Modern Physics

Kenneth S. Krane

Chapter 15

Cosmology: The Origin and Fate of the Universe - all with Video Answers

Educators

Chapter Questions

Problem 1

Use Hubble's law to estimate the wavelength of the $590.0 \mathrm{nm}$ sodium line as observed emitted from galaxies whose distance from us is $(a) 1.0 \times 10^{6}$ light-years; $(b) 1.0 \times 10^{9}$ light-years.

Eduard Sanchez

Eduard Sanchez

Numerade Educator

Problem 2

The light from a certain galaxy is red-shifted so that the wavelength of one of its characteristic spectral lines is doubled. Assuming the validity of Hubble's law, calculate the distance to this galaxy.

Eduard Sanchez

Numerade Educator

Problem 3

(a) Taking $u(E)=E N(E)$ as the energy density of the thermal radiation, with $N(E)$ given in Eq. $15.6,$ differentiate to find the energy at which the maximum of the radiation energy spectrum occurs. (b) Evaluate the peak photon energy of the 2.7 - $\mathrm{K}$ microwave background.

Eduard Sanchez

Numerade Educator

Problem 4

Starting with Eqs. 15.7 and 10.41 , show how to evaluate the numerical constants that appear in Eqs. 15.8 and $15.9 .$

Eduard Sanchez

Numerade Educator

Problem 5

Suppose an observer in a distant galaxy were observing the light from our Sun as the Sun moves directly toward the observer. Neglecting any net relative motion of the two galaxies, calculate the change in wavelength of the $121.5-\mathrm{nm}$ Lyman series line due to the rotation of our galaxy.

Eduard Sanchez

Numerade Educator

Problem 6

In Example 15.1 we calculated the change in wavelength of the Lyman $\alpha$ line due to the gravitational red shift. Compare this value with $(a)$ the special relativistic Doppler shift due to the rotation of the Sun and $(b)$ the thermal Doppler broadening (see Eq. 10.30). The Sun's radius is $6.96 \times 10^{8} \mathrm{~m}$, its rotational period is 26 days, and it surface temperature is $6000 \mathrm{~K}$

Eduard Sanchez

Numerade Educator

Problem 7

A satellite is in orbit at an altitude of $150 \mathrm{~km}$. We wish to communicate with it using a radio signal of frequency $10^{9} \mathrm{~Hz} .$ What is the gravitational change in frequency between a ground station and the satellite? (Assume $g$ doesn't change appreciably.)

Eduard Sanchez

Numerade Educator

Problem 8

According to the uncertainty principle, what is the minimum time interval necessary to measure a change in frequency of the magnitude observed in the Pound and Rebka experiment?

Eduard Sanchez

Numerade Educator

Problem 9

By drawing analogies between the Coulomb force law and the gravitational force law, use Eq. 6.8 for the deflection in Rutherford scattering to obtain Eq. 15.22 for the deflection of photons. Assume the photon behaves as if it has a mass $m=E / c^{2}$. (Hint: Write Eq. 6.8 in terms of the velocity of the particle instead of kinetic energy.)

Eduard Sanchez

Numerade Educator

Problem 10

In the binary star system known as PSR $1913+16,$ two neutron stars move about their common center of mass in highly elliptical orbits. Locate the orbital parameters for this motion, and add a row to Table 15.1 showing the precession angle expected from general relativity. (Hint: In Eq. 15.25 , $M$ is the total mass of the orbiting body and the central body.)

Eduard Sanchez

Numerade Educator

Problem 11

(a) Show that Eq. 10.57 for the radius of a neutron star of mass $M$ can be written $R=(12.3 \mathrm{~km})\left(M / M_{\odot}\right)^{-1 / 3}$ where $M_{\odot}$ is the mass of the Sun. ( $b$ ) Consider a star 1.5 times as massive as the Sun with a radius of $7 \times 10^{5} \mathrm{~km}$ (equal to the present radius of the Sun), rotating on its axis about once per year. (This is quite a slow rate of rotation-our Sun rotates about once per month.) If angular momentum is conserved in the collapse, what will be the final angular velocity? Assume the star can be represented as a sphere of uniform density, with rotational inertia $I=\frac{2}{5} M R^{2}$.

Eduard Sanchez

Numerade Educator

Problem 12

The rate of change of the cosmic expansion can be described in terms of a deceleration parameter $q=$ $-R\left(d^{2} R / d t^{2}\right) /(d R / d t)^{2} .(a)$ Evaluate $q$ for the matterdominated universe (Eq. 15.29 ) and the radiation-dominated universe (Eq. 15.31 ). ( $b$ ) By differentiating Eq. 15.28 , show that in a matter-dominated universe $q=4 \pi G \rho_{\mathrm{m}} / 3 H^{2}$. (Hint: Use $\rho_{\mathrm{m}} \propto R^{-3}$ to relate $d \rho_{\mathrm{m}} / d t$ to $\left.d R / d t .\right)$

Manik Pulyani

Numerade Educator

Problem 13

Derive Eq. 15.34

Eduard Sanchez

Numerade Educator

Problem 14

At what age did the universe cool below the threshold temperature for $(a)$ nucleon production; $(b)$ pi meson production?

Eduard Sanchez

Numerade Educator

Problem 15

(a) At what temperature was the universe hot enough to permit the photons to produce $\mathrm{K}$ mesons $\left(\mathrm{mc}^{2}=500 \mathrm{MeV}\right) ?$ (b) At what age did the universe have this temperature?

Eduard Sanchez

Numerade Educator

Problem 16

Derive Eqs. 15.36 and $15.37 .$

Eduard Sanchez

Numerade Educator

Problem 17

Suppose the difference between matter and antimatter in the early universe were 1 part in $10^{8}$ instead of 1 part in $10^{9}$.
(a) Evaluate the temperature at which deuterium begins to form. (b) At what age does this occur? (c) Evaluate the temperature and the corresponding time of radiation decoupling (when hydrogen atoms form).

Eduard Sanchez

Numerade Educator

Problem 18

What was the age of the universe when the nucleons consisted of $60 \%$ protons and $40 \%$ neutrons?

Eduard Sanchez

Numerade Educator

Problem 19

Assuming that the density of the universe is equal to its critical value and that $4.6 \%$ of the universe is baryonic matter, calculate the average number of baryons (nucleons) per cubic meter in the universe.

Eduard Sanchez

Numerade Educator

Problem 20

(a) Suppose the baryonic matter in the universe were composed of uniformly distributed stars of the mass of the Sun $\left(2.0 \times 10^{30} \mathrm{~kg}\right) .$ What would be the average spacing between the stars? Express your answer in light-years.
(b) Suppose instead that the baryonic matter were composed of uniformly distributed galaxies of the mass of the Milky Way $\left(1.2 \times 10^{42} \mathrm{~kg}\right) .$ Expressed in light-years, what would be the average distance between the galaxies?

Eduard Sanchez

Numerade Educator

Problem 21

Suppose the non-baryonic dark matter consists entirely of neutrinos. What is the average rest energy of the neutrinos that could account for this part of the mass of the universe? As a rough estimate, assume that the neutrino density is the same as the present photon density.

Penny Riley

Numerade Educator

Problem 22

Photons of visible light have energies between about 2 and $3 \mathrm{eV}$. ( $a$ ) Compute the number density of photons from the 2.73-K background radiation in that interval. (It is sufficient to characterize the visible region as $E=2.5 \mathrm{eV}$ with $d E=1.0 \mathrm{eV} .$ ) (b) Assume the eye can detect about 100 photons $/ \mathrm{cm}^{3}$. At what temperature would the background radiation be visible? At what age of the universe would this have occurred?

Eduard Sanchez

Numerade Educator

Problem 23

Consider the universe at a temperature of $5000 \mathrm{~K}$. ( a) $\mathrm{At}$ what age did this occur, and during which stage of the evolution of the universe? (b) Evaluate the average photon energy at that time. ( $c$ ) If there are $10^{9}$ photons per nucleon, evaluate the ratio between the radiation density and the mass density at that time.

Eduard Sanchez

Numerade Educator

Problem 24

The early universe was radiation dominated, and the present universe is matter dominated. ( $a$ ) At what temperature were the radiation and matter densities equal? (b) What was the age of the universe when this occurred?

Eduard Sanchez

Numerade Educator

Problem 25

A neutron star of 2.00 solar masses is rotating at a rate of 1.00 revolutions per second. ( $a$ ) What is the radius of the neutron star? (See Problem 11.) ( $b$ ) Find its rotational kinetic energy. (c) If its rotational speed slows by 1 part in $10^{9}$ per day, find the loss in rotational kinetic energy per day. ( $d$ ) Assuming that the entire energy loss goes into radiation, find the radiative power. $(e)$ If the star is $10^{4}$ light-years from Earth, what would be the average power received by an antenna of area $10 \mathrm{~m}^{2}$ if the star's energy were distributed uniformly in space instead of concentrated in a narrow beam?

Eduard Sanchez

Numerade Educator

Problem 26

Because we don't yet have a quantum theory of gravity, we cannot analyze the properties of the universe before the Planck time, about $10^{-43} \mathrm{~s}$. If we assume that the properties of the universe during that era were determined by quantum theory, relativity, and gravity, the Planck time should be characterized by the fundamental constants of those three theories: $h, c,$ and $G .$ We can therefore write $t \propto h^{i} c^{j} G^{k}$ where $i, j,$ and $k$ are exponents to be determined. $(a)$ Using
(b) Assuming a dimensional analysis, determine $i, j,$ and $k$. the proportionality parameter is of order unity, evaluate $t$.
(c) What was the size of the observable universe at the Planck time?

Manik Pulyani

Numerade Educator

Problem 27

Show that the spacetime interval given by Eq. 15.18 is invariant with respect to the Lorentz transformation. That is, show that $(d s)^{2}=\left(d s^{\prime}\right)^{2},$ where $\left(d s^{\prime}\right)^{2}=\left(c d t^{\prime}\right)^{2}-\left(d x^{\prime}\right)^{2}$

Manik Pulyani

Numerade Educator

Problem 28

Light from star $S$ in Figure 15.30 passes a distance $b$ from a galaxy $L$, where it is deflected by an angle $\alpha$ and then reaches the observer $O,$ who sees an image of the star at $I$. (For simplicity, assume the gravitational deflection takes place at a single point, and assume all angles in the figure are very small.) The galaxy (of mass $M$ ) is a distance $d_{L}$ from the observer, and the star is a distance $d_{S}$ from the observer. ( $a$ ) For small angles, show that $\theta d_{S}=\beta d_{S}+\left(4 G M / \theta d_{L} c^{2}\right)\left(d_{S}-d_{L}\right) .$ (Hint: Use Eq.
15.22 for the deflection angle $\alpha$ when the impact parameter is $b$ instead of $R,$ but double the value to account for the difference between the special and general relativity predictions.) ( $b$ ) Solve the resulting quadratic equation for $\theta$ and show that there are two image positions whose locations differ by $\Delta \theta=\sqrt{\beta^{2}+4 \theta_{E}^{2}},$ where the Einstein angle $\theta_{\mathrm{E}}$
is $\sqrt{4 G M\left(d_{S}-d_{L}\right) / c^{2} d_{S} d_{L}}$. This is an example of gravitational lensing, an effect of general relativity that has been observed for distant objects that appear in multiple images when their light travels a path through spacetime that is curved by an intervening galaxy. ( $c$ ) When the star, lensing galaxy, and observer lie along a single line, something other than two images appears. Given the symmetry of the figure when $\beta=0$, what do you expect to be observed in this case?

Manik Pulyani

Numerade Educator