Answers to True/False and Multiple Choice Questions of the Final Exam

Answers to the True/False Questions

1F Planetary nebula are created by low-mass stars that would leave a white dwarf at their center.
2F Not enough time by tens of thousands of years for Pioneer to get to a star at its slow speed.
3T Light reflecting from a mirror or refracted by a lens would be an example of accelerating light in the sense that the direction of motion has been changed, even though its speed has not. A subtlety though: light being "deflected" by a gravitational mass would not be considered acceleration since the path of light is a straight line (a geodesic) in four-dimensional space time.
4F Nearby galaxies like Andromeda can have a blue shift, local gravitational forces in the Local Cluster can overcome the expansion of space and so some galaxies can approach the Earth.
5T Atoms move with a higher average speed in hotter objects and so are more massive by the Einstein's formula for relativistic mass.
6F White dwarfs can have different colors as they cool down and move diagonally down to the right in a H-R diagram.
7T
8F Nearly all He was created during the Big Bang, only a tiny fraction has been created by fusing H in stars.
9T
10F Electrical (chemical) and gravitational forces can overcome the expansion of space. You don't expand nor do stars in a galaxy, not even galaxies within the Local Cluster.
11F In a white-dwarf supernova, the carbon nuclei in a white dwarf fuse to other elements (oxygen and silicon), essentially no carbon is left over. Carbon stars (red giants) are what spew lots of carbon into space.
12T Because electrons have a mass that is 1/2,000-th the mass of a proton, gamma rays produced by electron-positron annihilation do not have enough energy via E=mc² to produce a proton, nevertheless a proton-antiproton pair.
13T
14T If the strong interaction became so weak, then two or more protons in a nucleus would repel each other electrically with a force greater than what the strong interaction could hold together so all nuclei with two or more protons would fall apart. So atoms would consist of only one proton and possibly several neutrons.
15F The rotation curve for our solar system shows no significant dark matter, the speeds of planets are completely accounted for by the mass of the Sun and the masses of the other planets. Dark matter mainly accumulates toward the outer part of the galaxy, beyond the location of our solar system.
16T Hydrogen is steadily transformed into He and other elements and is not replaced so, eventually, stars can no longer form,
17T When four H atoms fuse to form a single He nucleus via the proton-proton chain, the number of particles in the Sun's core decreases, which causes the pressure to drop, which causes the core to compress, which causes the core to heat up, which causes the energy generated by fusion to increase. The net result is that the radius and temperature of the surface of the Sun slowly increase over time (the Sun is more luminous now than in the times of the dinosaurs).
18T The universe was much smaller six billion years while the number of galaxies was about the same, so all galaxies were closer together making collisions more likely.
19T A planet that rotates once on its axis while orbiting once around a star (just as the Moon rotates once on its axis during one orbital period) will have a day that is infinitely long, which is definitely longer than its sidereal year.
20T We have seen that stars bend light toward them. Thus two parallel light beams that approach either side of a star or galaxy are bent toward each other which is the hallmark of a spherical geometry.
21F The seasons on Earth arise from the tilt of the Earth's axis, the eccentricity of the orbit is so small that it plays a minor role in the seasons.

Answers to the Multiple Choice Questions

1(d)
2(d)
3(d)
4(d)
5(b)
6(d)
7(c)
8(c)
9(a)
10 The Sun will be in the same constellation Gemini at sunset. The Sun moves very slowly w.r.t. the celestial sphere (one trip around the ecliptic every 365 days) so effectively rotates rigidly with respect to the stars on any give day. Gemini and the Sun both end up together near the western horizon at sunset.
11(c)
12(a)
13(d)
14(b)
15(b)
16(e) The wavelength lambda_1' is not shifted at all while the wavelength lambda_2' is blue-shifted compared to lambda_2. So the correct answer is lambda_1' = lambda_1, lambda_2' < lambda_2.
17(c) This is the Algol paradox.
18(b) The more massive nuclei are denser and so lie closer to the center of the core.
19(a)
20(d)
21(c)
22(d)
23(d) A star with RA = 18^h is in the same part of the celestial sphere as the Sun itself so would appear during the day (if we could see past the glare of the Sun) and won't appear at all at night since it is below your horizon.
24(b)
25(a)
26(a)
27(e) All nuclei with atomic numbers greater than the atomic number of iron, Z=26, are created during a supernova.
28(c)
29(e) Members of the Oort cloud are believed to have formed near the gas giants but were flung inwards toward the Sun and outward beyond the Kuiper belt by the gravitational forces of the gas giants.
30(d)
31(b)
32(a)
33(e) The Moon moves along the ecliptic with respect to the stars of the celestial sphere. Thus as the celestial sphere seems to rotate around the Earth, the Moon slowly lags behind the background stars as it follows the ecliptic. Since the Moon completes its trip around the ecliptic in 28 days, in one day (midnight to midnight), the Moon moves (1 day / 28 day) x 360^o = ~12 degrees with respect to the stars in a direction opposite to the movement of the celestial sphere. So the following night, the Moon will be about 10 degrees away from the meridian and has not yet crossed the meridian. Since a solar day (midnight to midnight) is 4 minutes longer than a sidereal day, the following midnight the star will already have crossed the meridian. I encourage you to verify these facts using the SkyGazer software.
34(d)
35(b)
36(c)
37(b)
38(c) The larger the Hubble constant, the smaller the age of the universe since 1/H₀ is the age of the universe for constant expansion. Thus Senator Clinton believes the universe is younger than what President Bush believes, which means further that Senator Clinton believes the universe has expanded more rapidly to achieve its current size in a lesser amount of time. It would be very interesting to find out how much astronomy either the Senator or President actually know.

Some Miscellaneous Answers

Open Question 1: This is explained on pages 553-554 of the text, regarding how a red giant first forms.

1(a): increases
1(b): decreases
1(c): increases
1(d): increases
1(e): increases

Open Question 2: Answer is 23 degrees altitude with direction in the south (23^oS).

Open Problem 4: Many students gave Hubble's law, i.e., the expansion of space, as an example supporting the Big Bang. This is not correct. The cosmic microwave radiation, He abundance, and deuterium abundance are all consequences of a hot dense beginning that later cooled off, but the expansion of space itself says nothing about the thermal or physical properties of the early universe. In fact, until the cosmic background radiation and He abundance were measured, there was competing theory called the "steady state universe" which postulated that the universe had always been expanding, with matter spontaneously appearing to keep the density constant. This theory was attractive because it got rid of the question of a beginning and end for the universe, there was no beginning.

Open Question 6: Part (a), many students realized they needed to divide a distance, 200 ly, by a speed, 0.8c, but took the long path of converting everything to meters or seconds. Here you can see the answer quickly with a little thinking. At the speed of light c, it would take 200 years to go 200 light-years by definition of a light year. So if you are traveling a bit slower at 0.8c, it would (1/.8) times longer, or 200/.8 = 250 years.

Open Question 7: Answer to one significant digit is 2,000 K. The new temperature is way above the temperature of boiling water (373 K) so is bad news since all the oceans will turn to steam and Earth will become like Venus.

The subtle part of this problem was to deduce the distance d of Earth to the Sun after the Sun has swelled to about Mercury's orbit. Some students incorrectly chose d to be the distance of Earth to the edge of the swollen Sun. You needed to recall that at large distances between masses, a big mass like the Sun acts like a point mass at the center of the Sun. Thus whether the Sun swells to be a red giant or gets squished to a 3 km diameter black hole, Earth continues to remain 1 AU from the center of the Sun and this is the distance to use in this problem.

Open Question 8: Most students sadly missed the elegant insight here, there was evidently not enough practice during the course with the equivalence principle. The first step was to imagine an isolated rocket (no gravity present!) that is accelerating in space, say in the upwards direction of this page. Someone is holding two objects (say an apple and a steel ball) and these two objects are accelerating with the person and rocket because of the force exerted by the person's hands on the objects.

Now the person lets go of the two objects at exactly the same time and at exactly the same height above the floor. As soon as the person let's go, the objects no longer have any force acting on them and they move upward with identical constant speeds. (This is Newton's first law, if there is no force, you move at constant speed in a constant direction.) But the rocket is accelerating which means that the floor of the rocket is rising faster and faster and will catch up and hit the two objects, and it will hit the objects at exactly the same time provided the objects were dropped from the same height at the same time. You can see that the fact that the floor catches up with the two objects has absolutely nothing to do with the shape or chemical properties or masses of the objects: you have two points at the same height above the floor moving upward with a constant speed, and the floor as a plane catches up and touches both objects at the same time.

Finally, here is where you apply the equivalence principle. According to Einstein's insight, what happens in an accelerating rocket in the absence of gravity is physically undistinguishable from the same events happening in a stationary room in the presence of a vertical gravity field pointing downwards. Thus we must expect two objects to fall at the same rate and hit the ground at exactly the same time when dropped at the same time and height in a vertical gravitational field, and this fact can not depend on any physical properties such as mass or chemical composition because it is essentially a geometrical argument.

I should point out that this problem has everything backwards historically. Einstein used the experimental fact that objects fall with the same acceleration independently of mass to guess the equivalence principle which he then applied to other problems such as the slowing down of clocks or the bending of light. But this is a nice problem because it shows that you can go backwards, starting from the equivalence principle and then deduce that objects falling under the influence of gravity do so with an acceleration that is independent of any property of the object.

Open Questions 9 and 10 Many students unfortunately confused special relativity with general relativity and answered these questions as if I had asked about special relativity. General relativity deals with gravity and distortions of space and time. Laboratory tests of general relativity include the bending of light by a star's mass (verified in 1919), the precession of Mercury's orbit by 43'' per century, and gravitational time dilation (clocks at tops of towers tick more rapidly than identical clocks at the base of towers, also the gravitational red shift). General relativity is essential for cosmology when trying to understand how space itself can expand or contract during the Big Bang (Hubble's data), also essential for understanding black holes that power quasars, radio sources like Sag A^*, and some X-ray binaries (where does all the energy come from in such compact regions of space?).

Examples like lack of simultaneity, length contraction, time dilation, increase of mass, E=mc², and the modified addition rule for velocities all belong to special relativity. They are important in day-to-day thinking about astronomy, but are not as essential as general relativity for trying to understand the current astronomical mysteries such black holes, dark matter, beginning of the Big Bang, far future of the universe.

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