Problem Set 6

ASTR 3730

Read Chapters 15, 17, 18

1. Calculate the moment of inertia of a neutron star (I=2MR^{2}/5).
Calculate the energy of spin of the Crab Pulsar (E=Iw^{2}/2) where the period is 0.033 seconds. The Crab is
slowing at a rate that will stop its spin in 2500 years. Calculate its power
lost.

2. The acceleration g of a freely falling body near a black hole is given by:

Show that far from the event horizon this reduces to the usual Newtonian form. Show the force becomes infinite at the event horizon.

3. Let dt_{¥}
be the time interval elapsed for an observer at infinity. The elapsed time for
a local observer is:

Show that a local observer far from the black hole ages at the same rate as the observer at infinity, and that as the local observer approaches the event horizon aging stops as observed from infinity.

4. A particle falling from infinity takes time

to fall from radius r to the event horizon at radius R. Calculate dt for a particle falling from 10km onto a one solar mass black hole.

5. Using:

as derived by Hawking, calculate the evaporation timescale for a one solar mass black hole.

6. If you were to take a spectrum of a reflection nebula, would you see absorption lines, emission lines, or no spectral lines? How would this help to show that the illumination is by reflection from the central star?

7. To calculate the size of a Stromgren sphere,
idealize the problem by considering a pure hydrogen gas of uniform density
which surrounds a single hot star. Let N_{*} be the number of
ultraviolet photons beyond the Lyman limit which leave per unit time from the
central star. Assume that each such photon will ultimately ionize one and only
one hydrogen atom. Let R be the number or recombinations
of protons and electrons into hydrogen atoms per unit volume per unit time. In
a steady state, the total number of recombinations in
the Stromgren sphere of radius r must balance the
total number of ionizations:

R(4pr^{3}/3)=N_{*}

Given R and N_{*}, this equation would allow us to find r. To obtain
R, let us note that recombination at interstellar densities is a two-body
process (involving for each recombination one proton and electron). Thus the
number of recombinations per unit volume R must be
proportional to the product of the number densities of protons and electrons, n_{p}n_{e}. The proportionality factor is
denoted by a, and it is called the
"recombination coefficient." Thus

R=an_{p}n_{e}=an_{e}^{2}

Where we require n_{p}=n_{e} for
overall charge neutrality. Show now that the Stromgren
radius r is given by the formula

r = (3N_{*}/4pan_{e}^{2})^{1/3}

The "recombination coefficient" a
is a function of the temperature of the hydrogen plasma. For temperatures
characteristic of Galactic HII regions, a
has the approximate value 3x10^{-13} cm^{3}sec^{-1}.
Assume n_{e}=10cm^{-3}; compute r when N_{*}= 3x10^{49}
sec^{-1} (O5 V star), N_{*} = 4x10^{46}sec^{-1}
(B0 V star) and N_{*} = 1x10^{39} sec^{-1} (G2V star).
Convert your answer to light-years. What types of main-sequence stars have
appreciable HII regions?

8. Suppose a spherical shell identified as a supernova remnant is observed
with radius *r* and with outward expansion speed *v*. Assume the mass
density of the ambient medium to have the uniform value r_{0}; then the supernova remnant must have swept up
mass M = r_{0}4pr^{3}/3. Let the original mass M_{0}
be ejected at speed v_{0}. If we ignore communication between different
parts of the shell (via the thermal pressure of the hot interior), and suppose
that each piece of the shell preserves its outward linear momentum as it sweeps
up more material initially at rest, we have the snowplow model. Show that the
snowplow model implies

(M+M_{0})v=M_{0}v_{0}

The original kinetic energy E_{0} of the ejected material equals M_{0}v_{0}^{2}/2.
The present kinetic energy E of the shell equals (M+M_{0})v^{2}/2. Show that the ratios E/E_{0} and
v/v_{0} are given by

E/E_{0}=v/v_{0}=M_{0}/(M+M_{0})

In a typical supernova explosion, M_{0}=4M_{¤} worth of matter might be ejected
from the central star at speeds v_{0}=5000km/s. What is the original
kinetic energy E_{0}? How much mass M would have to be swept up to
bring the supernova shell speed *v* to a value of 10km/s, which is typical
of the random velocities of t interstellar clouds? In the process, by what
factor is E reduced from E_{0}? What must happen to the lost energy in
the snowplow model?

What is the radius *r* when *v* =10km/s if r_{0}=2x10^{-24}g/cm^{3}? Convert your
answer to parsecs.

For M>> M_{0}, show that *v* is inversely proportional to
r^{3}. With the time-rate change of radius proportional to r^{-3},
show the age t of the remnant will four
times shorter than had it expanded to radius *r* at a constant speed.
Compute t for our hypothetical
supernova remnant.

9. a) At what velocity must a hydrogen atom hit an ice cube to sputter off a
water molecule, given that the boiling point of water is 400K?

b) How far from a one solar mass star does the orbital velocity equal the
velocity of part a?

c) The temperature of Saturn is about 100K. How massive would an accretion core
have to be (at 100K) to hold a hydrogen atom against thermal escape? Assume a
core density of 3g/cc. How does this mass compare to Saturn's mass?