Download Homework #3 10 points Question #1 (2 pts) The brightest star in the

Homework #3
10 points
Question #1 (2 pts)
The brightest star in the night sky, Sirius, is actually a double star: Sirius A is a main
sequence star about twice as massive as the Sun, and Sirius B is a white dwarf. Which of
the two stars was born more massive? Please, explain your answer, don’t just say A or B.
(Just to make clear: since it is a double star, both Sirius A and B were born together.)
Since the both stars were born together, and Sirius B is already a white dwarf (a dead star),
it had a shorter life and, hence, was more massive in the beginning. What a change of
fortunes!
Question #2 (2 pts)
Thermonuclear supernova is an explosion of a white dwarf, which is less massive than a
progenitor of a core-collapse supernova. Which produces a larger fractional abundance of
iron? You can think the following way: let’s imagine that in a region of space A we found
one massive star that exploded as a core collapse supernova, and in a region of space B we
found several white dwarfs that all exploded as thermonuclear supernovae. The number
of white dwarfs in the region B is such that in both regions, A and B, exactly the same
amounts of carbon and oxygen are produced. In which region, A or B, will there be more
iron produced?
In a core collapse supernova the core goes into a neutron star or a black hole, while in a
thermonuclear supernova the whole core (i.e. the remaining white dwarf ) burns into iron.
On top of that a good fraction of iron produced in a massive star actually gets dissociated
back into hydrogen in a core collapse supernova. Hence, thermonuclear supernova are much
more efficient iron producers.
This is a non-trivial fact. Fractional iron abundance is used in Astronomy as an indirect
clock - since core collapse supernovae occur in young stars and thermonuclear supernovae
occur in the old stellar population, the older the stars the large their fraction iron abundance.
Question #3 (2 pts)
Helium balloons fly up in the air, because helium is lighter than air, and its buoyancy
(buoyancy is a force, also sometimes called the Archimedes’ force) may overcome gravity,
if the balloon is light enough. The first flying balloons were not filled with helium, however
(they did not know how to produce enough helium in 19th century), but instead were hot
air balloons. Explain why hot air balloons fly. You need to give me a detailed answer,
not just say “they are lighter than air”, as Wikipedia does - you also need to explain why
they are lighter than air (Wikipedia misses an important piece of physics that you need to
find).
Hot air balloons fly because they are hot. The answer may sound silly, but since the air
inside them is hotter than the air outside, it would have higher pressure at the same density.
Pressures inside and outside must be balanced, or the balloon would burst, so it expands to
reduce the density inside until it gets into the pressure balance with the outside air. The
pressure balance is that extra piece of physics that is important for understanding why hot
air balloons fly.
Question #4 (2 pts)
In an empty, perfectly sealed and insulated room there stands a working fridge. The fridge
is plugged into a wall outlet and is set to the coldest setting. The door of the fridge is
propped open. In the long run, how does the temperature in the room change? Please,
justify your answer.
The energy is conserved. Because the room is sealed and insulated, and the electricity is
flowing into the room via the outlet, the temperature must be increasing in the room, no
matter how powerful the refrigerator is.
In fact, the temperature increase only depends on the power consumption of the device.
If, for example, the fridge consumes 100 watts of power, and you replace it with a 100
watt room heater, the temperature increase in the room would be exactly the same, even
if a fridge is supposed to cool and a heater to heat. This is just another example that the
energy conservation is absolute, it does not depend on the technical details of what actually
uses that energy.
Question #5 (1 pts)
Radiation exerts pressure on matter, since when photons are absorbed or bounce off atoms,
they transfer their momentum to the atoms, and the change of momentum is a force.
Compute the Eddington luminosity for the Sun, i.e. the luminosity at which radiation
pressure force is equal the Sun’s gravity. For reference, the radiation pressure force is
FRAD =
κJ
,
c
where J is the radiation flux at a distance R from the center of the Sun,
J=
L
,
4πR2
L is the stellar luminosity, R is the solar radius, c is the speed of light, and κ is the opacity
of the gas per unit mass; you should use the value κ = 0.04 m2 /kg. Does the Eddington
luminosity depend on the value of the distance from the center of the Sun that you use?
If we balance gravity and radiation pressure (the value for the radiation force given is per
unit mass),
κL
GM⊙
=
,
2
R
c4πR2
we notice an amazing fact - both go as inverse square law, hence the balance does not
depend on the distance - at all distances from the star the balance either exists or not.
For the balance to exist, the luminosity of a star should be
L = (4πcG/κ)M⊙ = 1.3 × 1038 erg/s = 1.3 × 1031 W = 30,000L⊙ .
Question #6 (1 pts)
Luminosities of stars of different masses vary disproportionately to their masses. In fact,
as a function of stellar mass, the luminosity changes as, approximately, L ∝ M 3 (i.e. a star
10 times more massive than the Sun has a luminosity about 1000 times larger). Assuming
that stellar cores contain the same fraction of stellar mass for stars of all masses, how does
the main sequence lifetime (the time a star spends on the main sequence) depend on its
mass? To be specific, if the main sequence lifetime for the Sun is about 10 Gyr, what is
the main sequence lifetime for a star 10 times more massive than the Sun? You need to
compute this number, not to find it online.
If the core always contains the same fraction the stellar mass, then its mass (i.e. the
available fuel) scales with the stellar mass,
Mfuel ∝ M,
while the luminosity (i.e. the power, i.e. the rate with which the fuel is consumed) scales
as M 3 , hence the time to consume all the fuel should scales as
tage ∝ Mfuel /L ∝ M −2 .
Hence, a 10 solar mass star will live 10−2 = 0.01 of the solar age, or about 100 Myr.