Astronomy 102, Exam #2

Astronomy 102, Midterm Exam #2

Thursday November 17, 2005

2.00 pm – 3.15 pm

Do not turn the pages of the exam until you are instructed to do so.

You are responsible for reading the following rules carefully before beginning.

Exam rules: You may use only a writing instrument and a calculator while taking this test. You may not consult any computers, books, notes – neither on paper nor stored in a calculator – nor each other. All of your work must be written on the attached pages, using the reverse sides if necessary. Important equations, numbers and conversion factors, used in the problems, are found in the last pages of the exam, in the form of the Useful Equations sheet and the How Big Is That sheet. The final answers must be indicated clearly. Exams are due at 3:15, and will be available to be reclaimed in recitations after Thanksgiving.

The questions are each worth five (5) points. Partial credit is available for those questions involving essays, short answers, drawings or explicit calculations, and for multiple-choice questions indicated possibly to have more than one correct answer (e.g. “check all answers that apply”).

Name: ___________________________________

ID number: _______________________________

Recitation: ________________________________

1. The formula for the Schwarzschild singularity can be used correctly in the following calculations (check all correct calculations):

c Given the mass of a white dwarf or neutron star, calculate its circumference.

c Given the circumference of an object supported by degeneracy pressure, calculate its mass.

c Given the mass an object supported by degeneracy pressure, calculate its circumference.

c Given the circumference of a black hole, calculate its mass.

c Given the mass of a star, calculate its circumference.

2. Which of the following statements describe the observed characteristics of quasars (check all correct statements)?

c The luminosity is much larger than that of an entire galaxy, and originates in two clouds on either side of a visible galaxy.

c The motion of the surrounding stars, seen in Doppler shifts, indicates the imminent swallowing of an entire galaxy by a super massive black hole.

c Most of the radio waves originate from two lobes on either side of a visible galaxy.

c Extremely regularly pulsed radio emission.

c The luminosity is much larger than that of an entire galaxy, and originates in a central region vastly smaller than a galaxy.

c Two narrow jets are evident in the radio emission pattern.

c Most of the visible light appears to come from a star-like object.

c One narrow jet is evident in radio emission pattern.

c Superluminal (apparently faster-than-light) motions are observed.

c Most of the radio waves appear to come from a star-like object.

3. Which of the following statements describe the observed characteristics of radio galaxies (check all correct statements)?

c The luminosity is much larger than that of an entire galaxy, and originates in two clouds on either side of a visible galaxy.

c The motion of the surrounding stars, seen in Doppler shifts, indicates the imminent swallowing of an entire galaxy by a super massive black hole.

c Most of the radio waves originate from two lobes on either side of a visible galaxy.

c Extremely regularly pulsed radio emission.

c The luminosity is much larger than that of an entire galaxy, and originates in a central region vastly smaller than a galaxy.

c Two narrow jets are evident in the radio emission pattern.

c Most of the visible light appears to come from a star-like object.

c One narrow jet is evident in radio emission pattern.

c Superluminal (apparently faster-than-light) motions are observed.

c Most of the radio waves appear to come from a star-like object.

4. Consider two white dwarf stars with the same mass, one perfectly normal and the other in which all the electrons have been replaced by particles with half the mass of the electron but which are otherwise the same. Which white dwarf is larger in circumference, and why?

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5. The mass of a white dwarf with the same circumference as the Earth is

c 0.1 M_sun

c 0.9 M_sun

c 1.3 M_sun

c 1.4 M_sun

c None of above.

6. The mass of a neutron star with the same circumference as the Earth is

c 0.1 M_sun

c 0.9 M_sun

c 1.3 M_sun

c 1.4 M_sun

c None of above.

7. Supernovae happen when

c The outer parts of a rapidly-collapsing dead star bounce off of the surface of the neutron star suddenly formed at its core.

c The outer parts of a rapidly-collapsing dead star bounce off of the horizon of a black hole suddenly formed at its core.

c The outer parts of a rapidly-collapsing dead star are slung gravitationally around the horizon of a black hole suddenly formed at its core, and are ejected at speeds close to the speed of light..

c Matter and antimatter annihilate each other in the center of the star, converting all of their mass into a vast amount of light energy (according to E = mc²) and exploding the star.

c None of above.

8. The mass of a black hole with the same circumference as the Earth is

c 0.1 M_sun

c 0.9 M_sun

c 1.3 M_sun

c 1.4 M_sun

c None of above.

9._ The following figure shows a schematic sketch of an elliptical galaxy that harbors a super massive black hole, with twin jets and an accretion disk. Draw and label appropriately the positions of three observers, and their lines of sight to the central object, who would classify the galaxy as a blazar, a quasar, and a radio galaxy.

10. Astronomers find a star that seems to be in orbit about an invisible, 1.9 M_sun companion. At radio wavelengths, bright pulses of light are detected every 0.01 sec from the companion. The companion is

c A normal star.

c A white dwarf.

c A neutron star.

c A black hole.

c Unknown; the evidence is ambiguous.

11. Degeneracy pressure is due to

c The particle properties of the elementary constituents of matter.

c The wave properties of the elementary constituents of matter.

c The relativity of mass.

c The relativity of time.

c The absolute nature of the speed of light.

12. In a few complete sentences, explain why a higher-mass object supported by degeneracy pressure is smaller than a lower-mass one.

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13. In the view of a distant observer, time dilation appears extreme for in-falling objects when they reach a distance from a black hole corresponding to a circular orbit with circumference equal to

c 0.15 horizon circumferences.

c 1.5 horizon circumferences.

c 15 horizon circumferences.

c 150 horizon circumferences.

c Time dilation is extreme no matter what the distance from the black hole.

14. The black hole in a certain quasar accretes matter at a constant rate of 0.6 M_sun per year. If it turns 10% of the mass into energy in the form of light, what is its luminosity?

c 9.0´10¹¹ L_sun.

c 9.0´10¹⁰ L_sun.

c 9.0´10⁹ L_sun.

c 9.0´10⁸ L_sun.

c None of above.

15. The black hole at the center of the Milky Way has a mass of 2.6´10⁶ M_sun, but the central object has a luminosity of 10⁵ L_sun at most. Assuming an efficiency of 10% for converting mass into radiated energy, what is the maximum rate at which the black hole can be accreting matter?

c 9.1´10⁷ gm/sec.

c 4.2´10¹⁸ gm/sec.

c 2.0´10³⁶ gm/sec.

c 3.1´10⁵⁴ gm/sec.

c None of above.

16. The BeppoSAX satellite was constructed in

c The Netherlands.

c Holland.

c Nederland.

c All of the above.

17. A white dwarf is composed of

c Hydrogen nuclei and degenerate electrons.

c Helium nuclei and normal electrons.

c Carbon and oxygen nuclei and degenerate electrons.

c Degenerate iron nuclei.

c A helium burning core and a hydrogen burning shell.

18. The density of a neutron star is

c About the same as that of a white dwarf.

c About the same as that of the sun.

c About the same as that of an atomic nucleus.

c About that same as that of a water molecule.

c None of the above.

19. What is the mass of the black hole that is located in the center of the Milky Way?

c 2.6 x 10⁶ M_sun.

c 2.6 x 10⁴ M_sun.

c 2.6 x 10⁵ M_sun.

c 2.6 x 10⁸ M_sun.

c 2.6 x 10⁷ M_sun.

20. If a gamma-ray burster were to occur in Rochester, it would probably destroy life within

c Rochester.

c Western New York.

c About 3 light-years of Earth.

c About 3000 light-years of Earth.

c None of above.

21. The mass of a black hole with circumference the same as the Sun’s is

c 1 M_sun.

c 2.4´10⁵ M_sun.

c 3.4´10⁵ M_sun.

c 4.4´10⁵ M_sun.

c None of above.

22. In the spectrum of a certain star, an absorption line with rest wavelength of 5.00000 x 10^-5 cm is seen to vary periodically in wavelength between 5.00167 x 10^-5 cm and 4.99833 x 10^-5 cm. Assume that the star is in an orbit, observed edge on, about an unseen companion, and calculate the star’s orbital speed. Express your answer in km/sec, and show all of your work.

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23. A certain star is in orbit with a much less luminous companion. Its orbital plane is EDGE-ON to the line of sight. Its speed in orbit is 90 km/s. The overall velocity of the two-star system (that is, the velocity of their center of mass) along the line of sight is 90 km/s. What is the minimum wavelength (in centimeters) of a Doppler-shifted absorption line which, seen at rest, has a wavelength of 5.0000 x 10^-5 cm?

c 5.0000 x 10^-5 cm

c 5.0015 x 10^-5 cm

c 4.9970 x 10^-5 cm

c 4.9985 x 10^-5 cm

c 5.0030 x 10^-5 cm

24. Which of the following black-hole symptoms are observed in GRO J1655-40, one of the best known “stellar” black-hole candidates (check all correct symptoms)?

c Ejection of material at speeds near the speed of light.

c Details of the structure of its massive accretion disk.

c Enormous luminosity emitted from an extraordinarily small space.

c X- and g-ray emission.

c Gravitational deflection of the light of more distant stars.

25. Which of the following black-hole symptoms are observed in the quasar 3C 273 (check all correct symptoms)?

c Ejection of material at speeds near the speed of light.

c Details of the structure of its massive accretion disk.

c Enormous luminosity emitted from an extraordinarily small space.

c X- and g-ray emission.

c Gravitational deflection of the light of more distant stars.

26. Suppose your neighbor’s mass is 7.5´10⁴ gm. If your neighbor were to collapse to form a black hole during this exam, his or her circumference would be

c 7.0´10^-23 cm.

c 3.5´10^-3 cm.

c 4.3 cm.

c 18.6 km.

c 7.5´10⁴ km.

27. The mass of an object appears to approach infinity as its speed with respect to an observer approaches the speed of light. As a consequence (check all correct consequences):

c All massive objects must eventually slow down.

c Massive objects can replicate themselves, by going fast enough to double their mass.

c Tidal forces are infinite at the horizon of very massive black holes.

c The speed of a massive object cannot exceed or equal the speed of light.

c All of above.

28. Describe briefly one way in which a black hole could be used to generate energy in a form useful to today’s human society.

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29. X-rays are produced efficiently by black holes because of

c Radiation by electrically-charged particles that are about to fall through the black hole’s horizon, and have thus been given extremely large accelerations gravitationally.

c The swirl of space-time just outside the black hole’s horizon, where 10-30% of the hole’s total energy can be stored.

c Electrically-charged particles that have been ejected by the black hole in the form of a pair of jets, traveling at very high speeds.

c Large pulsations of the black hole’s horizon.

c None of above.

30. X-rays are not produced efficiently by normal stars and white dwarfs because

c The gravity of these objects is insufficient to impart large accelerations to charged particles, which thus cannot emit X-rays.

c The thick accretion disks that normally surround these objects absorb all of the X-rays.

c They lack the relativistic jets possessed by massive black holes.

c The gas pressure (in stars) or degeneracy pressure (in white dwarfs) inhibits the production of X-rays.

c None of above.

31. The following signatures would indicate that a black hole is spinning (check all that apply):

c Features on the event horizon that can be seen by a distant observer to rotate.

c Features in the ergo sphere that can be seen by a distant observer to rotate.

c Matter in stable orbits with a circumference of 3.5 C_S.

c Matter in stable orbits with a circumference of less than 3.0 C_S.

c Photons in orbits with a circumference different from 1.5 C_S.

32. A bright star is seen to be orbiting with a companion that is a strong source of visible light. From the orbital speed and mass of the bright star, it is inferred that the mass of the companion is 1.6 M_sun. The companion object is most likely to be

c A normal star.

c A white dwarf.

c A neutron star.

c A black hole.

c Unknown; the evidence is ambiguous.

33. A certain star is in orbit with a much less luminous companion. Its orbital plane is PERPENDICULAR to the line of sight. Its speed in orbit is 90 km/s. The overall velocity of the two-star system (that is, the velocity of their center of mass) along the line of sight is 90 km/s. What is the maximum wavelength (in centimeters) of a Doppler-shifted absorption line which, seen at rest, has a wavelength of 5.0000 x 10^-5 cm?

c 5.0000 x 10^-5 cm

c 5.0015 x 10^-5 cm

c 4.9970 x 10^-5 cm

c 4.9985 x 10^-5 cm

c 5.0030 x 10^-5 cm

34. The Chandrasekhar limit tells us that

c Accretion disks can grow hot through friction.

c Neutron stars with a mass of more than 3 M_sun are not stable.

c White dwarfs must have a mass of more than 1.4 M_sun.

c Not all stars will end up as white dwarfs.

c Stars with a mass of less than 0.5 M_sun will become black holes.

35. As a white dwarf cools, its radius does not change because

c Pressure due to nuclear reactions in a shell just below the surface keeps it from collapsing.

c Pressure does not depend on temperature for a white dwarf because the electrons are degenerate.

c Pressure does not depend on temperature because the white dwarf is too hot.

c Pressure does not depend on temperature because the star has exhausted all its nuclear fuels.

c Materials accreting onto the white dwarf from a companion, maintain its constant radius.

36. Quasars must be small because they

c Have high radial velocities

c Are very luminous.

c Are surrounded by a quasar fuzz.

c Radiate huge amounts of energy

c Fluctuate rapidly.

37. Which of the following types of galaxies were discovered by radio astronomers (check all that apply):

c Quasars.

c Radio galaxies.

c Seyfert galaxies.

c Blazars.

c Normal galaxies.

38. Which of the following statements about black holes are true (check all correct statements)?

c More energy would be released by dropping a ton of coal or gasoline into a black hole than by dropping a ton of water into the same black hole.

c A space traveler, descending vertically and hovering very close to the equator of a rotating black hole, would be seen to rotate with the black hole by a distant observer.

c Energy can be obtained from the ergo sphere outside the horizon of a rotating black hole.

c Energy cannot be released by accretion onto a black hole because nothing can escape a black hole, once it is inside the horizon.

c Time stops at the horizon of a black hole, from the viewpoint of a distant observer.

39. Two 1.8 M_sun neutron stars, orbiting each other at close range, suddenly spiral into each other and coalesce. This produces (check only one option):

c A black hole and a gamma-ray burst.

c A 3.6 M_sun neutron star.

c A very massive accretion disk and two jets of relativistic particles.

c A supernova.

c None of the above.

40. A spinning black hole with the same mass as the Sun has 30% of its total mass stored outside its horizon, in the form of rotating space-time. How much available energy does this represent?

c 5.4´10⁵⁴ erg.

c 5.4´10⁵³ erg.

c 5.4´10⁵¹ erg.

c 5.4´10¹⁶ erg.

c None of above.

41. Extra Credit: Match the pictures of the ball parks to the team names listed below.


A	B

C	D

E

___ 1. Yankees

___ 2. White Sox

___ 3. Mets

___ 4. Cubs

___ 5. Red Sox

Circumferences of white dwarfs, neutron stars and black-hole event horizons

Useful Equations

Length contraction, time dilation and velocity addition:

Minkowski absolute interval:

Useful rearrangements of the Absolute Interval formula:

Schwarzschild circumference:

Mass-energy equivalence:

Doppler shift:

How big is that?

Diameter of hydrogen atom	1.06 ´ 10^-8 cm
Diameter of the Moon	3.5 ´ 10³ km
Diameter of the Earth	1.3 ´ 10⁴ km
Diameter of the Sun	1.4 ´ 10⁶ km
Diameter of the Milky Way galaxy	1.7 ´ 10⁵ ly

Distance to the Moon	3.8 ´ 10⁵ km
Distance to the Sun	1.5 ´ 10⁸ km
Distance to the next nearest star	4 ly
Distance to the center of the Milky Way	2.7 ´ 10⁴ ly
Distance to the nearest galaxy	1.7 ´ 10⁵ ly

Mass of hydrogen atom	1.67 ´ 10^-24 gm
Mass of the Moon	7.4 ´ 10²⁵ gm
Mass of the Earth	6.0 ´ 10²⁷ gm
Mass of the Sun	2.0 ´ 10³³ gm (1 M_sun)
Mass of the Milky Way galaxy	5 ´ 10¹⁰ M_O

Luminosity of the Sun	3.8 ´ 10³³ erg/s (1 L_sun)
Luminosity of the largest stars	10⁵ L_O
Luminosity of the Milky Way galaxy	10¹⁰ L_O
Luminosity of quasar 3C 273	10¹² L_O

Earth’s rotation period	8.64 ´ 10⁴ s (1 day)
Moon’s revolution period	28 days
Earth’s revolution period	365.25 days (1 year)
Sun’s revolution period within Milky Way	2.4 ´ 10⁸ years
Age of the solar system	4.6 ´ 10⁹ years
Expected life span of the Sun	1.5 ´ 10¹⁰ years
Age of the Universe	1.3 ´ 10¹⁰ years

Earth’s equator rotation speed	0.47 km/s
Earth’s revolution speed	30 km/s
Sun’s speed within the Milky Way	220 km/s
Milky Way’s speed within the local Universe	500 km/s

Typical lengths:
Normal star diameter	10⁶ km
Distance between stars	a few ly
Normal galaxy diameter	10⁵ ly
Distance between galaxies	10⁶ ly

Typical masses:
Smallest star	0.1 M_sun
Normal star	1 M_sun
Giant star	10 M_sun
Normal galaxy	10¹⁰ - 10¹¹ M_sun
Galaxy cluster	10¹⁴ - 10¹⁵ M_sun

Typical luminosities:
Normal star	1 L_sun
Giant star	10³ - 10⁵ L_sun
Normal galaxy	10⁹ - 10¹⁰ L_sun
Quasar	10¹² - 10¹³ L_sun

Typical time spans:
Planetary revolution	1 year
Galaxy rotation	10⁷ - 10⁹ years
Life of giant stars	10⁶ - 10⁹ years
Life of normal star	10¹⁰ years

Typical speeds:
Planetary orbits	10 km/s
Stellar motion in galaxy	100 km/s
Between nearby galaxies	100 km/s

Other important constants:
1 ly = 9.46 ´ 10¹² km = 9.46 ´ 10¹⁷ cm	1 Mly = 10⁶ ly
1 km = 10⁵ cm	1 erg = 1 gm cm²/s²
1 hour = 3600 seconds	1 year = 3.16 ´ 10⁷ seconds
π = 3.14159265359	Hubble’s constant: H₀ = 20 km/(sec Mly)
Speed of light: c = 2.99792458 ´ 10⁵ km/s = 2.99792458 ´ 10¹⁰ cm/s = 1 ly/year	Newton’s gravitational constant: G = 6.67 ´ 10^-8 cm³/(gm s²)

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