Modern Physics

Paul A. Tipler, Ralph A. Llewellyn

Chapter 13

Astrophysics and Cosmology - all with Video Answers

Educators

Chapter Questions

Problem 1

Measurement of the Doppler shift of spectral lines in light from the east and west limbs of the Sun at the solar equator reveal that the tangential velocities of the limbs differ by $4 \mathrm{~km} / \mathrm{s}$. Use this result to compute the approximate period of the Sun's rotation $\left(R_{\odot}=6.96 \times 10^{5} \mathrm{~km}\right)$

Narayan Hari

Narayan Hari

Numerade Educator

Problem 2

The gravitational potential energy $U$ of a self-gravitating spherical body of mass $M$ and radius $R$ is a function of the details of the mass distribution. For the Sun, $U_{\odot}=-2 G M_{\odot}^{2} R_{\odot}$. What would be the approximate lifetime of the Sun, radiating at its present rate, if the source of its emitted energy were entirely derived from gravitational contraction? $\left(M_{\odot}=1.99 \times 10^{30} \mathrm{~kg} .\right)$

Narayan Hari

Numerade Educator

Problem 3

Lithium, beryllium, and boron $(Z=3,4,$ and $5,$ respectively) have very low abundances in the cosmos compared to many heavier elements (see Figure $13-33$ ). Considering the fusion of He to $\mathrm{C},$ explain these low abundances.

Nicole Smina

Numerade Educator

Problem 4

The Sun is moving with speed $2.5 \times 10^{5} \mathrm{~m} / \mathrm{s}$ in a circular orbit about the center of the Galaxy. How long (in Earth years) does it take to complete one orbit? How many orbits has it completed since it was formed?

Narayan Hari

Numerade Educator

Problem 5

The reason that massive neutrinos were considered as a candidate for solving the missing mass problem is that, at the conclusion of the lepton era, the universe contained about equal numbers of photons and neutrinos. They are still here, for the most part. The former can be observed and their density is measured to be about 500 photons $/ \mathrm{cm}^{3} ;$ thus, there must be about that number density of neutrinos in the universe, too. If neutrinos have a nonzero mass and if the cosmological expansion has reduced their average speed so that their energy is now primarily mass, what would be the individual neutrino mass (in eV/c $^{2}$ ) necessary to account for the missing mass of the universe? Recall that the observed mass of the stars and galaxies (including the dust and gas) accounts for only about 4 percent of that needed to close the universe

Narayan Hari

Numerade Educator

Problem 6

Using data from Table $13-3,$ construct a graph that demonstrates the validity of Equation $13-17$.

Deandre Johnson

Deandre Johnson

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Problem 7

Recalling that the light-year $c \cdot y$ is the distance light travels in one year, compute in meters the distance equivalent to 1 light-second, 1 light-minute, 1 light-hour, and I light-day.

Narayan Hari

Numerade Educator

Problem 8

A unit of length often used by astronomers to measure distances in "nearby" space is the parsec (pc), defined as the distance at which a star subtends a parallax angle of one arc second due to Earth's orbit around the Sun (see Equation $13-11$ and Example $13-4)$. The practical limit of such measurements is 0.01 arc second. ( $a$ ) How many light-years is $1 \mathrm{pc} ?(b)$ If the density of stars in the Sun's region of the Milky Way is $0.08 \mathrm{star} / \mathrm{pc}^{3},$ how many stars could, in principle, have their distances from us measured by the trigonometric parallax method?

Narayan Hari

Numerade Educator

Problem 9

Astronomers often use the apparent magnitude $m$ as a means of comparing the visual brightness of stars and relating the comparison to the luminosity and distance to "standard" stars, such as the Sun (see Equation $13-9$ ). The difference in the apparent magnitudes of two stars $m_{1}$ and $m_{2}$ is defined as $m_{2}-m_{1}=2.5$ log $\left(f_{1} / f_{2}\right),$ a relation based on the logarithmic response of the human eye to the brightness of objects. Pollux, one of the "twins" in the constellation Gemini, has apparent magnitude 1.16 and is 12 pc away.

Eduard Sanchez

Numerade Educator

Problem 9

Astronomers often use the apparent magnitude $m$ as a means of comparing the visual brightness of stars and relating the comparison to the luminosity and distance to "standard" stars, such as the Sun (see Equation $13-9$ ). The difference in the apparent magnitudes of two stars $m_{1}$ and $m_{2}$ is defined as $m_{2}-m_{1}=2.5 \log \left(f_{1} / f_{2}\right),$ a relation based on the logarithmic response of the human eye to the brightness of objects. Pollux, one of the "twins" in the constellation Gemini, has apparent magnitude 1.16 and is 12 pe away.

Andrew Duncan

Numerade Educator

Problem 10

Using the H-R diagram (Figure $13-17$ ), determine the effective temperature and the luminosity of a star whose mass is $(a) 0.3 M_{\odot}$ and $(b) 3 M_{\odot}(c)$ Compute the radius of each star. ( $d$ ) Determine their expected lifetimes relative to that of the Sun.

Sarah Mccrumb

Numerade Educator

Problem 11

Two stars in a binary system are $100 \mathrm{c} \cdot \mathrm{y}$ from Earth and separated from each other by $10^{8} \mathrm{~km}$. What is the angular separation of the stars in arc seconds? In degrees?

Andrew Duncan

Numerade Educator

Problem 12

Compute the energy required (in MeV) to produce each of the photodisintegration reactions in Equations $13-18$ and $13-19$.

Andrew Duncan

Numerade Educator

Problem 13

The gas shell of the planetary nebula shown in Figure $13-18$ is expanding at $24 \mathrm{~km} / \mathrm{s}$. Its diameter is $1.5 \mathrm{c} \cdot \mathrm{y}$. ( $a$ ) How old is the gas shell? $(b)$ If the central star of the planetary nebula is 12 times as luminous as the Sun and 15 times hotter, what is the radius of the central star in units of $R_{\odot} ?$

Andrew Duncan

Numerade Educator

Problem 14

Calculate the Schwarzschild radius of a star whose mass is equal to that of $(a)$ the Sun, $(b)$ Jupiter, (c) Earth. (The mass of Jupiter is approximately 318 times that of Earth.)

Narayan Hari

Numerade Educator

Problem 15

Consider a neutron star whose mass equals $2 M_{\odot} .(a)$ Compute the star's radius. ( $b$ ) If the neutron star is rotating at $0.5 \mathrm{rev} / \mathrm{s}$ and assuming its density to be uniform, what is its rotational kinetic energy? (c) If its rotation slows by 1 part in $10^{8}$ per day and the lost kinetic energy is all radiated, what is the star's luminosity?

Andrew Duncan

Numerade Educator

Problem 16

If the 90 percent of the Milky Way's mass that is "missing" resides entirely in a large black hole at the center of the Galaxy, what would be the black hole's $(a)$ mass and (b) radius?

Narayan Hari

Numerade Educator

Problem 17

Redshift measurements for a particular galaxy indicate that it has a recession velocity of $72,000 \mathrm{~km} / \mathrm{s}$. ( $a$ ) Compute the distance to the galaxy. ( $b$ ) The value of Hubble's constant depends critically on calibration distance measurements, which are difficult to make. If the calibration distance measurements are in error by 10 percent, by how much is the age calculated from Equation $13-28$ in error?

Narayan Hari

Numerade Educator

Problem 18

The bright core of a certain Seyfert galaxy had a luminosity of $10^{10} L_{\odot}$. The luminosity increased by 100 percent in a period of 18 months. Show that this means that the energy source of the core is less than $9.45 \times 10^{4} \mathrm{AU}$ in diameter. How does this compare to the diameter of the Milky Way?

Narayan Hari

Numerade Educator

Problem 20

Evaluate Equation $13-33$ for the critical density of the universe.

Narayan Hari

Numerade Educator

Problem 21

Cosmological theory suggests that the average separation of galaxies, that is, the scale of the universe, is inversely proportional to the absolute temperature. If that is true, relative to the present size, how large was the universe compared to the scale today
( $b$ ) $10^{6}$ years ago,
(a) 2000 years ago,
(c) $t=10 \mathrm{~s}$ after the Big Bang, $(d)$ when $t=1 \mathrm{~s},$ and
(e) when $t=10^{-6} \mathrm{~s} ?$

Eduard Sanchez

Numerade Educator

Problem 22

Determine the value of the mass density of the universe for $t=$ Planck time. How does this compare to the density of the proton? Of osmium?

Eduard Sanchez

Numerade Educator

Problem 23

At what wavelength is the blackbody radiation distribution of the cosmic microwave background at a maximum?

Andrew Duncan

Numerade Educator

Problem 24

How long after the Big Bang did it take the universe to cool to the threshold temperature for the formation of muons? What would be the mass of a particle-antiparticle pair that could be formed by the average energy of the current $2.725 \mathrm{~K}$ background radiation?

Eduard Sanchez

Numerade Educator

Problem 25

Show that the present mass density of the universe $\rho_{0}=R(t) \rho(t)$

John Nicolle

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Problem 26

If Hubble's law is true for an observer in the Milky Way (i.e., us), prove that it must also be true for observers in other galaxies. (Hint: Use the vector property of the velocity.)

Robert Zaballa

Numerade Educator

Problem 27

Find the minimum magnitude of the radius $a$ that a dust particle in orbit around the Sun may have in order to avoid being blown out of the solar system by the Sun's radiation pressure. Assume that the particle is a sphere of mass $m$ with the same density $\rho$ as Earth, $5500 \mathrm{~kg} / \mathrm{m}^{3}$. Ignore the solar wind and the solar magnetic field.

Bettina Hanlon

Numerade Educator

Problem 28

Show that the mass density of the universe at redshift $z$ is given by $\rho(z)=\rho(1+z)^{3}$

Farhanul Hasan

Numerade Educator

Problem 29

When the Sun was formed, about 75 percent of its mass was hydrogen, of which only about 13 percent ever becomes available for fusion. (The rest is in regions of the Sun where the temperature is too low for fusion reactions to occur.) $M_{\odot}=2 \times 10^{30} \mathrm{~kg}$ and the Sun fuses about $6 \times 10^{11} \mathrm{~kg} / \mathrm{s}$. ( a) Compute the total mass of hydrogen available for fusion during the Sun's lifetime. (b) How long (in years) will the Sun's initial supply of hydrogen last? $(c)$ Since the solar system is currently about $4.6 \times 10^{9} \mathrm{y}$ old, when should we begin to worry about the Sun running out of hydrogen for fusion?

Anand Jangid

Numerade Educator

Problem 30

Supernova SN1987A was first visible at Earth in 1987 . ( $a$ ) How many years B.P. (before present) did the explosion occur?
(b) If protons with $100 \mathrm{GeV}$ of kinetic energy were produced in the event, when should they arrive at Earth?

Andrew Duncan

Numerade Educator

Problem 31

Assume that the Sun when it first formed was composed of 70 percent hydrogen. How many hydrogen nuclei were there in the Sun at that time? How much energy would ultimately be released if all of the hydrogen nuclei fused into helium? Astrophysicists have predicted that the Sun can radiate energy at its current rate until about 23 percent of the hydrogen has been "burned." What total lifetime for the Sun does that prediction imply? Compare these results with the corresponding ones from Problem $13-29 .$

Andrew Duncan

Numerade Educator

Problem 32

Kepler's third law states that the square of a planet's orbital speed is proportional to the cube of its average orbital radius. Use Kepler's third law to answer each of the following questions. ( $a$ ) The Moon's orbital radius is $3.84 \times 10^{5} \mathrm{~km}$ and it orbits Earth once every 27.3 d. Neglecting the moon's mass, compute the mass of Earth. ( $b$ ) Io (one of Jupiter's moons) orbits Jupiter once every $42.5 \mathrm{~h}$ in a near-circular orbit of average radius $4.22 \times 10^{5} \mathrm{~km}$. Neglecting Io's mass, compute the mass of Jupiter. ( $c$ ) Compute the orbital period of the International Space Station as it orbits $300 \mathrm{~km}$ above Earth's surface. (d) Charon, a moon of Pluto, orbits that body once every $6.4 \mathrm{~d}$ at an average distance of $1.97 \times 10^{4} \mathrm{~km} .$ Compute the total mass of Pluto and Charon. What fraction of Earth's mass is this? (e) Using the data for the star S2, compute the volume (upper limit) that confines the black hole at the center of the Milky Way, Compare the result with the volume of the Sun.

Vishal Parmar

Numerade Educator

Problem 33

Consider an eclipsing binary whose orbital plane is parallel to our line of sight. Doppler measurements of the radial velocity of each component of the binary are shown in Figure $13-36 .$ Assume that the mass $m_{1}>m_{2}$ and that the orbits of each component about the center of mass are circular. ( $a$ ) What is the period $T$ and the angular frequency $\omega$ of the binary? ( $b$ ) Show that in this case $\left(m_{1}+m_{2}\right)=\left(\omega^{2} r^{3}\right) / G,$ where $r=$ separation of the binary. ( $c$ ) Compute the values of $m_{1}, m_{2}$, and $r$ from the data in the $v$ versus $t$ graph.

Andrew Duncan

Numerade Educator

Problem 34

Prove that the total energy of Earth's orbital motion $E=\left(m v^{2} / 2\right)+\left(-G M_{\odot} m / r\right)$ is equal to one-half of its gravitational potential energy $\left(-G M_{\odot} m / r\right),$ where $r$ is Earth's orbit radius.

Andrew Duncan

Numerade Educator

Problem 35

Given the currently accepted value of the Hubble constant and the fact that the average matter density of the universe is one $\mathrm{H}$ atom $/ \mathrm{m}^{3},$ what creation rate of new $\mathrm{H}$ atoms would be necessary in a steady-state model to maintain the present mass density, even though the universe is expanding? (Give your answer in $\mathrm{H}$ atoms $/ \mathrm{m}^{3}$ per $10^{6}$ years.) Would you expect such a spontaneous creation rate to be readily observable?

Zachary Warner

Numerade Educator

Problem 36

The ability of a planet to retain particular gases in an atmosphere depends on the temperature that its atmosphere has (or would have) and the escape velocity for the planet. In general, if the average speed of a particular gas molecule exceeds $1 / 6$ of the escape velocity, that gas will disappear from the atmosphere in about $10^{8}$ years. ( $a$ ) Graph the average speed of $\mathrm{H}_{2} \mathrm{O}, \mathrm{CO}_{2}, \mathrm{O}_{2}, \mathrm{CH}_{4}, \mathrm{H}_{2},$ and He from $50 \mathrm{~K}$ to $1000 \mathrm{~K}$. On the same graph show the points representing $1 / 6$ of the escape velocity versus average temperature of the atmosphere for the planets in Table $13-5$ below. $(b)$ Show that the escape speed $v$ from a planet is given by
$$
\frac{v}{v_{\text {Earth }}}=\sqrt{\frac{\left(M / M_{\text {Earth }}\right)}{\left(R / R_{\text {Earth }}\right)}}
$$
(c) Which of the six gases plotted probably would and would not currently be found in the atmospheres of the solar system bodies in the table? Explain each answer briefly.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 37

Using the parallax technique, compute the distance to ( $a$ ) Alpha Centauri (parallax angle 0.742 arc second) and $(b)$ Procyon (parallax angle 0.0286 arc second). Express each answer in both light-years and parsecs.

Andrew Duncan

Numerade Educator

Problem 38

As the Sun evolves into a red giant star, suppose that its luminosity increases by a factor of $10^{2}$. Show that Earth's oceans will evaporate, but that the water vapor will not escape from the atmosphere

Andrew Duncan

Numerade Educator

Problem 39

The approximate mass of dust in the Galaxy can be computed from the observed extinction of starlight. Assuming the mean radius of dust grains to be $R$ with a uniform number density $n$ grains $/ \mathrm{cm}^{3},(a)$ show that the mean free path $d_{0}$ of a photon in interstellar dust is given by $d_{0}=1 /\left(n \pi R^{2}\right) .$ (b) Starlight traveling toward an Earth observer a distance $d$ from the star has intensity
$$
I=I_{0} e^{-d / d_{0}}
$$
In the vicinity of the Sun a measurement of $I$ yields $d_{0}=3000 \mathrm{c} \cdot \mathrm{y}$. If $R=10^{-5} \mathrm{~cm},$ calculate $n$. (c) The average mass density of solid material in the Galaxy is $2 \mathrm{~g} / \mathrm{cm}^{3}$ and in the disk the density of stars is about $1 M_{\odot} / 300(c \cdot y)^{3}$. Compute the ratio of the mass density of dust to the mass density of stars, assuming $1 M_{\odot}$ in $300(c \cdot y)^{3}$.

Salamat Ali

Numerade Educator

Problem 40

The supernova SN1987A certainly produced some heavy elements. Compared to the energy released in fusing $56{ }^{1} \mathrm{H}$ atoms into one ${ }^{56} \mathrm{Fe}$ atom starting from the protonproton cycle, how much energy would be required to fuse two ${ }^{56} \mathrm{Fe}$ atoms into one ${ }^{112} \mathrm{Cd}$ atom?

Bettina Hanlon

Numerade Educator

Problem 41

Current theory suggests that black holes evaporate by the emission of Hawking radiation in a time $t$ that depends on the mass $M$ of the black hole according to the following relation:
$$
t=\left(1.024 \times 10^{4} \pi^{2} \mathrm{~m}^{3} / \mathrm{s}^{2}\right) G^{2} M^{2} / h c^{4}
$$
(a) Explain without calculating anything why the formula implies that high-mass black holes have longer lifetimes than low-mass ones and why the rate of evaporation accelerates as the black hole loses mass. ( $b$ ) Compute the lifetime of a black hole whose mass equals $1 M_{\odot} .$ Compare this time with the current age of the universe. (c) According to some theories, the largest black hole that could conceivably form would have a mass $10^{12} M_{\odot},$ of the order of the mass of an entire galaxy. What would be the lifetime of a black hole that large?

Manish Jain

Numerade Educator