Astronomy 102, Midterm Exam #2: Solutions

November 13, 2003

 

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.         Occasionally Cygnus X-1 emits short bursts of light seen at ultraviolet wavelengths, in the form of a train of pulses that dies off towards the end.  One such burst, seen with the Hubble Space Telescope by Joe Dolan (2001, PASP 113, 974), is shown in the Figure below.

Describe what effect produces these bursts and what does the period of these bursts correspond to?

 

The sequence of bursts is caused by material breaking off from the accretion disk and falling towards the horizon.  When the material leaves the accretion disk, it will spiral towards the horizon.  In this orbit, the light emitted by the material will be blocked when it passes the horizon on the far side.

The pulse period is close to the orbital period of the last stable orbit around this black hole.

 

2.         Match the pictures to the names listed below.

A

B

C

D

E

_E_  Tim Wakefield

_A_  Johnny Damon

_B_  Pedro Martinez

_C_  Manny Ramirez

_D_  Jason Varitek

 

3.         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 130 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 5.0000E-5 cm?

c      4.99850E-5 cm

c      5.00367E-5 cm

c      4.99783E-5 cm

g      5.00067E-5 cm

c      5.00150E-5 cm

c      4.99933E-5 cm

c      4.99633E-5 cm

c      5.00217E-5 cm

 

4.         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 130 km/s.   What is the minimum wavelength (in centimeters) of a Doppler-shifted absorption line which, seen at rest, has a wavelength of 5.0000E-5 cm?

c      4.99850E-5 cm

c      5.00367E-5 cm

c      4.99783E-5 cm

g      5.00067E-5 cm

c      5.00150E-5 cm

c      4.99933E-5 cm

c      4.99633E-5 cm

c      5.00217E-5 cm

 

5.         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.

 

6.         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 130 km/s.   What is the minimum wavelength (in centimeters) of a Doppler-shifted absorption line which, seen at rest, has a wavelength of 5.0000E-5 cm?

c      4.99850E-5 cm

c      5.00367E-5 cm

c      4.99783E-5 cm

c      5.00067E-5 cm

c      5.00150E-5 cm

c      4.99933E-5 cm

c      4.99633E-5 cm

g      5.00217E-5 cm

 

7.         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.

 

8.         When was the last time the Red Sox won the World Series?

c      1776

c      2004

c      1915

g      1918

c      2003

 

9.         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.

 

10.   A certain star is in orbit with a much less luminous companion.  Its orbital plane is edge-on to the line of sight.  The Figure shows the time-dependence of a Doppler-shifted absorption line emitted by the star, which, seen at rest, has a wavelength of 5.0000E-5 cm.

AppleMark

      What is the orbital velocity of the star (in km/s)?

 

The observed wavelength of the absorption line varies between 4.9925E-5 cm and 5.0025E-5 cm.  The variation in observed wavelength is due to the oribital motion of the star.  Assuming the star has an orbital velocity of V and assuming the center-of-mass of the binary star system moves with a velocity Vcm with respect to our line out sight, we can calculate the maximum and minimum observed wavelengths:

 

 

 

These two equations show that the difference between the maximum and minimum wavelength depends only on the orbital velocity of the star:

 

 

In this problem, the difference in wavelength is 0.01E-5 cm, and the orbital velocity is thus equal to

 

 

11.   What is the overall velocity of the two-star system discussed in Question # 10 (that is, the velocity of their center of mass) along the line of sight.  Express your answer in terms of km/s.

The average of the maximum and minimum wavelength is related to the velocity of the center-of-mass of the two-star system:

 

 

From the figure in Problem 10 we see that the average wavelength is 4.9975E-5 cm.  This information can be used to determine the velocity of the center-of-mass of the star system:

 

 

 

12.    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.

 

13.   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 130 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 5.0000E-5 cm?

g      4.99850E-5 cm

c      5.00367E-5 cm

c      4.99783E-5 cm

c      5.00067E-5 cm

c      5.00150E-5 cm

c      4.99933E-5 cm

c      4.99633E-5 cm

c      5.00217E-5 cm

 

14.   What is the circumference of a neutron star with a mass of 0.2 Msun?

c      4 km

c      110 km

g      200 km

c      1000 km

c      95000 km

 

15.   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.

 

16.   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.

 

17.   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.

 

18.   The BeppoSAX satellite was constructed in

c      The Netherlands.

c      Holland.

c      Nederland.

g      All of the above.

 

19.   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.

 

20.   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.

 

21.   You observe a student, who rest mass is 75 kg, traveling with 75% of the speed of light?  What is the mass of this student in your reference frame?

c      75 kg

g      113 kg

c      99 kg

c      94 kg

c      150 kg

 

22.   What is the ergosphere of a black hole?

The ergosphere is the region just outside the horizon of a spinning black hole were space rotates so fast that it becomes impossible for a body to hover in such a way that it would appear stationay to a distance observer.  A significant fraction of the total energy of the black hole can be present in this region.

 

23.   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.

 

24.   A star has a luminosity of 0.216 Lsun, and a total lifetime of 1.16E+11 years.  How much energy does the star produce in its lifetime?

 

The energy released by the star per second is equal to

 

 

The lifetime of the star is 1.16E11 yr = 3.67E18 s (since 1 year = 3.16E7 s).  The total energy release by the star during its lifetime is equal to

 

 

25.   Consider a stellar core that collapses and forms a black hole.  Which of the following physical quantities, which can be used to describe the black hole or the stellar core, will not change as a result of this collapse?  Check all quantities that remain unchanged.

g      Mass.

c      Magnetic field.

g      Electric Charge.

c      Particle properties.

c      Stellar type.

 

26.   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.

 

27.   A black hole with a mass of 0.5 Msun, and spinning near its maximum rate, has 23 percent of its total mass stored outside its horizon, in the form of rotating space-time.  How much energy does this represent?

c      1.0E+20 erg

c      2.0E+19 erg

c      3.9E+52 erg

g      2.1E+53 erg

c      9.0E+53 erg

c      2.1E+55 erg

 

28.   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.

 

29.   The black hole in a certain quasar swallows matter at an average rate of 1.5 Msun/year. If it turns 13 percent of the mass into energy in the form of light, what is its luminosity in solar luminosities?

c      2.2E+13 Lsun

c      1.5E+13 Lsun

g      2.9E+12 Lsun

c      2.7E+54 Lsun

c      3.5E+53 Lsun

c      1.8E+54 Lsun

 

30.   A black-hole candidate is found to have a luminosity, in the form of X-rays and gamma-rays, of 115 Lsun.  Assuming an efficiency of 18 percent for converting mass into radiated energy, how many grams of mass must it consume per second to generate its luminosity?

 

The luminosity of the of the black hole system is

 

 

This luminosity is generated by the accretion of mass by the black hole.  This process has an efficiency of 18%, and the rest energy of the mass accreted by the black hole must thus be equal to

 

 

Using EinsteinÕs mass-energy relation we can now express this rate in terms of g/s:

 

 

31.   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.

 

32.   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.

 

33.   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.

 

34.   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.

 

35.   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.

 

36.   Consider the active galaxies shown in the pictures.  Match the picture to the type of galaxies listed below.

 

 

A

B

 

 

C

D

_D_  Quasar Galaxy

_A_  Blazar Galaxy

_C_  Radio Galaxy

_B_  Seyfert Galaxy

 

37.   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.

 

38.   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.

 

39.   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.

 

40.   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.