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% (C) 2010 by Eugeniy Mikhailov, <evgmik@gmail.com>
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\hw{12}

\problem{Problem 1 (5 points)}
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Using Sch\"onberg-Chandrasekhar limit estimate the mass of the iron core
assuming that the mass of the star is 100~M$_\odot$ and that outside of the
core material consist of ionized hydrogen only.

\problem{Problem 2 (5 points)}
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Assume that outer layers of the star from above are blown away and all we
left is the isothermal iron core with temperature $10^8$~K.
At what size of the core electrons degeneracy
pressure overcomes the ideal gas pressure?
Make the same calculation for neutron degeneracy pressure.

\problem{Problem 3 (5 points)}
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Assuming ingoing collapse, will the core from the above problem able to
resist gravitational collapse?

\problem{Problem 4 (5 points)}
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Derive the exact formula for degeneracy pressure due to electrons
(eq.~16.12). Do not use prior text book derivation based on uncertainty
principal (it gives wrong numerical factor). Assume temperature of the gas
to be zero.

Hints: It is convenient to derive using concentration of electrons $n_e$ and
arrive to the formula between eqs 16.11 and 16.12.  You will need to know
number of electron with given energy $n(E)dE$, we derived such formula for
total number of fermions with energy $\leq$ to a given one (i.e.
$\int_0^{E_f} n(E) dE = N(E_f)$). You will need to
differentiate it. Find first total energy of such gas, then use formula
$P V = (2/3) E$, where E is total energy of the gas.

\problem{Problem 5 bonus (5 points)}
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The section 15.3 of the text book describes observations of SN 1987A
neutrinos arrival. Neutrinos arrive to Earth 3 hours before photons hit the
Earth. How would you explain that light which is supposedly the fastest was beaten
by neutrinos?


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