29:62 General Astronomy
Second Hour Exam
March 12, 1997
All problems are worth 10 points. Write
legibly, preferably in pen. Good luck and no whining.
(1) Calculate the density of a neutron gas at which the Fermi energy
equals the rest mass energy of a neutron. The rest mass energy is defined
as the amount of energy which would be released if an object were entirely
converted to energy. What would be the radius of an object which had this
density, and the mass of the Sun? Compare this object to relevant
objects discussed in class.
Answer: Use expression for which the Fermi energy = rest mass energy.
so
Assume the mass of a neutron is the same as that of a proton (it is actually
slightly larger), which in turn is equal to that of a hydrogen atom
kg. We then have a density of
neutrons/m 
.
Second part of problem: What is the size of a sphere with this density
and the mass of the Sun? The mass of a sphere is:
where 
is the mass density. 
in the previously defined
units. We then have for the radius of such an object,
where M is the mass of the Sun, which is equal to 
kg.
Plugging in these numbers, we get 
meters, which is
3 kilometers. This is not much different than the radius of a neutron star
which we talked about in class. Thus the material in a neutron star is not
much different from being relativistic.
(2) Attached is a diagram representing an image (picture) of the radio
source 3C166. Higher contours correspond to brighter parts of the source.
The observations were made by a radio telescope at a frequency of 4885 Megahertz.
What was the size of the radio telescope which made these observations? Given
this estimate of the size of the telescope, do you have guess as to what
kind of radio telescope it was? Hint The angular scale on the
y axis of the plot is in arcseconds. Sixty arcseconds equal one
arcminute, and 60 arcminutes equals 1 degree.
Answer: One can see that the bright source near the center of the picture
has an angular size of 
arcseconds. One can reasonably assume that
this is a point source, and for the purposes of this problem assume that
4.4 arcseconds is the beamwidth of the radiotelescope, 
. 4.4
arcseconds = 
degrees = 
radians. We
then use the formula from the formula sheet,
in radians. The wavelength of observation corresponding to a frequency of
4885 MHz is 6 centimeters = 0.06 meters, so
/ Rearranging to find D, we have
meters = 3.4 kilometers for the effective diameter of
the radio telescope. This is larger than any solid single dish, and so the
telescope must be an interferometer of connected radiotelescopes. In
fact, the image was made with the Very Large Array radiotelescope.
(3) Below is question from my Modern Astronomy class from last
semester. Choose the correct answer and describe why it is the correct
answer.
``When coming out into the Hillcrest dining room tonight for supper, you notice
a space alien eating by himself/herself/itself (assume for the sake of the problem
that you can distinguish between a space alien and the other residents).
He/she/it tells you that one of the following famous stars is going to become
a supernova within the next year. Which is it most likely to be?''
The choices are: (a) Antares (a red supergiant), (b) Vega (a main sequence
A star), (c) T Tauri (a star in the process of formation, approaching the
Main Sequence), (d) Sirius B (white dwarf), (e) Gliese 229A (a red dwarf).
Answer: Antares is the only possible choice. Supernovae occur in massive,
post-Main Sequence stars. Antares is a supergiant, and thus a fairly late
post-Main sequence star, and thus a good candidate. Vega is a Main Sequence
star, as is Gliese 229A. Sirius B has gone through its Main Sequence evolution
and became a white dwarf instead of a supernova, and T Tauri hasn't even
gotten to the Main Sequence yet.
(4) Consider a red dwarf star with half the mass of the Sun. Assume that
25 % of its mass is in the form of Helium (an accurate number). Now assume
somehow that it is able to undergo the triple 
process rather than the
proton-proton cycle (this is a suspension of the laws of physics). How
long would the star maintain its luminosity requirements via this process?
Show your work and clearly indicate all assumptions.
Answer: This is another two step problem. First, calculate the total
amount of energy available for this process to go. The mass in Helium is
kg. Not
all of this is converted to energy. One must multiply by the efficiency of
the triple process, i.e. what fraction of the mass is converted to
energy. The triple process, for the purposes of this problem, is
. From the attached table of nucleides, the mass
of He 
= 4.002603, so three of them will have a mass of 12.00781 atomic
mass units. One Carbon 12 nucleus has a mass of 12.00000 amu, so the difference,
which must be converted to energy, is 0.00781 atomic mass units, or
of the total mass involved. Thus the total mass ``lost''
to energy if all of the Helium in this red dwarf is cycled through the
triple process is 
kg.
The energy released by this is then 
Joules.
Second part: Calculate the luminosity. Employ the mass-luminosity relationship. An accurate approximation to this is
although any plausible exponent would be acceptable for the purposes of this
problem. With the mass of 0.5 solar masses, the luminosity of this star should
be 
that of the Sun, or 
Watts.
Thus the length of time the star could shine by this process is given by
E = L T, so 
seconds, or about 14 billion years.
(5) The oldest star clusters are believed to be about 12 billion years old.
Using the ideas discussed in class about stellar evolution, describe what you
would expect to find in such a cluster.
Answer: The star cluster would have been formed with stars of all masses.
Since it is so old, all stars with masses greater than a bit less than one
solar mass would have evolved off the Main Sequence. The massive ones would
have gone through the entire post-Main sequence evolution, produced supernovae,
and now have nothing but neutron star remnants. We would expect to find lots of
old pulsars in this star cluster. The less massive ones would have gone through
the red giant and red supergiant phases, and we would expect to find stars of
that sort as well, and also white dwarfs. Finally, still on the Main Sequence
would only by stars somewhat less massive than the Sun. An energetic student
might quickly calculate out what the limiting mass was.
(6) Explain why a burst of neutrinos is expected as a massive star core collapses.
Answer: As the core collapses, the process of ``neutronization'' will occur, in which
where 
indicates a neutrino. Thus every proton in the collapsing core
is converted into a neutron in a very short instant of time. Every neutron-
production event is accompanied by a neutrino, so there will be a burst of
such neutrinos. This neutrino burst was in fact seen in the case of supernova
1987a.