# Solutions for Chapter 18 Problems

1.) (Problem adapted from textbook:) Which one of the following is sensible:
a.) Most white dwarf stars have masses that are similar to that of our Sun,
but a few white dwarf stars are up to 3 times more massive than the Sun.
b.) If you want to find a pulsar, a good place to look would be near a
supernova remnant.
c.) The interstellar matter that surrounds black holes is gravitationally
attracted to the black holes, rapidly falls into black holes, and makes
X-rays that we can detect from Earth when the matter hits the black holes.
Answer: "b" is sensible. When a massive star explodes in a supernova
explosion, it can leave a pulsar behind. So, it is sensible to look for
pulsars near supernova remnants. An example of a pulsar found near the
remnant of an observed supernova is the Crab pulsar,
which was found near the Crab nebula. Note that massive stars aren't the
only kind of stars that can explode in supernova explosions. White dwarfs
in binaries can also explode in supernova explosions. These do not leave
behind pulsars. So, not all supernova remnants will contain pulsars.

2.) There are two ways to make supernova explosions. White dwarfs
make Type Ia explosions. Massive stars make Type Ib and Type II
explosions. Describe the differences in conditions between these two ways
to make supernova explosions.
1.) A white dwarf is very different from a massive star. A white dwarf
has already burned up its available fuel, collapsed, and shed the
outer parts of the star. In contrast, a massive star (which is about to explode)
is finishing off all of its available fuel. It has not yet collapsed.
2.) In order for a white dwarf to have a supernova explosion, it has to receive
material from a companion star. It has to receive so much material that its
total mass rises above 1.4 x MSun. In contrast, a massive star does not require
mass transfer from a binary companion in order to make it explode immediately.
3.) A white dwarf star explodes for a different reason than a massive star explodes.
As the white dwarf receives mass from its companion, its temperature rises.
Eventually, it gets high enough to suddenly burn all the material in the white
dwarf. The nuclear burning releases a huge amount of energy. (Note: An
explosion is equal to the sudden release of a huge amount of energy.)
In contrast, when a massive star explodes, it does so because the core suddenly
collapses (under the weight of gravity) and so releases enormous amounts of
gravitational potential energy.

3.) What is the idea behind the Chandrasekhar limit on the mass of a white dwarf?
Answer: The Chandrasekhar limit (Mstar cannot exceed approximately 1.4 x MSun)
is imposed because the electrons inside of white dwarfs cannot move faster than
the speed of light. Here is the logic: More massive white dwarfs are smaller than
less massive white dwarfs. Thus, their electrons are more tightly packed. The tighter
the electrons are packed, the faster they must move. (The reason for this was discussed
in Chapter S4, which was not assigned.) The electron speed would be as large as the
speed of light for a white dwarf having about 1.4 x MSun. Since the speed of light
is the limit, the electrons do not go faster than that limit. As a result, if you squeeze
such a white dwarf, you do not increase its electron degeneracy pressure. Thus,
we have found the limit to the amount of electron degeneracy pressure. If a companion
star were to dump enough material onto a white dwarf to push its mass above 1.4 x MSun,
then the white dwarf would have more gravitational pull inwards than the electron
degeneracy pressure could counteract. Thus, the white dwarf would collapse.

4.) What 'holds up' a neutron star?
a.) neutron degeneracy pressure
b.) thermal pressure
d.) none of the above

5.) Why do we see pulses from pulsars?
a.) we see light released after periodic nuclear burning of material which flowed
onto the neutron star from a companion star
b.) we see light from one of the pulsar's magnetic poles when the pulsar's rotation
brings one of its magnetic poles into our view
c.) we see light made when the neutron star periodically contracted and converted
gravitational potential energy into radiative energy
d.) none of the above

6.) How are novas similar to X-ray bursts?
1.) Both novas and X-ray bursts happen on
degenerate stars. Novas happen on white dwarfs, which are supported by
electron degeneracy pressure, while X-ray bursts happen on neutron stars,
which are supported by neutron degeneracy pressure.
2.) Both novas and X-ray bursts are due to flashes of nuclear burning on
the surface of the degenerate star.
3.) The fuel for the flash of nuclear burning is due to material flowing from
an accretion disk onto the star.
4.) The accretion disk that provides the fuel that burns in a flash that makes
the nova or X-ray burst is made up of material that was gravitationally
pulled off of a binary companion star.

7.) Regarding the size of a black hole:
a.) What is the physical reasoning behind the "Schwarzschild radius" (i.e. event horizon)?
Answer: The Schwarzschild radius is the "dividing line" between the region nearer to the
black hole, where even something as fast as a photon cannot escape, and the region further
from the black hole where light can escape the black hole's gravitational pull. In order
to escape the black hole's gravitational pull, the photon or object has to be moving faster
than the escape velocity.
b.) What is the equation for a "Schwarzschild radius"?
Answer: RS = (2 G M) / (c2)
c.) If we could compress the Sun so that it would fit into the trunk of a car
(for the sake of argument, let us say that is 1 m3), how big would its event horizon be?
You can either use this formula:
RS = (2 G M) / (c2)
= 2 x 6.67 x 10-11 (m3/(kg sec)) x 1.97 x 1030 kg / (3 x 108 m)2
= 3000 meters = 3.0 kilometers
or, you can use another formula in the book, one in which they have already converted
most of the units for you:
RS = 3.0 x M / MSun kilometers
= 3.0 x (MSun / MSun) kilometers
= 3.0 kilometers
d.) Would the Schwarzschild radius be inside of or outside of the newly-squished Sun?
Answer: The event horizon would be outside of the newly-squished Sun.

8.) Regarding black holes:
a.) there are theoretical explanations for black holes, but no observational evidence
b.) there is observational evidence for black holes, but no theoretical explanations
c.) there are both observational evidence and theoretical explanations for black holes