Physics for Scientists and Engineers with Modern Physics

Douglas C. Giancoli

Chapter 44

Astrophysics and Cosmology - all with Video Answers

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Chapter Questions

Problem 1

(I) The parallax angle of a star is $0.00029^{\circ} .$ How far away is the star?

Jacob Shpiece

Jacob Shpiece

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Problem 2

(I) A star exhibits a parallax of 0.27 seconds of arc. How far away is it?

Jacob Shpiece

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Problem 3

(I) What is the parallax angle for a star that is 65 ly away? How many parsecs is this?

Jacob Shpiece

Numerade Educator

Problem 4

(I) A star is $56 \mathrm{pc}$ away. What is its parallax angle? State (a) in seconds of arc, and ( $b$ ) in degrees.

Jacob Shpiece

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Problem 5

(I) If one star is twice as far away from us as a second star, will the parallax angle of the farther star be greater or less than that of the nearer star? By what factor?

Matthew Miranda

Matthew Miranda

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Problem 6

(II) A star is $85 \mathrm{pc}$ away. How long does it take for its light to reach us?

Jacob Shpiece

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Problem 7

(II) What is the relative brightness of the Sun as seen from Jupiter, as compared to its brightness from Earth? (Jupiter is 5.2 times farther from the Sun than the Earth is.)

Matthew Miranda

Matthew Miranda

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Problem 8

(II) We saw earlier (Chapter 19) that the rate energy reaches the Earth from the Sun (the "solar constant") is about $1.3 \times 10^{3} \mathrm{~W} / \mathrm{m}^{2} .$ What is $(a)$ the apparent brightness $b$ of the Sun, and ( $b$ ) the intrinsic luminosity $L$ of the Sun?

Jacob Shpiece

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Problem 9

(II) When our Sun becomes a red giant, what will be its average density if it expands out to the orbit of Mercury $\left(6 \times 10^{10} \mathrm{~m},\right.$ from the Sun $) ?$

Jacob Shpiece

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Problem 10

(II) Estimate the angular width that our Galaxy would subtend if observed from the nearest galaxy to us (Table $44-1$ ). Compare to the angular width of the Moon from Earth.

Jacob Shpiece

Numerade Educator

Problem 11

(II) Calculate the $Q$ -values for the He burning reactions of Eq. $44-2 .$ (The mass of the very unstable ${ }_{4}^{8} \mathrm{Be}$ is $\left.8.005305 \mathrm{u} .\right)$

Jacob Shpiece

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Problem 12

(II) When our Sun becomes a white dwarf, it is expected to be about the size of the Moon. What angular width would it subtend from the present distance to Earth?

Jacob Shpiece

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Problem 13

(II) Calculate the density of a white dwarf whose mass is equal to the Sun's and whose radius is equal to the Earth's. How many times larger than Earth's density is this?

Matthew Miranda

Matthew Miranda

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Problem 14

(II) A neutron star whose mass is 1.5 solar masses has a radius of about $11 \mathrm{~km} .$ Calculate its average density and compare to that for a white dwarf (Problem 13) and to that of nuclear matter.

Matthew Miranda

Matthew Miranda

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Problem 15

(III) Stars located in a certain cluster are thought to be about the same distance from us. Two such stars have spectra that peak at $\lambda_{1}=470 \mathrm{nm}$ and $\lambda_{2}=720 \mathrm{nm},$ and the ratio of their apparent brightness is $b_{1} / b_{2}=0.091$. Estimate their relative sizes (give ratio of their diameters) using Wien's law and the Stefan-Boltzmann equation, Eq. $19-17 .$

Lainey Roebuck

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Problem 16

(III) Suppose two stars of the same apparent brightness $b$ are also believed to be the same size. The spectrum of one star peaks at $750 \mathrm{nm}$ whereas that of the other peaks at $450 \mathrm{nm}$. Use Wien's law and the Stefan-Boltzmann equation (Eq. $19-17$ ) to estimate their relative distances from us.

Elliott Walker

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Problem 17

(I) Show that the Schwarzschild radius for a star with mass equal to that of Earth is $8.9 \mathrm{~mm}$.

Jacob Shpiece

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Problem 18

(II) What is the Schwarzschild radius for a typical galaxy (like ours)?

Matthew Miranda

Matthew Miranda

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Problem 19

(II) What is the maximum sum-of-the-angles for a triangle on a sphere?

Matthew Miranda

Matthew Miranda

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Problem 20

(II) Calculate the escape velocity, using Newtonian mechanics, from an object that has collapsed to its Schwarzschild radius.

Jacob Shpiece

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Problem 21

(II) What is the apparent deflection of a light beam in an elevator (Fig. 44-13) which is $2.4 \mathrm{~m}$ wide if the elevator is accelerating downward at $9.8 \mathrm{~m} / \mathrm{s}^{2}$ ?

Jacob Shpiece

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Problem 22

(I) The redshift of a galaxy indicates a velocity of $1850 \mathrm{~km} / \mathrm{s} .$ How far away is it?

Zhaojie Xu

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Problem 23

(I) If a galaxy is traveling away from us at $1.5 \%$ of the speed of light, roughly how far away is it?

Jacob Shpiece

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Problem 24

(II) A galaxy is moving away from Earth. The "blue" hydrogen line at $434 \mathrm{nm}$ emitted from the galaxy is measured on Earth to be $455 \mathrm{nm}$. (a) How fast is the galaxy moving? (b) How far is it from Earth?

Jacob Shpiece

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Problem 25

(II) Estimate the wavelength shift for the $656-\mathrm{nm}$ line in the Balmer series of hydrogen emitted from a galaxy whose distance from us is $(a) 7.0 \times 10^{6} \mathrm{ly},(b) 7.0 \times 10^{7} \mathrm{ly}$

Elliott Walker

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Problem 26

(II) If an absorption line of calcium is normally found at a wavelength of $393.4 \mathrm{nm}$ in a laboratory gas, and you measure it to be at $423.4 \mathrm{nm}$ in the spectrum of a galaxy, what is the approximate distance to the galaxy?

Susan Hallstrom

Susan Hallstrom

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Problem 27

(II) What is the speed of a galaxy with $z=0.060 ?$

Jacob Shpiece

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Problem 28

(II) What would be the redshift parameter $z$ for a galaxy traveling away from us at $v=0.075 \mathrm{c} ?$

Jacob Shpiece

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Problem 29

(II) Starting from Eq. $44-3$, show that the Doppler shift in wavelength is $\Delta \lambda / \lambda_{\text {rest }} \approx v / c$ (Eq. 44-6) for $v \ll c$. [Hint: Use the binomial expansion.

Lainey Roebuck

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Problem 30

(II) Radiotelescopes are designed to observe $21-\mathrm{cm}$ waves emitted by atomic hydrogen gas. A signal from a distant radio-emitting galaxy is found to have a wavelength that is $0.10 \mathrm{~cm}$ longer than the normal $21-\mathrm{cm}$ wavelength. Estimate the distance to this galaxy.

Jacob Shpiece

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Problem 31

(I) Calculate the wavelength at the peak of the blackbody radiation distribution at $2.7 \mathrm{~K}$ using Wien's law.

Matthew Miranda

Matthew Miranda

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Problem 32

(II) Calculate the peak wavelength of the CMB at $1.0 \mathrm{~s}$ after the birth of the universe. In what part of the EM spectrum is this radiation?

Jacob Shpiece

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Problem 33

(II) The critical density for closure of the universe is $\rho_{\mathrm{c}} \approx 10^{-26} \mathrm{~kg} / \mathrm{m}^{3} .$ State $\rho_{\mathrm{c}}$ in terms of the average number
of nucleons per cubic meter.

Matthew Miranda

Matthew Miranda

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Problem 34

(II) The scale factor of the universe (average distance between galaxies) at any given time is believed to have been inversely proportional to the absolute temperature. Estimate the size of the universe, compared to today, at
(a) $t=10^{6} \mathrm{yr}$
(b) $t=1 \mathrm{~s}$
(c) $t=10^{-6} \mathrm{~s},$ and
(d) $t=10^{-35} \mathrm{~s}$

Lainey Roebuck

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Problem 35

(II) At approximately what time had the universe cooled below the threshold temperature for producing (a) kaons $\left(M \approx 500 \mathrm{MeV} / c^{2}\right)$ (b) $\Upsilon\left(M \approx 9500 \mathrm{MeV} / c^{2}\right)$ and $(c)$ muons $\left(M \approx 100 \mathrm{MeV} / c^{2}\right) ?$

Lainey Roebuck

Numerade Educator

Problem 36

(II) Only about $5 \%$ of the energy in the universe is composed of baryonic matter. (a) Estimate the average density of baryonic matter in the observable universe with a radius of 14 billion light-years that contains $10^{11}$ galaxies, each with about $10^{11}$ stars like our Sun. $(b)$ Estimate the density of dark matter in the universe.

Jacob Shpiece

Numerade Educator

Problem 37

The evolution of stars, as discussed in Section $44-2,$ can lead to a white dwarf, a neutron star, or even a black hole, depending on the mass. (a) Referring to Sections $44-2$ and $44-4,$ give the radius of (i) a white dwarf of 1 solar mass, (ii) a neutron star of 1.5 solar masses, and (iii) a black hole of 3 solar masses. (b) Express these three radii as ratios $\left(r_{i}: r_{\mathrm{ii}}: r_{\text {iii }}\right) .$

Lainey Roebuck

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Problem 38

Use conservation of angular momentum to estimate the angular velocity of a neutron star which has collapsed to a diameter of $16 \mathrm{~km},$ from a star whose radius was equal to that of our $\operatorname{Sun}\left(7 \times 10^{8} \mathrm{~m}\right)$. Assume its mass is 1.5 times that of the Sun, and that it rotated (like our Sun) about once a month.

Elliott Walker

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Problem 39

By what factor does the rotational kinetic energy change when the star in Problem 38 collapses to a neutron star?

Jacob Shpiece

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Problem 40

Assume that the nearest stars to us have an intrinsic luminosity about the same as the Sun's. Their apparent brightness, however, is about $10^{11}$ times fainter than the Sun. From this, estimate the distance to the nearest stars. (Newton did this calculation, although he made a numerical error of a factor of $100 .)$

Narayan Hari

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Problem 41

Suppose that three main-sequence stars could undergo the three changes represented by the three arrows, $\mathrm{A}, \mathrm{B},$ and $\mathrm{C}$ in the H-R diagram of Fig. 44-35. For each case, describe the changes in temperature, intrinsic luminosity, and size.

Lainey Roebuck

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Problem 42

A certain pulsar, believed to be a neutron star of mass 1.5 times that of the Sun, with diameter $16 \mathrm{~km}$, is observed to have a rotation speed of $1.0 \mathrm{rev} / \mathrm{s}$. If it loses rotational kinetic energy at the rate of 1 part in $10^{9}$ per day, which is all transformed into radiation, what is the power output of the star?

Elliott Walker

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Problem 43

The nearest large galaxy to our Galaxy is about $2 \times 10^{6} \mathrm{ly}$ away. If both galaxies have a mass of $3 \times 10^{41} \mathrm{~kg}$, with what gravitational force does each galaxy attract the other?

Matthew Miranda

Matthew Miranda

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Problem 44

Estimate what neutrino mass (in $\mathrm{eV} / \mathrm{c}^{2}$ ) would provide the critical density to close the universe. Assume the neutrino density is, like photons, about $10^{9}$ times that of nucleons, and that nucleons make up only (a) $2 \%$ of the mass needed, or (b) $5 \%$ of the mass needed.

Lainey Roebuck

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Problem 45

Two stars, whose spectra peak at $660 \mathrm{nm}$ and $480 \mathrm{nm}$, respectively, both lie on the main sequence. Use Wien's law, the Stefan-Boltzmann equation, and the $\mathrm{H}-\mathrm{R}$ diagram (Fig. $44-6$ ) to estimate the ratio of their diameters.

Lainey Roebuck

Numerade Educator

Problem 46

(a) In order to measure distances with parallax at 100 parsecs, what minimum angular resolution (in degrees) is needed? (b) What diameter mirror or lens would be needed?

Elliott Walker

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Problem 47

What is the temperature that corresponds to $1.96-\mathrm{TeV}$ collisions at the Fermilab collider? To what era in cosmological history does this correspond? [Hint: See Fig. $44-30 .$ ]

Lainey Roebuck

Numerade Educator

Problem 48

Astronomers have measured the rotation of gas around a possible supermassive black hole of about 2 billion solar masses at the center of a galaxy. If the radius from the galactic center to the gas clouds is 68 light-years, estimate the measured value of $z$

Lainey Roebuck

Numerade Educator

Problem 49

In the later stages of stellar evolution, a star (if massive enough) will begin fusing carbon nuclei to form, for example, magnesium:
$$
{ }_{6}^{12} \mathrm{C}+{ }^{12} \mathrm{C} \rightarrow{ }_{12}^{24} \mathrm{Mg}+\gamma
$$
(a) How much energy is released in this reaction (see Appendix F)? (b) How much kinetic energy must each carbon nucleus have (assume equal) in a head-on collision if they are just to "touch" (use Eq. $41-1$ ) so that the strong force can come into play? $(c)$ What temperature does this kinetic energy correspond to?

Lainey Roebuck

Numerade Educator

Problem 50

Consider the reaction
$$
{ }_{8}^{16} \mathrm{O}+{ }_{8}^{16} \mathrm{O} \rightarrow{ }_{14}^{28} \mathrm{Si}+{ }_{2}^{4} \mathrm{He}
$$
and answer the same questions as in Problem $49 .$

Lainey Roebuck

Numerade Educator

Problem 51

Calculate the Schwarzschild radius using a semi-classical (Newtonian) gravitational theory, by calculating the minimum radius $R$ for a sphere of mass $M$ such that a photon can escape from the surface. (General Relativity gives $\left.R=2 G M / c^{2} .\right)$

Jacob Shpiece

Numerade Educator

Problem 52

How large would the Sun be if its density equaled the critical density of the universe, $\rho_{\mathrm{c}} \approx 10^{-26} \mathrm{~kg} / \mathrm{m}^{3}$ ? Express your answer in light-years and compare with the Earth-Sun distance and the diameter of our Galaxy.

Jacob Shpiece

Numerade Educator

Problem 53

The Large Hadron Collider in Geneva, Switzerland, can collide two beams of protons at an energy of $14 \mathrm{TeV}$. Estimate the time after the Big Bang probed by this energy.

Jacob Shpiece

Numerade Educator

Problem 54

(a) Use special relativity and Newton's law of gravitation to show that a photon of mass $m=E / c^{2}$ just grazing the Sun will be deflected by an angle $\Delta \theta$ given by
$$
\Delta \theta=\frac{2 G M}{c^{2} R}
$$
where $G$ is the gravitational constant, $R$ and $M$ are the radius and mass of the Sun, and $c$ is the speed of light.
(b) Put in values and show $\Delta \theta=0.87^{\prime \prime} .$ (General Relativity predicts an angle twice as large, $\left.1.74^{\prime \prime} .\right)$

Lainey Roebuck

Numerade Educator

Problem 55

Astronomers use an apparent magnitude $(m)$ scale to describe apparent brightness. It uses a logarithmic scale, where a higher number corresponds to a less bright star. (For example, the Sun has magnitude $-27,$ whereas most stars have positive magnitudes.) On this scale, a change in apparent magnitude by +5 corresponds to a decrease in apparent brightness by a factor of $100 .$ If Venus has an apparent magnitude of -4.4 whereas Sirius has an apparent magnitude of $-1.4,$ which is brighter? What is the ratio of the apparent brightness of these two objects?

Jacob Shpiece

Numerade Educator

Problem 56

Estimate the radius of a white dwarf whose mass is equal to that of the Sun by the following method, assuming there are $N$ nucleons and $\frac{1}{2} N$ electrons (why $\left.\frac{1}{2} ?\right):(a)$ Use Fermi-Dirac statistics (Section $40-6$ ) to show that the total energy of all the electrons
$$
E_{\mathrm{e}}=\frac{3}{5}\left(\frac{1}{2} N\right) \frac{h^{2}}{8 m_{\mathrm{e}}}\left(\frac{3}{\pi} \frac{N}{2 V}\right)^{\frac{2}{3}}
$$
[Hint: See Eqs. $40-12$ and $40-13$; we assume electrons fill energy levels from 0 up to the Fermi energy.] (b) The nucleons contribute to the total energy mainly via the gravitational force (note that the Fermi energy for nucleons is negligible compared to that for electrons - why?). Use a gravitational form of Gauss's law to show that the total gravitational potential energy of a uniform sphere of radius $R$ is
$$
-\frac{3}{5} \frac{G M^{2}}{R}
$$
by considering the potential energy of a spherical shell of radius $r$ due only to the mass inside it (why?) and integrate from $r=0$ to $r=R .$ (See also Appendix D.) $(c)$ Write the total energy as a sum of these two terms, and set $d E / d R=0$ to find the equilibrium radius, and evaluate it for a mass equal to the Sun's $\left(2.0 \times 10^{30} \mathrm{~kg}\right)$.

Lainey Roebuck

Numerade Educator

Problem 57

Determine the radius of a neutron star using the same argument as in Problem 56 but for $N$ neutrons only. Show that the radius of a neutron star, of 1.5 solar masses, is about $11 \mathrm{~km} .$

Lainey Roebuck

Numerade Educator

Problem 58

Use dimensional analysis with the fundamental constants $c$, $G,$ and $\hbar$ to estimate the value of the so-called Planck time. It is thought that physics as we know it can say nothing about the universe before this time.

Zhaojie Xu

Numerade Educator