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PHYS 175, FALL 2014
HW #9 S OLUTION
1
21.11 Should we worry about Earth being pulled into a black hole? Why or why not?
No, we should not. Black holes only exert the same gravitational attraction as other
massive bodies. While it would be incredibly dense, a black hole of, say 1 M at a
distance of 1 AU from Earth (sound familiar?), would not ”pull Earth in” any more
than an ordinary star of the same mass and distance. In fact, we’d remain in orbit.
We should sleep easy tonight in this knowledge.
21.15 Describe two ways in which a member of a binary star system could become a
black hole.
I’ll describe three here: 1) One of the binary pair is greater than or about 25 M and
undergoes a Type II supernova, resulting in a core remnant that is greater than about
3 M – it thus becomes a black hole; 2) the two stars are a red giant and either of a
white dwarf or neutron star, the giant swells beyond its Roche lobe and the smaller,
more dense companion takes on inflating matter from its bigger companion, this
causes the mass of the core remnant to exceed about 3 M , thus forming a black
hole; and finally 3) two neutron stars are in close orbit with one another and due to
gravitational wave radiation, their orbits gradually decay and they collide/coalesce,
thus forming a black hole.
21.19 How do astronomers locate supermassive black holes in galaxies?
There are three primary techniques that astronomers use to conclude that supermassive black holes exist at the center of many, if not most, large galaxies. 1) Optical and
radio telescopes are used to observe massive discs (V) around and highly directed,
energetic jets (R) emanating from galaxies, 2) Doppler shifts showing rotational rates
of stars in the central regions of galaxies, and 3) gravitational effects on gas and dust
surrounding a galaxy. In each of these cases, the only plausible explanation is that a
very large black hole exists at the center of the galaxy under observation.
21.31 Summarize what leads to Hawking radiation.
The curvature of spacetime at an event horizon and the Heisenberg Uncertainty
Principle describe how energy and time can give rise to virtual pair production.
This is the continuous process of the production (and quick annihilation) of pairs
of antiparticles. If this happens right at the event horizon, one particle is lost to the
back hole, while the other carries mass away from it (albeit a very small amount),
and no annihilation takes place. As this process continues, the black hole ”evaporates” eventually. The smaller it is, the faster it experiences this Hawking radiation.
Still, the process would take an extremely long time for a star of more than about a
solar mass.
PHYS 175, FALL 2014
HW #9 S OLUTION
2
21.46 Find the Schwarzschild radius for an object having a mass equal to that of the
planet Saturn.
We know that when the escape speed of a massive object equals the speed of light
that we are at the Schwarzschild radius. For a Saturn-mass, this is a very small sized
black hole! We find this via the equation
RSch =
2GM
2 · 6.67 × 10−11 · 5.685 × 1026
=
c2
(3 × 108 )2
≈ 0.84 m .
22.8 What must happen to a hydrogen atom for it to emit a photon of wavelength 21
cm?
The initial atom must be in a spin-parallel configuration, where the proton and electron spins are aligned. This will spontaneously transition to the slightly lower energy configuration of anti-aligned spins. The expected lifetime of such a transition
is very long, nearly 107 yr; however, there is a lot of hydrogen out there (see §§22.32,
below).
22.13 Many old black-and-white photographs of spiral galaxies were taken using film
that was most sensitive to blue light. Explain why the spiral arms were particularly prominent in such photographs.?
The spiral arms of a galaxy are regions of intense star formation and have large
numbers of luminous, but short-lived type O and B stars. Each of these types of
stars give off a significant amount of light in the blue part of the spectrum, so they
would show up especially well on these early black and white films or plates.
22.18 How do astronomers conclude that vast quantities of dark matter surround our
Galaxy? How is this dark matter distributed in space?
The observed rotational rates of stars in our galaxy do not correspond to what Keplerian orbits would look like when we take into account only the stars we can ”see”
(i.e., observe by means of electromagnetic radiation). In order to explain the observed rotational rates in the various part of the galaxy, astronomers need to invoke
a roughly spherical cloud of dark matter (dark, in the sense that it has no signature
in the electromagnetic spectrum that we are capable of observing). This spherical
cloud of dark matter is many times the size of the visible part of the galaxy and
comprises roughly 90% of its mass.
PHYS 175, FALL 2014
HW #9 S OLUTION
3
22.20 What is the difference between dark matter and dark nebulae?
The former is the name given for the ”missing” 90% of the mass of the galaxy that
we need to explain the observed rotational behavior of the visible 10% (gas, dust
and stars). The latter is observed as an opaque region of space, in which protostars
are forming. The former are very difficult to observe (if not impossible), while the
latter are commonly observed in regions of star formation.
22.32 A typical hydrogen atom in interstellar space undergoes a spin-flip transition
only once every 107 years. How, then, is it possible to detect the 21-cm radio
emission from interstellar hydrogen?
Here we recall that roughly 74% of the entire universe is made up of H – there is so
much hydrogen out there in interstellar space that these transitions, while individually rare, occur with great frequency on the whole.