Astronomy 102, Midterm Exam #2: Solutions

November 17, 2005

 

If any of these answers seems obscure, please ask us questions until we clear it up for you. Correct answers are indicated in bold type and filled squares (g).

 

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.

g   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.

g   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.

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

g   One narrow jet is evident in radio emission pattern.

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

g   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.

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

c   Extremely regularly pulsed radio emission.

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

g   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?

 

      The one in which the electrons are less massive is larger in circumference than the ÒnormalÓ one. This is because a smaller confinement space is necessary to reveal wave properties in larger-mass particles, and thus for those entities to exert a given degeneracy pressure. Smaller confinement space for each particle in the system leads to smaller volume and circumference for the whole thing; thus the ÒnormalÓ white dwarf is smaller than the one made of ÒlightweightÓ electrons.

 

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

c   0.1 Msun

g   0.9 Msun

c   1.3 Msun

c   1.4 Msun

c   None of above.

 

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

c   0.1 Msun

c   0.9 Msun

c   1.3 Msun

c   1.4 Msun

g   None of above.

 

7.   Supernovae happen when

g   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 Msun

c   0.9 Msun

c   1.3 Msun

c   1.4 Msun

g   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 Msun 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. 

g   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.

g   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.

 

Larger pressure is required to balance the weight of a larger mass. To increase the degeneracy pressure one must confine each degenerate particle to a smaller volume; if each of the object's particles has a smaller volume, the object has decreased in size.

 

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.

g   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 Msun per year.  If it turns 10% of the mass into energy in the form of light, what is its luminosity?

g   9.0«1011 Lsun.

c   9.0«1010 Lsun.

c   9.0«109 Lsun.

c   9.0«108 Lsun.

c   None of above.

 

15. The black hole at the center of the Milky Way has a mass of 2.6«106 Msun, but the central object has a luminosity of 105 Lsun 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.

g   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.

g   All of the above.

 

17. A white dwarf is composed of

c   Hydrogen nuclei and degenerate electrons.

c   Helium nuclei and normal electrons.

g   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.

g   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?

g   2.6 x 106 Msun.

c   2.6 x 104 Msun.

c   2.6 x 105 Msun.

c   2.6 x 108 Msun.

c   2.6 x 107 Msun.

 

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.

g   About 3000 light-years of Earth.

c   None of above.

 

21. The mass of a black hole with circumference the same as the SunÕs is

c   1 Msun.

g   2.4«105 Msun.

c   3.4«105 Msun.

c   4.4«105 Msun.

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 starÕs orbital speed.  Express your answer in km/sec, and show all of your work.

 

      The velocities corresponding to the two observed wavelengths are

      This is what you would see for a star in circular orbit at a constant speed of 100 km/sec, viewing the orbit edge on.

 

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?

g   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 ÒstellarÓ black-hole candidates (check all correct symptoms)?

g   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.

g   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)?

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

c   Details of the structure of its massive accretion disk.

g   Enormous luminosity emitted from an extraordinarily small space.

g   X- and g-ray emission.

c   Gravitational deflection of the light of more distant stars.

 

26. Suppose your neighborÕs 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

g   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.

g   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 todayÕs human society.

 

___ Here are two, either of which would do:

 

___ Throw matter into a nearby black hole. As it nears the black hole, the matter will emit high-energy light such as X rays and gamma rays.  Use solar panels to absorb the high-energy light and create electric currents thereby.  Adjust the rate at which matter is thrown into the black hole to produce electric currents sufficient for the need.

 

      Or take a (very) long shaft or two, and arrange (with rockets to provide thrust) for one end to hover within the ergosphere of a spinning black hole.  The black holeÕs rotation will move its shaft end around in a circle, dragging our end along for the ride.  This rotary motion can be used to drive any sort of machine that can convert motion to other sorts of power.

 

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

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

c   The swirl of space-time just outside the black holeÕs horizon, where 10-30% of the holeÕs 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 holeÕs horizon.

c   None of above.

 

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

g   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.

g   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 CS.

g   Matter in stable orbits with a circumference of less than 3.0 CS.

g   Photons in orbits with a circumference different from 1.5 CS.

 

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 Msun.  The companion object is most likely to be

g   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

g   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 Msun are not stable.

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

g   Not all stars will end up as white dwarfs.

c   Stars with a mass of less than 0.5 Msun 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.

g   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

g   Fluctuate rapidly.

 

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

g   Quasars.

g   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.

g   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.

g   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.

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

 

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

g   A black hole and a gamma-ray burst.

c   A 3.6 Msun 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.

g   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

 

 

_B_   1. Yankees

_E_   2. White Sox

_A_   3. Mets

_D_   4. Cubs

_C_   5. Red Sox