ASTR 10, Vista College, Spring 2003 - Instructors: Drs. Sallmen & Korpela


Solutions to HW #4 for Chapters 9, 10, and 11


Explain ALL answers.

    ISM and Star Formation (15 points)

  1. Interstellar Medium (5 points): What evidence do we have that the interstellar medium contains gas and dust? (Give AT LEAST one for each).

    The gas of the interstellar medium can be detected from the absorption lines present in the light coming from distant stars. The lines from the interstellar medium are narrower than those from the stars themselves, indicating that they are produced by material at a much lower density and temperature than the surface of the stars. Interstellar gas is also revealed by the 21-cm radiation we receive from large clouds of hydrogen. Some clouds also produce emission spectra, because they are excited and/or ionized.

    Dust in the interstellar medium is evident in the extinction (dimming) and reddening of light from distant stars. Atoms and individual molecules do not efficiently extinct or redden light. Dust is also apparent in reflection nebulae, in that dust scatters the blue light from distant or embedded objects.

  2. Energy Generation (10 points): What is the energy source for the following two objects? (In other words, where does the energy come from that makes them shine?)
    1. a protostar (a glowing, collapsing cloud of gas that hasn't yet become a star)?

      The energy source for a protostar is gravitational energy. During the collapse from a cloud of dust and gas, the gravitational energy is converted to thermal energy: the material heats up, and so it emits radiation.

    2. a main sequence star?

      The energy source for a main sequence star is nuclear fusion. 4 hydrogen nuclei combine in a chain of reactions to produce one helium nucleus which is more tightly bound (slightly less massive) than the original atoms, thus releasing energy. The amount of energy released is the equivalent energy of the difference in mass between the initial products (4 H) and the end product (He).

    Stellar Evolution (20 points)

  3. How (5 points): Mass is the physical quantity that determines how (and when) each star evolves. Why does this single property affect stellar evolution so strongly? Explain in terms of simple physical concepts. (Hint: What explains the Mass-Luminosity relation for main sequence stars? What causes the differing main-sequence lifetimes for various stars?)

    Mass is the property of a star that determines how and when each star evolves. More Massive stars have more weight pressing down on the centers, so require higher pressures to support this weight (gravitational equilibrium). As a result, more massive Main Sequence stars are more luminous, and so use up the hydrogen in their cores more quickly, and as a result have much shorter main-sequence lifetimes. Once a star leaves the main sequence, its subsequent life cycle also depends on its mass, again due to the requirements of gravitational equilibrium (balance between weight and pressure).

  4. Evidence (10 points): How can star clusters confirm our theories of stellar evolution? (Hint: What do we expect the HR diagrams of both young and old clusters to look like? How does this help?)

    Star clusters contain stars of a wide variety of masses that all formed at about the same time. We can take models of many different masses of stars and determine what kinds of stars a cluster should contain after a given time. We can then compare the HR diagram from our model cluster with that of the observations of an actual cluster to see how well they agree.

    According to our models, young star clusters should have main-sequence stars of all luminosities, but almost no red supergiants, and no red giants or white dwarfs. Old star clusters should have no hot, luminous main-sequence stars, but lots of red giants/supergiants and at least some white dwarfs, as well as low-mass (dim, red) main-sequence stars. One clear test that our models are at least on the right track, is that we find no star clusters with both hot/luminous main-sequence stars AND a large number of giants and supergiants. All of the star clusters show us that when giants and supergiants are present in the clusters, there are few if any hot/luminous stars on the main sequence. Put another way, some clusters have HR diagrams that match our models for young clusters, while others match our models for older clusters of varying ages. For the most part, the agreement between the model HR diagrams and the cluster HR diagrams is quite good.

  5. Time-scales (5 points): Red Giants have proceeded further in the evolutionary sequence than main sequence stars, but is it correct to say that all red giants are older than all main sequence stars? Why or why not? As an example, which would be older, the Sun or a Red Giant of 10 times the mass of the Sun? (Table 9-2 might help you here).

    Remember that the most massive stars use up their hydrogen and leave the main sequence MUCH faster than lower-mass stars. The Sun will spend about 1010 (10 billion) years on the main sequence fusing hydrogen (just a little longer than the 1.1 solar mass star in Table 9-2), but a 10 solar mass star will spend only about 30 million years on the main sequence (closer to the 11 million years of a 15 solar mass star than the 440 million years of a 3.5 solar mass star in table 9-2). Since a star spends 90% of its life on the main sequence, a red giant/supergiant with 10 times the mass of the Sun is only a little over 30 million years old, while the Sun (on the main sequence) is about 5 billion years old. So red giants can be younger than lower-mass main sequence stars, even though they are further in their evolutionary sequence. An analogy would be a mayfly that is near death, but only 1 day old, while a human still in childhood could be 3 years old. The human is not as far on in its life cycle as the mayfly, but is older.

    End products of Stellar Evolution (35 points)

  6. Massive-Star Supernovae (10 points):
    1. What are two predictions of the theory of massive-star supernovae? (Don't explain the theory, but rather tell me two things which should be observed if it is true.) A massive-star supernova is one that occurs at the end of the life of a high-mass star.

      This kind of supernova occurs when a massive star has an iron core. Since neither fission nor fusion of iron produces energy, the core collapses into a black hole or a neutron star and energy is transferred to the outer layers of the star in a massive explosion.

      We expect these supernovae to (you only needed to come up with 2 items!)

      1. suddenly make a previously normal star MUCH more luminous
      2. result in a burst of neutrinos as the core forms a neutron star
      3. result in an expanding "supernova remnant" (evidence of the explosion in the interstellar medium) at the site of the visible supernova
      4. result in a neutron star or black hole in or near supernova remnants
      5. have both lines of hydrogen (from the outer layers of the star) and elements heavier than iron (formed during the explosion) in their spectra
      6. occur in populations of young stars (since massive stars are short-lived)

    2. What observations have confirmed each of these theoretical predictions?

      1. In 1054 AD, a "guest star" was suddenly bright enough to see during the day, and many ancient astronomers noted its appearance. In 1987, a similar event (SN 1987A) was seen in a nearby galaxy.
      2. A neutrino burst was observed at the time of SN 1987A
      3. We see expanding supernova remnants in both of these cases. The supernova of 1054 AD resulted in the "Crab Nebula".
      4. The Crab Pulsar (a spinning neutron star) has been seen in the Crab Nebula. Pulsars have been seen in other supernova remnants as well.
      5. Hydrogen lines are seen in the spectra of this type of supernova, and the infrared spectra of SN 1987A exhibited lines of Cobalt and Nickel (both heavier than Iron).
      6. Although we didn't discuss it in detail, this type of supernova occurs in areas with hot, blue, luminous (and therefore young) stars.

  7. White Dwarfs (5 points): Why does a white dwarf shine? What (if any) is its internal energy source? (Hint: It isn't the same as for a protostar or main-sequence star.)

    White dwarfs shine because they are hot, and hot objects emit radiation. White dwarfs have no internal energy source at this time, and are cooling as they emit radiation. (They do not shrink significantly because they are supported by electron degeneracy, which does not depend on internal temperature).

  8. Determining Final Fates (5 points): What property of a star determines whether it will end its life as a white dwarf or a neutron star? Why?

    The mass of a star determines whether it will end its life as a white dwarf or a neutron star. This is because the conditions in the core are governed by the balance between gravity and pressure. More massive stars have more weight pushing down on their cores, and need to have higher central pressures in order to support themselves.

    Here are the results of this competition between pressure and gravity in low-mass and high-mass stars. The core of a low-mass star is supported by degenerate electrons before the core temperature rises enough for elements heavier than Helium to undergo nuclear fusion. This core will be smaller than 1.4 solar masses, and will become a white dwarf when the outer layers puff gently away. The core of a more massive star will become hot enough to fuse elements up to Iron. This core will collapse at this time because energy generation ends, material will continue to fall in until the core is larger than 1.4 solar masses, and so electron degeneracy cannot support it. The electrons and protons are forced together into neutrons, and a neutron star is formed.

  9. Neutron Stars (10 points): Astronomers concluded that pulsars MUST be rotating neutron stars through a process of elimination - it's the only theory that worked!
    1. What is one other explanation that didn't work (and why)?

      The following contains more detail than I expected, as I only asked you to discuss ONE of the other possible explanations.

      Pulsars could have been due to i) orbital motion of two objects (but some of the pulse periods were too short), ii) rotation, or iii) vibration of a) main-sequence stars, b) white dwarfs, or c) neutron stars. We've already given one reason why orbital motion is unlikely. In addition, an orbit that short would be losing energy due to gravitational radiation VERY quickly, and would be unstable. The short pulse duration indicates that the emitting region is smaller than 300 km, telling us it is likely to be a neutron star. In addition, a rotating main-sequence star or white dwarf would fly apart if they were rotating that fast. So it must be a vibrating white dwarf or a vibrating or rotating neutron star. The range of pulse periods is difficult to explain with vibration (if the stars are ringing like bells, why do similar stars give different periods/frequencies, or "pitches"?). Vibrating objects speed up as they lose energy, while rotating ones slow down as they lose energy. Pulsars are slowing down, so they must be rotating neutron stars.

    2. What supporting evidence is there for the winning theory (2 things we observe that match the predictions) ?

      The theory that pulsars are rotating neutron stars is supported by several things.

      • Some pulsars are found in supernova remnants, where we expect neutron stars (but not white dwarfs) to form.
      • Polarization observations are consistent with the theory that the radio beam is coming from the magnetic pole of an object, and we believe neutron stars have large magnetic fields.
      • The lighthouse model of pulsars (involving large magnetic fields) can explain the energy required to keep the Crab Nebula emitting radiation 50 years after the supernova of 1054 AD.
      • The mass estimates of neutron stars are consistent with the predicted range of 1.4 to 2 or 3 solar masses.
      Observations that were part of the original process of elimination (and here I was hoping for ADDITIONAL evidence) include:
      • The full range of pulse periods can be explained by rotation
      • The short duration of pulses is explained by the theory that the radiation from pulsars comes from a beam that sweeps past us like a lighthouse.
      • Pulsars are slowing down, as expected for rotating objects

  10. Neutron Stars (5 points): A burst of neutrinos was observed from Supernova 1987A, indicating that a neutron star formed. No pulsar has yet been detected at the position of SN 1987A. Does this mean that there is no pulsar there? Why or why not?

    There might still be a pulsar at the position of SN 1987A, even if we can't see it. This is because the radiation from pulsars is beamed, and the pulsar is only visible to observers if the beam points to them at some point during the pulsar's rotation. The beam of a pulsar formed in SN 1987A simply might not point to us.

    Relativity (15 points)

    If you missed class, you should read the relativity notes before doing this section.

  11. A Simpson Classic (5 points)

    Bart builds himself a rocket-powered car and on its maiden trip he crashes into a mountain and dies. (Don't worry, he'll be back again next episode.) Before crashing, Bart is travelling (with his headlights on) at 90% of the speed of light. To Bart, the photons from his headlights seem to travel at the speed of light (c = 300,000 km/s). Lisa is watching the whole disaster. How fast does Lisa think the light from Bart's headlights is travelling (and why)? (Hint: You don't need ANY fancy math to solve this question - a basic knowledge of relativity is enough!)

    One of the assumptions of Relativity is that the speed of light is the same for all observers. Therefore Lisa must think that the light from Bart's headlights is travelling at the speed of light, c = 300,000 km/s.

  12. Einstein Was Right (10 points)

    What OBSERVATIONS have been made that confirm predictions made by Einstein's theory of General Relativity (at least 3)? (Don't just tell me the predictions.)

    Many observations have been made that confirm preditions of General Relativity:

  13. Black Holes (15 points)

    1. How would the orbit of the Earth change if our Sun suddenly collapsed into a black hole? (Assume the mass stays the same.)

      It wouldn't. The elements of the equation that describes the force of gravity between the Earth and the black-hole-sun are just the mass of the Earth, the mass of the Sun, and the distance between them, and these (F = GMSME/d2) haven't changed! So the force of gravity acting on the Earth due to the Sun has not changed. Since it is gravity that determines the orbit, the orbit wouldn't change.

      (Bonus: Would humanity survive this change in the Sun? Explain your answer.)

      If the Sun were suddenly a black hole of the same mass, we would no longer receive any energy from the Sun! Temperatures here would drop a lot, so we'd all die horrible deaths, but .. at least we'd be in the same orbit.

    2. Nothing that falls beneath the surface of a black hole ever emerges, not even light. So how can we detect a black hole using X-rays?

      Black holes themselves may be invisible, but the material around it may emit X-rays. If a black hole is near another star or gas cloud, then it can draw some of that matter into itself. As this matter approaches the black hole, it will form an accretion disk around the black hole (due to conservation of angular momentum). The matter in the accretion disk will become very hot as it is squeezed together by the gravitational field of the black hole (frictional and tidal forces create heat). The temperature of this accretion disk can reach over 1 million K. Hot objects emit radiation with a shorter-wavelength, so very hot objects emit X-rays. So a black hole can't emit X-rays on its own, but an accretion disk of matter falling INTO the black hole can emit X-rays.

    3. The search for black holes has succeeded in finding a few strong candidates, but the problem is being sure a particular binary system contains a black hole and not a neutron star. What observations would you make of an X-ray binary system to distinguish between a black hole and a neutron star? (ie. What "stellar" quantity would you like to measure?)

      The most important observation would be one to determine the mass of the components of the binary system. If the compact object can be shown to have a mass in excess of 3 solar masses, then the object may well be a black hole. If it can be shown to have a mass greater than 10 solar masses, then most astronomers would accept it as a black hole.

      Bonus: How do you think you can measure this quantity?)

      To measure the mass of the binary system, we must try to determine the shape of the orbit. Measuring the Doppler shifts of lines in a companion star will give us some idea of the orbit, but as we discussed earlier in the class, this alone doesn't give us enough information to completely determine both masses. Usually some assumptions (like the mass of the companion star) must be made, resulting in increased uncertainty in the mass measurement of the compact object.

    4. Extra Credit The Sun has a Schwarzschild radius (Rs = 2GM/c2) of only 3 km. If the Sun has a Schwarzschild radius, why isn't it a black hole?

      Every object has a Schwarzschild radius. This is just the size the object would have to be squeezed in order for its surface gravity to become large enough that light cannot escape (and so the object would become a black hole). The Sun isn't a black hole because it is bigger than its Schwarzschild radius.