## Info

What is the evidence that the Lyman alpha forest arises in material between us and the quasar? 19.13.

In this chapter, we saw various examples of active galactic nuclei. List these types. What are the similarities among them? What are the differences that distinguish one type from another?

What makes us think that starburst galaxies are sites of very active star formation? 19.14.

Why is it unlikely that the current rate of star formation in a starburst galaxy can be sustained for a long time?

What does the detection of many supernova remnants tell us about starburst galaxies? 19.15.

In studying starburst galaxies, we make observations in various parts of the spec- 19.16.

trum. List them, and state briefly what we learn from each.

What is the suspected role of galaxy interactions in creating starbursts? 19.17. How do we know that the radio emission from radio galaxies is synchrotron? What does this tell us about the regions that are emitting? 19.18. In a radio galaxy, emission comes from a 19.19. very large area. What makes us think that the ultimate source of this energy is the 19.20. nucleus of the galaxy?

In radio galaxies, high energy particles are 19.21. transported from the inner parts of the galaxy to the lobes. Once at the lobes, why don't they quickly lose their energy? 19.22.

In a radio galaxy, how do we think that the observed jets are formed? 19.23.

Why do we think that the ultimate source of energy in a radio galaxy is matter falling onto a black hole? Why is it important for the black hole to be very massive? In calculating the energy we can extract by dropping matter into a black hole, we only considered the energy gained in falling to the Schwarzschild radius. However, matter will still accelerate after crossing RS. Why don't we add this extra energy to what we can extract?

If we have a particle and an antiparticle annihilating and producing energy, what is the efficiency of that reaction (in terms of the fraction of the mass converted into energy)?

What are the similarities between radio galaxies and quasars? Why were lunar occultations important in the discovery of quasars? How might the discovery of quasars have come sooner if radio interferometry had come sooner? What is the problem that was posed by quasars being at the distances indicated by their redshifts (that is, at cosmological distances)?

Why is the variability of quasars important? How would you carry out a search for radio-quiet quasars?

What do we mean by the 'energy-redshift problem'?

What evidence is there that quasars are truly at the distances implied by their redshifts?

How do quasars allow us to see the universe as it was in the distant past? Why is the discovery of gravitationally lensed quasars important?

Problems

[For all problems, unless otherwise stated, use H0 = 70 km/s/Mpc.]

19.1. (a) To the extent that we can ascribe a temperature to the large lobes of radio galaxies, it is about 106 K. For the numbers given in this chapter, estimate the thermal energy stored in one of these lobes. (b) Calculate the magnetic energy, and compare it with the answer in (a).

19.2. For the sizes, densities and speeds given in the chapter, estimate the kinetic energy in the flow of the jets of radio galaxies.

19.3. A radio galaxy has an angular extent of 2°, and the associated optical galaxy has its spectral lines shifted by 15% of their rest wavelength. What is the diameter of the radio galaxy?

19.4. What are wavelengths of the Hp and H7 lines in 3C273?

19.5. What are the wavelengths of the Ha and Hp lines in 3C48?

19.6. Using the same reasoning as was used for 3C273, estimate the distance to 3C48.

19.7. A quasar is observed with its Ha line at 800.0 nm. Estimate its distance.

19.8. Estimate the redshift of a quasar whose light has been traveling 1 Gyr to reach us.

19.9. What redshift would be needed to shift the Lya line into the visible part of the spectrum?

19.10. For the 103 km/s wide emission lines in Seyferts, what temperature would be required for thermal Doppler broadening?

19.11. For the superluminal source discussed in Example 19.2 (with p = 0.95), what is the range of angles about 18.2° for vapp to be greater than c?

19.12. What p is required , assuming the optimal angle, to produce a superluminal source with vapp = 10c?

19.13. If a quasar varies on a time scale of four months, how does the maximum size of the emitting region compare with the Schwarzschild radius for a 109 M0 black hole?

19.14. At what rate must mass be falling into a black hole to produce the luminosity of a typical radio galaxy?

*19.15. Suppose we try to explain the redshift of 3C273 as a gravitational redshift, rather than cosmological. (a) From what radius, relative to RS, is the radiation emerging? (b) If the width of the lines is 0.1% of their wavelength, what is the range of radii (relative to RS ) from which the radiation can be emitted?

19.16. What are the Schwarzschild radius and density for a 109 M0 black hole?

19.17. What is the energy density if the density of photons is 108 cm~3 ? Assume that the photons are all at the peak wavelength of a 105 K blackbody.

19.18. Compare the angular resolution for HST observations and for VLBI observations with the Schwarzschild radii for the M31 and M87 central objects at their respective distances.

19.19. For the galaxy NGC 3115 we find a rotational speed of 150 km/s and the dispersion for random motions of 270 km/s at an angular radius of 1.0 arc sec. What is the mass of the central object?

19.20. For the galaxy M87 we find negligible rotation and the dispersion for random motions of 350 km/s at an angular radius of 1.5 arc sec. What is the mass of the central object?

### Computer problems

19.1. You drop an object of mass m into a black hole. Calculate the energy it has acquired when it reaches 2 RS, 1.5 RS, 1.1 RS and 1.01 RS. Express your answer as a fraction of the maximum energy (when reaching RS ).

19.2. Plot a graph of vapp vs. 6 for v/c = 0.50, 0.90, 0.95, 0.99.

19.3. Estimate the angular sizes of the RS for

106 M0, 107 M0, 108 M0 , 109 M0 black holes for an object in the Virgo cluster and for an object at redshift 1.

19.4. What are the orbital speeds at 2 RS, for objects in circular orbits around 106 M0,

107 M0, 108 M0 and 109 M0 black holes?

Chapter 20

### Cosmology

Einstein once said that the most incomprehensible thing about the universe is that it is comprehensible. It is amazing that we can apparently describe the universe with what are very simple theories. We can ask truly fundamental questions of where we have come from and where we are going and expect scientific answers. In this chapter and the next, we will study cosmology, the large-scale structure of the universe. We can learn a great deal using only the physics we have introduced in this book. With the introduction of some more physics, namely elementary particle physics (in Chapter 21), we will see that even more fascinating concepts are within our grasp.

### 20.ll The scale of the universe

When we study the gas in a room, we must deal with it as a collection of molecules. We don't care about the fact that the molecules are made up of atoms or that the atoms are made up of protons, neutrons and electrons; or that the protons and neutrons are made up of other particles. All we care about is how the molecules interact with one another, and how that affects the large-scale properties of the gas. When we study the universe, we also treat it as a gas. The molecules of the gas are galaxies. In the big picture, stars, planets, etc., don't matter. Of course these smaller objects can still contain some hidden clues for us to learn about the larger structure. They simply don't affect the larger structure themselves.

How do we study cosmology? On the theoretical side, we look at the universe as large-scale fluid and ignore the lumps. The only force that currently affects the large-scale structure is gravity. We can apply gravity as described by general relativity, though for many things Newtonian gravitation is a sufficiently accurate approximation. The electromagnetic force is important in that electromagnetic radiation carries information, but it doesn't affect the large-scale structure. We will see in the next chapter that there was a time when radiation was dominant and all of the forces we know had an important effect on the structure of the universe.

Until recently, we have had very few observational clues about the large-scale structure of the universe. We will see that recent experiments, some characterized by great difficulty and resourcefulness, have greatly added to our knowledge. The field of observational cosmology is a growing one. In studying cosmology, as with other fields of astrophysics, we combine theory and observations to increase our insight into what is happening.

In making theoretical models of the universe, we start with an assumption, called the cosmological principle. It says that on the largest scales the universe is both homogeneous and isotropic. By homogeneous, we mean that, at any instant, the general properties, such as density and composition, are the same everywhere. By isotropic, we mean that, at any instant, the universe appears the same in all directions. We know that our everyday world is neither homogeneous nor isotropic, but for the universe, on the scales of many superclusters, this is a very good description.

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spherical shells, centered on the Earth, each of thickness dr. The volume of each shell is then dV = 4-nr2 dr

If there are n stars per unit volume, the number of stars in each shell is

The number of stars per shell goes up as r2. However, the brightness we see for each star falls as 1/r2. Therefore, the r2 and 1/r2 will cancel, and the brightness for each shell is the same. If there are an infinite number of shells, the sky will appear infinitely bright after we add up the contributions from each shell. Actually, this isn't quite the case. The stars have some extent, and eventually the nearer stars will block the more distant stars. However, this will not happen until the whole sky looks as if it is covered with stars, with no gaps in between.

There might appear to be some obvious ways out of this. You might say that our galaxy doesn't go on forever. Most lines of sight will leave the galaxy before they strike a star. Unfortunately, the argument can be recast in terms of galaxies instead of stars, and the same problem applies. Another possible solution is to invoke the absorption of distant starlight by interstellar dust. However, if the universe has been around forever (or even a very long time), the dust will have absorbed enough energy to increase its temperature to the same as that of the surface of a star. If the dust became any hotter than that, the dust would cool by giving off radiation which the stars would absorb. Even if the sky was not bright from the light of stars, it would be bright from the light of dust. (Scattering by dust wouldn't help because it would make the sky look like a giant reflection nebula, again like the surface of a star.)

The redshift due to the expansion of the universe is of some help. The energy of each photon is reduced in proportion to the distance it travels before we detect it. This adds an additional factor of 1/r to the apparent brightness of each shell, meaning that the brightness of each shell falls off as 1/r, instead of being constant. Note that this doesn't completely solve the problem. If we add up the contributions from all of the shells, we

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(a) How a finite size for the universe helps with Olbers's paradox. (b) How a finite age for the universe helps with Olbers's paradox.

have to take the integral of dr/r (see Problem 20.3). This gives us ln(r), so if r can be arbitrarily large the brightness can be also.

It is possible to get out of the problem if the universe has a finite size. This is illustrated in Fig. 20.2(a). If there is a finite size, then we cut off our integral at whatever that size is. There is another way to achieve the same effect. This is if the universe has a finite age t0, as illustrated in Fig. 20.2(b). We can only see stars that are close enough for their light to have reached us in this time. That is, we can cut off r at ct0. There may even be a cutoff before this because it took a certain amount of time for stars and galaxies to form. So, there is a finite cutoff to the number of shells that can contribute to the sky brightness, and the problem is solved. It is amazing that this simple observation - that the night sky is dark -leads to the conclusion that the universe has a finite size or age (or both).

### 20.2.2 Keeping track of expansion

We can show that Hubble's law follows from the assumption of homogeneity. In Fig. 20.3, suppose that P observes two positions, O and O', with distance vectors from P being r and r', respectively. The vector from O' to O is a, so that a = r - r'

Using equation (20.2) to eliminate r', this becomes v'(r - a) = v(r) - v(a)

The homogeneity of the universe means that the functional form of v and v' must be the same. (The functional form of the velocity cannot a

Vectors for locating objects in the universe.

depend on where you are in the universe.) This means that v'(r - a) = v(r - a) (20.5)

Using this, equation (20.4) becomes v(r - a) = v(r) - v(a )

This means that v(r) must be a linear function of r. The only velocity law that satisfies this relationship is v(r) = H(t) r

We let v be a function to give the rate of change of length vectors ending at O and v' the corresponding function for length vectors ending at O' . We then have

Note that we haven't required the expansion (H could be zero). However, if there is an expansion, it must follow this law, if the cosmological principle is correct.

When we want to keep track of the expansion of the universe, it is not convenient to think about the size of the universe. Instead, we introduce the scale factor which will keep track of the ratios of distances. We let t0 be the age of the universe at some reference epoch. (It doesn't matter how this reference is chosen.) We let r(t) be the distance between two points as a function of time. (The points must be far enough apart so that their separation is cosmologically significant.) We define ro = r(to) (20.8)

The scale factor R(t) is a scalar, defined from

Note from this definition that R(t0) = 1. If the universe is always expanding, R < 1 for t < t0, and R > 1 for t > t0.

We can rewrite Hubble's law in terms of the scale factor. We start by writing Hubble's law as dr/dt = H(t) r (20.10)

Using r(t) = R(t) r0 (equation 20.9) makes this r0 (dR/dt) = H(t) R(t) r0 Dividing by r0 gives dR/dt = H(t) R(t) (20.11)

Note that we now only have to deal with a scalar equation, instead of a vector equation. We a r p can solve equation (20.11) to give the Hubble parameter in terms of the scale factor:

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