Astronomy & Astrophysics Chapter 18 Homework Gravitational Lensing The Bending Beams Light Massive

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226 Bennett, Donahue, Schneider, Voit

Chapter 18. Dark Matter, Dark Energy, and the

Fate of the Universe

This chapter focuses on dark matter and dark energy and their presumed

significance in the universe. Students are likely to find this chapter particularly

relevant because these topics are so often in the astronomical news.

As always, when you prepare to teach this chapter, be sure you are familiar

Teaching Notes (by Section)

Section 18.1 Unseen Influences in the Cosmos

Students have probably heard the terms dark matter and dark energy and will

naturally assume that because we have names for these things we understand

them. Of course, the reality is that these are simply names given to whatever

might be causing particular patterns of observation: high orbital velocities in

Section 18.2 Evidence for Dark Matter

This section presents the evidence for dark matter in galaxies and galaxy

clusters and discusses the possible nature of dark matter, in particular the

evidence that suggests that it is not baryonic, or , We also

discuss the possibility that dark matter does not exist at all and that we are instead

misunderstanding the nature of gravity, but point out that the sheer weight of the

evidence at this point means that any such alternative interpretation of the data

will have to meet many observational constraints (e.g., alternative gravity theories

must do more than simply explain flat rotation curves of spiral galaxies the

current evidence for dark matter is far deeper and more diverse than that).

We include Cosmic Calculations 18.1, on estimating the mass-to–light

ratio in solar units.

In support of discussions of the nature of science, we have included an

Extraordinary Claims feature,

highlighting the history of the discovery of dark matter by Zwicky, Rubin,

and Ford.

Instructor Guide for The Essential Cosmic Perspective, Eighth Edition 227

For simplicity, we define ordinary matter to be baryonic matter (protons

and neutrons) and electrons and exotic matter to be nonbaryonic matter.

We prefer to speak of ordinary matter and exotic matter when possible

because these terms convey the proper impression to that large group of

students who have trouble remembering what a baryon is.

This section includes Figure 18.11, a plot that shows how the primordial

deuterium-to-hydrogen ratio puts constraints on the density of ordinary

Section 18.3 Structure Formation

This section covers large-scale structure in the universe and the role of dark

matter in creating it. The analogy for growth via gravity that we use in class is

that in certain economic systems, the rich get richer and the poor get poorer, and

the dichotomy grows with time. Places in the universe that were a little denser

Section 18.4 Dark Energy and the Fate of the Universe

This section explains how we try to assess the fate of the universe by measuring the

xpansion

rate has changed through time. We emphasize the potential role of dark energy,

presuming it exists.

In the 20th century, we could make a one-to-one correspondence between

three possible models of universal fate (recollapsing, critical, and coasting)

and three possible geometries (closed, flat, open). The introduction of dark

energy breaks this simple correspondence. We therefore find it easiest for

228 Bennett, Donahue, Schneider, Voit

Many popular articles and books have taken the evidence for an

accelerating expansion to suggest that we now know that the universe

will expand forever. The logic is simple: Evidence was already pointing to

an open universe before the acceleration was discovered, so acceleration

would seem only to further seal the case. We take a slightly different

Answers/Discussion Points for Think About It/See It for Yourself

Questions

The Think About It and See It for Yourself questions are not numbered in the

book, so we list them in the order in which they appear, keyed by section number.

Section 18.2

(p. 470) The graph of orbital speeds and distances for the moons orbiting

Jupiter would most resemble Figure 18.1b. The rotation curve for the moons

around Jupiter would decrease with increasing distance, just as the rotation

curve for the planets around the Sun decreases with increasing distance.

Section 18.3

(p. 478) Earth is held together by its own gravity; a hurricane is

not clear (at least as far as the content provided in the textbook) whether

or not the Orion Nebula is gravitationally bound. Certainly at one point

it was, but it loses gas as the stars evolve and now may or may not be

bound. A supernova is not gravitationally bound, because its shock wave

propagates out and does not return. (A piece of a massive star supernova

that may be left behind as a neutron star or a black hole is certainly itself

gravitationally bound.)

Instructor Guide for The Essential Cosmic Perspective, Eighth Edition 229

Section 18.4

(p. 483, SIFY) St graphs should have the approximate shape of

this figure (units are not required, but time is measured on the x-axis, and

relative height on the y–axis). This curve most closely resembles that of the

gravity were not as strong, the ball

would not slow down as fast, and

Solutions to End-of-Chapter Problems (Chapter 18)

Visual Skills Check

Review Questions

1. Dark matter is matter that gives off little or no light. Dark energy is the

name given to whatever it is that is causing the u

accelerate. While they have similar names, they are not similar in nature.

Dark matter is massive, behaves like something that gravitates, and is really

2. A rotation curve plots the rotational velocity of stars against their distances

from the center of the

as we move out from the center of the galaxy and then more or less levels

3. We construct rotation curves for other spiral galaxies by observing the

4. We measure the masses of elliptical galaxies by measuring how much the

spectral absorption lines of stars in the galaxy are broadened in a spectrum

that includes measurements of many stars at once. The broader the spectral

5. We can measure the masses of clusters of galaxies in three ways. The first

method is to measure the speeds and positions of the galaxies in the cluster

deduce the mass of the cluster. The second method for finding the mass of a

6. Gravitational lensing is the bending of beams of light by massive objects.

This bending occurs because masses distort spacetime, according to general

7. Because we have measured the masses of galaxies and clusters using our

current theories of gravity, it is possible that the masses of the galaxies and

clusters are wrong if our theories of gravity are wrong. However, the

8. Dark matter is dark in the sense that it does not emit enough light for us to

9. Calculations based on the abundance of deuterium, lithium, and helium-3

indicate that the density of baryonic matter in the universe should be about

10. We say that particles such as neutrinos are weakly interacting particles

because they interact only with other objects through gravity or the weak

11. WIMPs, or weakly interacting massive particles, are a form of dark matter

that is made up of particles that do not generally interact with baryonic

12. The large-scale structure of the universe shows galaxies arranged in huge

chains and sheets that span millions of light-years. Between the chains and

sheets are giant voids. This structure probably mirrors the original distribution

of dark matter in the early universe. The denser regions in the early universe

13. There are four possible patterns for the expansion of the universe:

Recollapsing: We get a recollapsing universe if there is enough mass in the

universe. If this were the case, eventually gravity will halt the expansion of

the universe and reverse it. All of the matter in the universe would

eventually be crushed back together again, re-creating the conditions of

14. The cosmic microwave background (CMB) studies indicate that the total

density of matter plus energy in the universe is very close to 100% of the

15. Current studies (and Figure 18.17) indicate that the actual density of the

16. Current evidence indicates that the universe will continue to expand forever.

If the acceleration of the expansion of the universe continues without

declining, dark energy will cause galaxies to eventually separate so far from

Instructor Guide for The Essential Cosmic Perspective, Eighth Edition 233

Does It Make Sense?

17. Strange as it may sound, most of both the mass and the energy in the

universe may take forms that we are unable to detect directly. This

18. A cluster of galaxies is held together by the mutual gravitational attraction

19. We can estimate the total mass of a cluster of galaxies by studying the

20. Clusters of galaxies are the largest structures that we have so far detected in

21. The primary evidence for an accelerating universe comes from observations

22. There is no doubt remaining among astronomers that the fate of the

23.

24. Dark energy is the energy associated with the motion of particles of dark

25. Evidence that the expansion of the universe is accelerating comes from

observations showing that the average distance between galaxies is increasing

26. If dark matter consists of WIMPs, then we should be able to observe photons

produced by collisions between these particles. This statement does not make

234 Bennett, Donahue, Schneider, Voit

Quick Quiz

27. c. we can observe its gravitational influence on visible matter

31. c. gravitational lensing

Process of Science

38. The case for dark energy as presented at this level is the case made by

39. At the minimum, a new theory of gravity should pass all of the tests that

our current theory of gravity (general relativity) has already passed: the

Group Work Exercise

40. a. Students might speculate that

Short Answer/Essay Questions

41. If the acceleration continues, the formation of large-scale structures in the

universe ceases almost completely over the next 10 billion years. The

42. Dark matter allows gravity to make galaxies in a reasonably sprightly time

and gives life a chance to develop (before the nasty boring phase of the

43. We can only describe the rotation curves here:

a. All mass concentrated in the center of the galaxy: Decreasing rotation

44. In this situation, the supernovae turned out to be fainter than expected. (The

45. One possible constituent of dark matter is weakly interacting massive

particles, which do not interact with light. We can detect them in ways

similar to those we use to detect neutrinos or by their gravitational effects on

46. To explain flat rotation curves without dark matter, gravity would have to be

stronger than expected at large distances, because you need more gravity for

Quantitative Problems

47. The mass-to-light ratio of a 1MSun white dwarf with a luminosity of

0.001LSun is the mass over the luminosity:

48. The mass-to-light ratio of a 30MSun supergiant with a luminosity of

300,000LSun is the mass over the luminosity:

49. To find the mass-to-light ratio of the solar system, we need the mass and the

luminosity. The luminosity is essentially just that of the Sun, because the

Sun is the only significant light source in the solar system. (Recall that

50. We can derive a general-purpose formula for this problem and for the next

problem by using the mass/velocity/radius relation and plugging in 100 km/s

for the velocity and 10,000 light-years for the radius. Then, for this and the

next problem, we need to do fewer computations:

51. For this problem, we assume that the gas in the cluster is all at one

temperature (isothermal) and that we have X-ray data from the cluster out to

at least 5.1 million light-years (4.8 1022 m). The equivalent velocity for the

particles in the gas, from the relationship between speed of the hydrogen

particles and the temperature in