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Astronomy 10: Introduction to General Astronomy
Instructor: Tony Piro, [email protected]
Homework #4: Solutions
Chapter 11
(1) page 273, question 4
From the Stefan-Bolzmann law, we know that L ∝ R2 T 4 . If the star is the same size, but has a
temperature 4 times as high, then L goes up by a factor of 44 = 256
(2) page 273, question 8
(a) Distance and parallax are inversely related. So if the parallax is 0.2, then the distance is 1/0.2 =
5 parsecs .
(b) Once again, using the inverse relationship, the parallax must be 1/100 = 0.01 seconds of arc
Make sure you also know how to convert parallax into light years!
(3) page 273, question 10
The inverse-square law tells us that apparent brightness goes as b ∝ L/d2 . So if the Sun were 10 times
further from us, then the apparent brightness would go down by a factor of
b ∝ 1/d2 = 1/(10)2 = 1/100
(4) page 273, question 16
(a) Yes. Since the spectral type is mainly determined by the temperature, if the stars have the same
spectral type, then they have similar temperatures.
(b) No. Just knowing the spectral type is not enough to know the difference in size. If one knows the
relative intrinsic luminosity of the stars, then the brighter one much have a larger surface area.
(c) No. Once again, just knowing the spectral type is not enough to know the relative distance without
further information. We need to compare the intrinsic to the apparent brightness, or have a direct distance
measurement (like parallax).
(5) page 273, question 28
(a) The luminosity-mass relation we found in lecture was that L ∝ M 3.5 . If one star is 3 times as massive,
then luminosity is 33.5 ≈ 47 times more. On a test you won’t have a calculator, but you should understand
how you can estimate this answer. For example, it should be between 33 = 27 and 34 = 81, or somewhere
around 50.
(b) Since the amount of fuel is proportional to M and the rate the fuel is burned is proportional to L,
then the time a star lives is
M
M
∝ 3.5 ∝ M −2.5
t∝
L
M
If a star is 3 times as massive then it live 3−2.5 ≈ 0.064 times as long. This should make sense–more
luminous stars live a shorter amount of time.
(6) page 274, question 35
The cluster that doesn’t have any O, B, A, and F stars (the second one) must be older. The lack of F
stars means that the cluster must be old enough for these stars to have evolved off the main sequence.
Chapter 12
(7) page 291, question 2
(a) Regions of star formation are usually shrouded by dust and gas, the material that the stars form from.
Since longer wavelengths are scattered less (just like in the atmosphere for sunsets), optical and ultraviolet
don’t make it easily out of these regions.
(b) Typically infrared and radio are best for viewing regions of star formation. They have longer wavelengths, which makes them less scattered by dust and gas.
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(8) page 291, question 8
(a) The kind of element you have is determined by the number of protons. If you remove one neutron,
you still have helium. This is what we call an isotope of helium.
(b) Now you have changed helium into a different element because you have changed the number of
protons. You now have hydrogen, and in particular, since there are still two neutrons, you have a particular
isotope of hydrogen called tritium.
(9) page 291, question 13
When a star is in “mechanical balance” it means the forces of gravity and pressure are balancing. This is
also called hydrostatic equilibrium. Also note that it has nothing to do with the presence of nuclear reactions.
Chapter 13
(10) page 321, question 2
When the helium core is first formed, the core is not hot enough fuse the helium into heavier elements.
Only once the red giant phase occurs, and the core contracts and heats up to a temperature of around 108 K
is the core hot enough to start burning helium.
(11) page 321, question 6
When a star sheds its surface in a planetary nebula, it keeps about the same luminosity, but the radius
has decreased dramatically. Since L ∝ R2 T 4 , this means that the temperature must go way up.
(12) page 321, question 7
Inside a white dwarf the inward pull of gravity is being balanced by the force of electron degeneracy.
(13) page 321, question 16
Type Ia supernovae are exploding white dwarfs, which have accreted a mass greater than the Chandrasekhar limit. Type II supernovae are massive stars, that have exhausted the fuel in their core, and
eventually collapsing then exploding as the surface envelope bounces off the collapsing iron core.
(14) page 321, question 20
The heavy elements in our body come from exploding stars–supernovae. In particular, elements such
as carbon and oxygen are formed during stable burning, but require the explosion to get into space and
eventually form other stars, planets, and life. The heavier elements, such as iron, are formed directly in the
burning during the explosion.
(15) page 322, question 29
The location of pulsars closely follows the plane of our Galaxy, demonstrating that they must be residing
in the Milky Way. This are spiral arms, which makes sense because this is where young, massive stars are
forming and eventually dying.
(16) page 322, question 31
We knew about pulsars before the Crab Nebula, but none were so directly connected to a site where we
thought a neutron star should be sitting. For the Crab Nebula, we knew that a supernova has occurred, so
we thought a neutron star should be present. Once a pulsar was found in the Crab Nebula, it confirmed
that pulsars are neutron stars.
Chapter 14
(17) page 341, question 2
A black piece of paper still has some light emitting from it (in particular, infrared radiation). But a black
hole does not let any light escape.
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(18) page 341, question 6
A black hole the mass of the Sun has a Schwarzchild radius of 3 km, and the radius is proportional to
the mass. So if the star is 10 solar masses, then the Schwarzchild radius is 10 × 3 km = 30 km
The photon sphere is slightly larger than the Schwarzchild radius, 4.5 km for the mass of the Sun. So for
a star that is 10 solar masses, the photon sphere is 10 × 4.5 km = 45 km
(19) page 342, question 14
If the Sun suddenly became a black hole, the orbits of the planets would not change. The extreme
effects of a black hole only occur near the horizon (3 km away for the Sun), and all the planets are much
further.
(20) page 342, question 19
The appearance of an X-ray source from a binary system shows that there is a compact object accreting
material, but it doesn’t necessarily imply that it’s a black hole. It could also be a neutron star. Doppler
shifts of the companion can help constrain the mass. If we get a mass greater than 3 solar masses, then
we’re pretty sure it can’t be a neutron star and assume it’s a black hole.
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