Show that the Schwarzschild radius for Earth is 8.9 mm

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

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

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?

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.)

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

(II) We saw earlier (Chapter 14) 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? Equation Transcription: Text Transcription: 1.3 \times 10^{3} W / m^2

Estimate the angular width that our Galaxy would subtend if observed from the nearest galaxy to us (Table 331). Compare to the angular width of the Moon from Earth

Assuming our Galaxy represents a good average for all other galaxies, how many stars are in the observable universe?

Calculate the density of a white dwarf whose mass is equal to the Suns and whose radius is equal to the Earths. How many times larger than Earths density is this?

A neutron star whose mass is 1.5 solar masses has a radius of about 11 km. Calculate its average density and compare to that for a white dwarf (Problem 9) and to that of nuclear matter

(II) A star is 56 pc away. What is its parallax angle? State (a) in seconds of arc, and (b) in degrees.

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

A star is 85 pc away. How long does it take for its light to reach us?

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 nm whereas that of the other peaks at 450 nm. Use Wiens law and the Stefan-Boltzmann equation (Eq. 146) to estimate their relative distances from us. [Hint: See Examples 334 and 335.

(III) Stars located in a certain cluster are assumed 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. 14–6.

Show that the Schwarzschild radius for Earth is 8.9 mm

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

What mass will give a Schwarzschild radius equal to that of the hydrogen atom in its ground state?

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

Describe a triangle, drawn on the surface of a sphere, for which the sum of the angles is (a) and

What is the apparent deflection of a light beam in an elevator (Fig. 3313) which is 2.4 m wide if the elevator is accelerating downward at

The redshift of a galaxy indicates a recession velocity of How far away is it?

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

A galaxy is moving away from Earth. The blue hydrogen line at 434 nm emitted from the galaxy is measured on Earth to be 455 nm. (a) How fast is the galaxy moving? (b) How far is it from Earth based on Hubbles law?

Estimate the wavelength shift for the 656.3-nm line in the Balmer series of hydrogen emitted from a galaxy whose distance from us is (a) (b) 7.0 * 107 7.0 * 10 ly

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

What is the speed of a galaxy with z = 0.060?

What would be the redshift parameter z for a galaxy traveling away from us at v = 0.075c?

Estimate the distance d from the Earth to a galaxy whose redshift parameter z = 1.

(II) Estimate the speed of a galaxy, and its distance from us, if the wavelength for the hydrogen line at 434 nm is measured on Earth as being 610 nm.

Radiotelescopes are designed to observe 21-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 cm longer than the normal 21-cm wavelength. Estimate the distance to this galaxy

(III) Starting from Eq. 33–3, show that the Doppler shift in wavelength is \(\Delta \lambda/\lambda_{rest}\ \approx\ v/c\) (Eq. 33-6) for v << c. [Hint: Use the binomial expansion.]

(I) Calculate the wavelength at the peak of the blackbody radiation distribution at 2.7 K using Wien’s law.

Calculate the peak wavelength of the CMB at 1.0 s after the birth of the universe. In what part of the EM spectrum is this radiation?

The critical density for closure of the universe is State in terms of the average number of nucleons per cubic meter.

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) (b) (c) and

At approximately what time had the universe cooled below the threshold temperature for producing (a) kaons (b) and

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 galaxies, each with about stars like our Sun. (b) Estimate the density of dark matter in the universe.

Use conservation of angular momentum to estimate the angular velocity of a neutron star which has collapsed to a diameter of 16 km, from a star whose core radius was equal to that of Earth Assume its mass is 1.5 times that of the Sun, and that it rotated (like our Sun) about once a month.

By what factor does the rotational kinetic energy change when the star in Problem 39 collapses to a neutron star?

Suppose that three main-sequence stars could undergo the three changes represented by the three arrows, A, B, and C, in the HR diagram of Fig. 3335. For each case, describe the changes in temperature, intrinsic luminosity, and size

Assume that the nearest stars to us have an intrinsic luminosity about the same as the Suns. Their apparent brightness, however, is about times fainter than the Sun. From this, estimate the distance to the nearest stars.

A certain pulsar, believed to be a neutron star of mass 1.5 times that of the Sun, with diameter 16 km, is observed to have a rotation speed of If it loses rotational kinetic energy at the rate of 1 part in per day, which is all transformed into radiation, what is the power output of the star?

The nearest large galaxy to our Galaxy is about \(2 \times 10^6\ Iy\) away. If both galaxies have a mass of \(4 \times 10^{41}\ kg\), with what gravitational force does each galaxy attract the other? Ignore dark matter.

How large would the Sun be if its density equaled the critical density of the universe, Express your answer in light-years and compare with the EarthSun distance and the diameter of our Galaxy.

Two stars, whose spectra peak at 660 nm and 480 nm, respectively, both lie on the main sequence. Use Wien’s law, the Stefan-Boltzmann equation, and the H–R diagram (Fig. 33–6) to estimate the ratio of their diameters.

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

(a) What temperature would correspond to 14-TeV collisions at the LHC? (b) To what era in cosmological history does this correspond? [Hint: See Fig. 3329.]

In the later stages of stellar evolution, a star (if massive enough) will begin fusing carbon nuclei to form, for example, magnesium: (a) How much energy is released in this reaction (see Appendix B)? (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. 301) so that the strong force can come into play? (c) What temperature does this kinetic energy correspond to?

Consider the reaction and answer the same questions as in Problem 49.

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.

Estimate the mass of our observable universe using the following assumptions: Our universe is spherical in shape, it has been expanding at the speed of light since the Big Bang, and its density is the critical density

Problem 1COQ How many cm3 are in 1.0 m3? (a) 10. (b) 100. (c) 1000. (d) 10,000. (e) 100,000. (f) 1,000,000.

Problem 1MCQ Which one of the following is not expected to occur on an H–R diagram during the lifetime of a single star? (a) The star will move off the main sequence toward the upper right of the diagram. (b) Low-mass stars will become white dwarfs and end up toward the lower left of the diagram. (c) The star will move along the main sequence from one place to another. (d) All of the above.

Problem 1P (I) The parallax angle of a star is 0.00029°. How far away is the star?

Problem 1Q The Milky Way was once thought to be “murky” or “milky” but is now considered to be made up of point sources. Explain.

Problem 1SL Estimate what neutrino mass (in eV/c2) would provide the critical density to close the universe. Assume the neutrino density is, like photons, about 109 times that of nucleons, and that nucleons make up only (a) 2% of the mass needed, or (b) 5% of the mass needed.

Problem 2MCQ When can parallax be used to determine the approximate distance from the Earth to a star? (a) Only during January and July. (b) Only when the star’s distance is relatively small. (c) Only when the star’s distance is relatively large. (d) Only when the star appears to move directly toward or away from the Earth. (e) Only when the star is the Sun. (f) Always. (g) Never.

Problem 2P (I) A star exhibits a parallax of 0.27 seconds of arc. How far away is it?

Problem 2Q A star is in equilibrium when it radiates at its surface all the energy generated in its core. What happens when it begins to generate more energy than it radiates? Less energy? Explain.

Problem 2SL Describe how we can estimate the distance from us to other stars. Which methods can we use for nearby stars, and which can we use for very distant stars? Which method gives the most accurate distance measurements for the most distant stars?

Problem 3MCQ Observations show that all galaxies tend to move away from Earth, and that more distant galaxies move away from Earth at faster velocities than do galaxies closer to the Earth. These observations imply that (a) the Earth is the center of the universe. (b) the universe is expanding. (c) the expansion of the universe will eventually stop. (d) All of the above.

Problem 3P (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?

Problem 3Q Describe a red giant star. List some of its properties.

The evolution of stars, as discussed in Section 33–2, can lead to a white dwarf, a neutron star, or even a black hole, depending on the mass. (a) Referring to Sections 33–2 and 33–4, 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 \(\left(r_{i}, r_{i i}, r_{i i i}\right)\) Equation Transcription: Text Transcription: (r_i, r_ii, r_iii)

Problem 4MCQ Which process results in a tremendous amount of energy being emitted by the Sun? (a) Hydrogen atoms burn in the presence of oxygen—that is, hydrogen atoms oxidize. (b) The Sun contracts, decreasing its gravitational potential energy. (c) Protons in hydrogen atoms fuse, forming helium nuclei. (d) Radioactive atoms such as uranium, plutonium, and cesium emit gamma rays with high energy. (e) None of the above.

Problem 4P (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.)

Does the H–R diagram directly reveal anything about the core of a star?

(a) Describe some of the evidence that the universe began with a “Big Bang.” (b) How does the curvature of the universe affect its future destiny? (c) How does dark energy affect the possible future of the universe?

Problem 5MCQ Which of the following methods can be used to find the distance from us to a star outside our galaxy? Choose all that apply. (a) Parallax. (b) Using luminosity and temperature from the H–R diagram and measuring the apparent brightness. (c) Using supernova explosions as a “standard candle.” (d) Redshift in the line spectra of elements and compounds.

Problem 5P (II) When our Sun becomes a red giant, what will be its average density if it expands out to the orbit of Mercury (6 X 1010 from the Sun)?

Problem 5Q Why do some stars end up as white dwarfs, and others as neutron stars or black holes?

Problem 5SL When stable nuclei first formed, about 3 minutes after the Big Bang, there were about 7 times more protons than neutrons. Explain how this leads to a ratio of the mass of hydrogen to the mass of helium of 3 : 1. This is about the actual ratio observed in the universe.

The history of the universe can be determined by observing astronomical objects at various (large) distances from the Earth. This method of discovery works because (a) time proceeds at different rates in different regions of the universe. (b) light travels at a finite speed. (c) matter warps space. (d) older galaxies are farther from the Earth than are younger galaxies.

Problem 6P (II) We saw earlier (Chapter 14) that the rate energy reaches the Earth from the Sun (the “solar constant”) is about 1.3 X 103 W/m2. What is (a) the apparent brightness b of the Sun, and (b) the intrinsic luminosity L of the Sun?

Problem 6Q If you were measuring star parallaxes from the Moon instead of Earth, what corrections would you have to make? What changes would occur if you were measuring parallaxes from Mars?

Problem 6SL Explain what the 2.7-K cosmic microwave background radiation is. Where does it come from? Why is its temperature now so low?

Problem 7MCQ Where did the Big Bang occur? (a) Near the Earth. (b) Near the center of the Milky Way Galaxy. (c) Several billion light-years away. (d) Throughout all space. (e) Near the Andromeda Galaxy.

Problem 7P (II) Estimate the angular width that our Galaxy would subtend if observed from the nearest galaxy to us (Table 33–1). Compare to the angular width of the Moon from Earth.

Problem 7Q Cepheid variable stars change in luminosity with a typical period of several days. The period has been found to have a definite relationship with the average intrinsic luminosity of the star. How could these stars be used to measure the distance to galaxies?

Problem 7SL We cannot use Hubble’s law to measure the distances to nearby galaxies, because their random motions are larger than the overall expansion. Indeed, the closest galaxy to us, the Andromeda Galaxy, 2.5 million light-years away, is approaching us at a speed of about 130 km/s. (a) What is the shift in wavelength of the 656-nm line of hydrogen emitted from the Andromeda Galaxy, as seen by us? (b) Is this a redshift or a blueshift? (c) Ignoring the expansion, how soon will it and the Milky Way Galaxy collide?

Problem 8MCQ When and how were virtually all of the elements of the Periodic Table formed? (a) In the very early universe a few seconds after the Big Bang. (b) At the centers of stars during their main-sequence phases. (c) At the centers of stars during novae. (d) At the centers of stars during supernovae. (e) On the surfaces of planets as they cooled and hardened.

Problem 8P (II) Assuming our Galaxy represents a good average for all other galaxies, how many stars are in the observable universe?

Problem 8Q What is a geodesic?What is its role in General Relativity?

We know that there must be dark matter in the universe because (a) we see dark dust clouds. (b) we see that the universe is expanding. (c) we see that stars far from the galactic center are moving faster than can be explained by visible matter. (d) we see that the expansion of the universe is accelerating.

Problem 9P (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?

If it were discovered that the redshift of spectral lines of galaxies was due to something other than expansion, how might our view of the universe change? Would there be conflicting evidence? Discuss.

Problem 10MCQ Acceleration of the universe’s expansion rate is due to (a) the repulsive effect of dark energy. (b) the attractive effect of dark matter. (c) the attractive effect of gravity. (d) the thermal expansion of stellar cores.

(II) A neutron star whose mass is 1.5 solar masses has a radius of about 11 km. Calculate its average density and compare to that for a white dwarf (Problem 9) and to that of nuclear matter.

Problem 10Q Almost all galaxies appear to be moving away from us. Are we therefore at the center of the universe? Explain.

Problem 11P (II) A star is 56 pc away. What is its parallax angle? State (a) in seconds of arc, and (b) in degrees.

Problem 11Q If you were located in a galaxy near the boundary of our observable universe, would galaxies in the direction of the Milky Way appear to be approaching you or receding from you? Explain.

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

Problem 12Q Compare an explosion on Earth to the Big Bang. Consider such questions as: Would the debris spread at a higher speed for more distant particles, as in the Big Bang? Would the debris come to rest? What type of universe would this correspond to, open or closed?

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

If nothing, not even light, escapes from a black hole, then how can we tell if one is there?

The Earth’s age is often given as about 4.6 billion years. Find that time on Fig. 33–29. Modern humans have lived on Earth on the order of 200,000 years. Where is that on Fig. 33–29?

(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 750mm whereas that of the other peaks at 50 mm. Use Wien’s law and the Stefan-Boltzmann equation (Eq. 14–6) to estimate their relative distances from us. [Hint: See Examples 33–4 and 33–5.]

(III) Stars located in a certain cluster are assumed to be about the same distance from us. Two such stars have spectra that peak at \(\lambda_{1}=470 \mathrm{~nm} \text { and } \lambda_{2}=720 \mathrm{~nm}\), 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. 14–6. Equation Transcription: Text Transcription: \lambda_1=470 nm and \lambda_2=720 nm b_1/b_2=0.091

Problem 15Q Why were atoms, as opposed to bare nuclei, unable to exist until hundreds of thousands of years after the Big Bang?

Problem 16P (I) Show that the Schwarzschild radius for Earth is 8.9 mm.

Problem 16Q (a) Why are Type Ia supernovae so useful for determining the distances of galaxies? (b) How are their distances actually measured?

Problem 17P (II) What is the Schwarzschild radius for a typical galaxy (like ours)?

Problem 17Q Under what circumstances would the universe eventually collapse in on itself?

Problem 18P (II) What mass will give a Schwarzschild radius equal to that of the hydrogen atom in its ground state?

Problem 18Q (a) Why did astronomers expect that the expansion rate of the universe would be decreasing (decelerating) with time? (b) How, in principle, could astronomers hope to determine whether the universe used to expand faster than it does now?

Problem 19P (II) What is the maximum sum-of-the-angles for a triangle on a sphere?

Problem 20P (II) Describe a triangle, drawn on the surface of a sphere, for which the sum of the angles is (a) 359o and (b) 179o.

(III) What is the apparent deflection of a light beam in an elevator (Fig. 33–13) which is \(2.4 \mathrm{~m}\) wide if the elevator is accelerating downward at \(9.8 \mathrm{~m} / \mathrm{s}^{2}\)? Equation Transcription: Text Transcription: 2.4 m 9.8 m/s2

Problem 22P (I) The redshift of a galaxy indicates a recession velocity of 1850 km/s. How far away is it?

Problem 23P (I) If a galaxy is traveling away from us at 1.5% of the speed of light, roughly how far away is it?

(II) A galaxy is moving away from Earth. The “blue” hydrogen line at 434 nm emitted from the galaxy is measured on Earth to be 455 nm. (a) How fast is the galaxy moving? (b) How far is it from Earth based on Hubble’s law?

Problem 25P (II) Estimate the wavelength shift for the 656.3-nm line in the Balmer series of hydrogen emitted from a galaxy whose distance from us is (a) 7.0 X 106 ly, (b) 7.0 X 107 ly.

Problem 26P (II) If an absorption line of calcium is normally found at a wavelength of 393.4 nm in a laboratory gas, and you measure it to be at 423.4 nm in the spectrum of a galaxy, what is the approximate distance to the galaxy?

Problem 27P (II) What is the speed of a galaxy with z = 0.060?

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

(II) Estimate the distance d from the Earth to a galaxy whose redshift parameter z = 1.

Probem 30P (II) Estimate the speed of a galaxy, and its distance from us, if the wavelength for the hydrogen line at 434 nm is measured on Earth as being 610 nm.

(II) Radiotelescopes are designed to observe 21-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 cm longer than the normal 21-cm wavelength. Estimate the distance to this galaxy.

(III) Starting from Eq. , show that the Doppler shift in wavelength is \(\Delta \lambda / \lambda_{\text {rest }} \approx v / c\) (Eq. ) for \(v \ll c\) [Hint: Use the binomial expansion.] Equation Transcription: Text Transcription: \Delta \lambda / \lambda_rest \approx v / c v \ll c

Estimate the distance to a 6000-K main-sequence star with an apparent brightness of \(2.0 \times 10^{-12}\ W/m^2\).

Problem 33EC What is the Schwarzschild radius for an object with 10 solar masses?

Problem 33ED A black hole has radius R. Its mass is proportional to (a) R, (b) R2 (c) R3. Justify your answer.

Problem 33P (I) Calculate the wavelength at the peak of the blackbody radiation distribution at 2.7 K using Wien’s law.

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

Problem 36P (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 = 106 yr, (b) t = 1s, (c) t = 10-6s, and (d) t = 10-35 s.

(II) At approximately what time had the universe cooled below the threshold temperature for producing kaons \(\left(M \approx 500 \mathrm{MeV} / \mathrm{c}^{2}\right)\), (b) \(Y\left(M \approx 9500 \mathrm{MeV} / \mathrm{c}^{2}\right)\), and (c) muons \(\left(M \approx 100 \mathrm{MeV} / \mathrm{c}^{2}\right)\)? Equation Transcription: Text Transcription: (M \approx 500MeV/c2) Y(M \approx 9500MeV/c2) (M \approx 100MeV/c2)

Problem 39GP Use conservation of angular momentum to estimate the angular velocity of a neutron star which has collapsed to a diameter of 16 km, from a star whose core radius was equal to that of Earth (6 X 106 m). Assume its mass is 1.5 times that of the Sun, and that it rotated (like our Sun) about once a month.

Problem 38P (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 1011 galaxies, each with about 1011 stars like our Sun. (b) Estimate the density of dark matter in the universe.

Problem 40GP By what factor does the rotational kinetic energy change when the star in Problem 39 collapses to a neutron star? 39. Use conservation of angular momentum to estimate the angular velocity of a neutron star which has collapsed to a diameter of 16 km, from a star whose core radius was equal to that of Earth (6 X 106 m). Assume its mass is 1.5 times that of the Sun, and that it rotated (like our Sun) about once a month.

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.

Problem 43GP A certain pulsar, believed to be a neutron star of mass 1.5 times that of the Sun, with diameter 16 km, is observed to have a rotation speed of 1.0 rev/s. If it loses rotational kinetic energy at the rate of 1 part in 109 per day, which is all transformed into radiation, what is the power output of the star?

The nearest large galaxy to our Galaxy is about \(2 \times 10^6\ Iy\) away. If both galaxies have a mass of \(4 \times 10^{41}\ kg\), with what gravitational force does each galaxy attract the other? Ignore dark matter.

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

Problem 46GP Two stars, whose spectra peak at 660 nm and 480 nm, respectively, both lie on the main sequence. Use Wien’s law, the Stefan-Boltzmann equation, and the H–R diagram (Fig. 33–6) to estimate the ratio of their diameters.

Problem 47GP (a) In order to measure distances with parallax at 100 ly, what minimum angular resolution (in degrees) is needed? (b) What diameter mirror or lens would be needed?

(a) What temperature would correspond to \(14-T e V\) collisions at the \(\mathrm{LHC}\)? (b) To what era in cosmological history does this correspond? [Hint: See Fig. 33–29.] Equation Transcription: Text Transcription: 14-TeV LHC

Consider the reaction \({ }_{8}^{16} O+{ }_{8}^{16} O \rightarrow{ }_{14}^{28} S i+{ }_{2}^{4} H e\) and answer the same questions as in Problem 49 . Equation Transcription: Text Transcription: _8^16 O+_8^16 O \rightarrow_14^28 S i+_2^4 H e

Use dimensional analysis with the fundamental constants c, G, and 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.

Estimate the mass of our observable universe using the following assumptions: Our universe is spherical in shape, it has been expanding at the speed of light since the Big Bang, and its density is the critical density.

The parallax angle of a star is 0.00029. How far away is the star?

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

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?

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.)

When our Sun becomes a red giant, what will be its average density if it expands out to the orbit of Mercury ( from the Sun)?

We saw earlier (Chapter 14) that the rate energy reaches the Earth from the Sun (the solar constant) is about What is (a) the apparent brightness b of the Sun, and (b) the intrinsic luminosity L of the Sun?

Estimate the angular width that our Galaxy would subtend if observed from the nearest galaxy to us (Table 331). Compare to the angular width of the Moon from Earth

Assuming our Galaxy represents a good average for all other galaxies, how many stars are in the observable universe?

Calculate the density of a white dwarf whose mass is equal to the Suns and whose radius is equal to the Earths. How many times larger than Earths density is this?

A neutron star whose mass is 1.5 solar masses has a radius of about 11 km. Calculate its average density and compare to that for a white dwarf (Problem 9) and to that of nuclear matter

(II) A star is 56 pc away. What is its parallax angle? State (a) in seconds of arc, and (b) in degrees.

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

A star is 85 pc away. How long does it take for its light to reach us?

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 nm whereas that of the other peaks at 450 nm. Use Wiens law and the Stefan-Boltzmann equation (Eq. 146) to estimate their relative distances from us. [Hint: See Examples 334 and 335.

(III) Stars located in a certain cluster are assumed 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. 14–6.

Show that the Schwarzschild radius for Earth is 8.9 mm

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

What mass will give a Schwarzschild radius equal to that of the hydrogen atom in its ground state?

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

Describe a triangle, drawn on the surface of a sphere, for which the sum of the angles is (a) and

What is the apparent deflection of a light beam in an elevator (Fig. 3313) which is 2.4 m wide if the elevator is accelerating downward at

(I) The redshift of a galaxy indicates a recession velocity of 1850 km/s. How far away is it?

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

(II) A galaxy is moving away from Earth. The “blue” hydrogen line at 434 nm emitted from the galaxy is measured on Earth to be 455 nm. (a) How fast is the galaxy moving? (b) How far is it from Earth based on Hubble’s law?

Estimate the wavelength shift for the 656.3-nm line in the Balmer series of hydrogen emitted from a galaxy whose distance from us is (a) (b) 7.0 * 107 7.0 * 10 ly

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

What is the speed of a galaxy with z = 0.060?

What would be the redshift parameter z for a galaxy traveling away from us at v = 0.075c?

Estimate the distance d from the Earth to a galaxy whose redshift parameter z = 1.

Estimate the speed of a galaxy, and its distance from us, if the wavelength for the hydrogen line at 434 nm is measured on Earth as being 610 nm.

Radiotelescopes are designed to observe 21-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 cm longer than the normal 21-cm wavelength. Estimate the distance to this galaxy

(III) Starting from Eq. 33-3, show that the Doppler shift in wavelength is \(\Delta \lambda / \lambda_{\text {rest }} \approx v / c\) (Eq. 33-6) for \(v \ll c\). [Hint: Use the binomial expansion.]

Calculate the wavelength at the peak of the blackbody radiation distribution at 2.7 K using Wiens law

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

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

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) (b) (c) and

At approximately what time had the universe cooled below the threshold temperature for producing (a) kaons (b) and

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 galaxies, each with about stars like our Sun. (b) Estimate the density of dark matter in the universe.

Use conservation of angular momentum to estimate the angular velocity of a neutron star which has collapsed to a diameter of 16 km, from a star whose core radius was equal to that of Earth Assume its mass is 1.5 times that of the Sun, and that it rotated (like our Sun) about once a month.

By what factor does the rotational kinetic energy change when the star in Problem 39 collapses to a neutron star?

Suppose that three main-sequence stars could undergo the three changes represented by the three arrows, A, B, and C, in the HR diagram of Fig. 3335. For each case, describe the changes in temperature, intrinsic luminosity, and size

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.

A certain pulsar, believed to be a neutron star of mass 1.5 times that of the Sun, with diameter 16 km, is observed to have a rotation speed of 1.0 rev/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?

The nearest large galaxy to our Galaxy is about \(2 \times 10^6\ Iy\) away. If both galaxies have a mass of \(4 \times 10^{41}\ kg\), with what gravitational force does each galaxy attract the other? Ignore dark matter.

How large would the Sun be if its density equaled the critical density of the universe, Express your answer in light-years and compare with the EarthSun distance and the diameter of our Galaxy.

Two stars, whose spectra peak at 660 nm and 480 nm, respectively, both lie on the main sequence. Use Wiens law, the Stefan-Boltzmann equation, and the HR diagram (Fig. 336) to estimate the ratio of their diameters.

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

(a) What temperature would correspond to 14-TeV collisions at the LHC? (b) To what era in cosmological history does this correspond? [Hint: See Fig. 3329.]

In the later stages of stellar evolution, a star (if massive enough) will begin fusing carbon nuclei to form, for example, magnesium: \(^{12} _6C+^{12} _6C \rightarrow ^{24} _{12} Mg + \gamma\). (a) How much energy is released in this reaction (see Appendix B)? (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. 30–1) so that the strong force can come into play? (c) What temperature does this kinetic energy correspond to?

Consider the reaction and answer the same questions as in Problem 49.

Use dimensional analysis with the fundamental constants c, G, and 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

Estimate the mass of our observable universe using the following assumptions: Our universe is spherical in shape, it has been expanding at the speed of light since the Big Bang, and its density is the critical density