stars?
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Finally, let us assume that only half of binaries survive a supernova
explosion, what is the rate of binary-neutron-star mergers in
the galaxy?
-
What is the mean star-formation rate of the universe?
Take Ω*=0.00245. You want a round number.
What is the rate of GRBs in the two models?
Problem 2 - Neutrino-Eddington Limit
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Use the cross section for neutrino pair-production as an estimate of
the cross for a neutrino to scatter of an electron. What is the
neutrino Eddington limit to the luminosity as a function of the mass
of the star in solar masses and the energy of the neutrinos?
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What is the Eddington limit to the accretion rate?
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Use this Eddington-limited accretion rate to estimate the maximum value
of Γ for a gamma-ray burst.
Problem 3 - The Millisecond Magnetar
A neutron star is born spinning with a period of 1.6 milliseconds. It has
a magnetic field of 1016 G.
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What is P-dot for the magnetar when it is born?
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What is the initial spin-down luminosity of the magnetar?
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Does the spin-down luminosity of the magnetar increase or decrease with
time? What does this mean in the context of the internal shocks model
for gamma-ray burst emission?
Reading List
- ``Spectral constraints on the redshift of the optical counterpart to the gamma-ray burst of 8 May 1997.''
[
ADS,
PDF
]
REF: Metzger, M. R., others, 1997, Nature, 387, 878-880 .
- ``Gamma-ray bursts from stellar mass accretion disks around black holes''
[
ADS,
PDF
]
REF: Woosley, S. E. 1993, Astrophys. J., 405, 273-277 .
Last modified: Tuesday, 06 April 2004 07:28:12
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