Suggested Exercises

1. The maximum possible size of a pulsar can be roughly determined from the fine structure of the pulses emitted, reasoning as follows.

If the sun were to turn off all over instantaneously, we would not see the light cease instantaneously, for the last light from the near side of the sun would reach the earth a little sooner than the last light from the far edge of the sun. Given that the sun's radius is 6 X 108 meters, how much time would elapse between the arrival of the two bits of light?

A flash from a pulsar may have a fine structure with typical time of rise and fall around 30 microseconds. Use this to get a rough estimate for the maximum possible size of whatever is emitting the flash.
2.Pulsars are thought to be rotating neutron stars. The pulses probably emanate from the magnetic poles, which rotate with the star (as with the earth, these poles may not be aligned with the axis of rotation). The light goes straight out from the poles, like a beam from a searchlight, and on every rotation, the beam may sweep across the earth.

What would we see if the magnetic poles of the pulsar were exactly aligned with its axis of rotation, and pointing toward the earth?

From this "lighthouse" model, would you conclude that there are many more pulsars than we have observed? Why? Construct an equation for the number of unobserved nearby pulsars, in terms of the number of nearby pulsars that have been observed. (Before writing the equation, you will need to define some symbols in terms of the geometry of a typical pulsar beam.)

3.The energy of a pulsar's emission is drawn from the pulsar's rotation, resulting in a slowing down of the rotation rate. This slowing down has been measured. The Crab pulsar, which has a rotation period of 33 milliseconds, is slowing at the rate of 4.2 X 10-13 seconds per second.

What is the rate of slowing in seconds per year? If you used this pulsar as a clock, how much time would pass before you were "slow" by one minute?

The age of a pulsar is given approximately by the equation

Age = half (T/T1)

where T1 is the period of rotation and 1/T is the rate of change in the period. Using this equation, how old is the Crab pulsar?

4. LABORATORY EXERCISE: measuring short time intervals using stroboscope.

The principle of the stroboscope can be demonstrated with a slowly blinking light, such as a spotlight on the street. If you blink your eyes at the same rate as the light, you can make it seem to yourself as if the light is always off, by having your eyes closed whenever the light is on. Or you can make it seem as if the light is always on. By controlling the way you look, you can make something that moves in a repetitive way "stand still." This is precisely like the way Taylor's electronics helped Cocke and Disney make the pulsar's blinking "stand still" so they could see it.

The hand stroboscope lets you do this for things that move more rapidly than the eye can blink. It is a disk with several equally spaced slits. As the disk is rotated in front of the eye, the eye catches a glimpse of the moving object, which can appear to be stopped. For example, you may look at a rotating circular saw with a red spot painted at one position. If you make the spot seem to stop by looking at the saw through a hand stroboscope of 6 slits, turning at 35 revolutions in 5 seconds, you can calculate: .,

(35 revolutions / 5 seconds) x (6 slits / revolution) = 42 slits per second

therefore the saw is rotating at 42 revolutions per second.

(a) Could the saw also be rotating at 84 revolutions per second? at 21 revolutions per second?

(b) If the spot on the circular saw appears to move slowly backward when seen by a stroboscope, is the stroboscope rotating a little faster or a little slower than the rate that will hold the spot still? Explain. (This is precisely similar to Cocke and Disney's concern with holding the pulse in the middle of their screen.)

(c) A television camera looked at the Crab pulsar (whose period is .033 seconds) through a stroboscope. If the strobe had 8 slits, at what rate should it rotate to make the pulsar seem always lit up?

(d) Use a hand stroboscope to measure the rotation rate of an electric fan; the vibration rate of a bell.

(e) What do you see if you look at a fluorescent light through a stroboscope? (The effect can sometimes be seen by looking at a rotating object, such as an electric fan or the gears on a rotary hand drill, under a fluorescent light.)
5. Much as students make applications for college, astronomers must make applications to use major telescopes. Read the "Application for Time on a Telescope" form used by a large observatory. Comment on each portion of the form. Specifically, why does the committee that decides who gets telescope time need an answer to each question? Imagine that you are Disney or Cocke and fill out the application in order to convince the committee to give you time to search for an optical pulsar. D,I
6. LABORATORY EXERCISE: Make a diaphragm. Cocke and Disney described a diaphragm that they needed to observe the pulsar (so their light-detecting device would not be swamped by the light of other nearby stars). They made it with a razor and aluminum foil.

(a) Comment on this aspect of a modern experiment. Should they have used more refined apparatus? How would their story have been different if they had required a special diaphragm that took a team of technicians a month to build?

(b) Make a diaphragm ten times the size of theirs: a square hole in foil exactly 1 mm x 1 mm. Then try for one 0.1 mm x 0.1 mm.

(c) Can you think of a better way for making such a diaphragm at home? (One possibility: coat a piece of glass with paint, let it dry, and scratch some of the paint away.) Try other methods.
7. R. Willstrop of Cambridge, England, was taking data from the Crab pulsar months before Cocke and Disney. However, Willstrop needed a large amount of computer analysis of his data, so he was not able to announce the "discovery" of an optical pulsar until after Cocke and Disney. Who should get credit for the discovery? What difference does it make who gets the credit? Some sociologists believe that credit for a discovery is the chief reward a scientist strives after. How important to progress in science was assignment of credit in the pulsar story?

A student, Jocelyn Bell, was first to observe a radio pulsar while analyzing data—she noticed something peculiar that nobody else had paid attention to, and hunted down its source. But a Nobel Prize was awarded to Anthony Hewish, who had assigned her the task and who had built the radio telescope. There has been some controversy over whether the prize award was fair. Compare this with the assignment of credit in the optical pulsar story. How much credit should go to the builder of an instrument, like Hewish or Taylor? How much to the person who designs a research program, like Hewish, and how much to the person who carries it out in a creative way, like Bell? Does it matter?
8. Consider the following circumstances of the optical pulsar discovery: (a) Cocke and Disney meet; (b) Taylor has a suitable apparatus; (c) a new fast pulsar has recently been discovered (it turns out that of the hundreds of pulsars observed with radio telescopes, only a handful are readily seen with ordinary light); (d) they get extra observing nights because a colleague's wife is ill; (e) they find a math error when they recompute their values; (f) they hold PhDs, representing years of hard training in science.

How much luck was involved in the discovery? How much hard work? In your answer, refer to the above circumstances as well as any others that seem relevent.

A favorite motto of scientists was stated by Louis Pasteur: "In the realm of observation, chance favors the prepared mind." In what way does or does not this saying apply to the pulsar story?
9. Optical pulsars help us understand neutron stars, a late stage in the life cycle of some stars. What is the significance of Cocke and Disney's work? If we understood the life cycle of stars better—so what? Should research of this type be supported with tax dollars? D,R
10. When Taylor got the excited phone call from Disney and Cocke, he was skeptical at first. Would most scientists have felt that way? How important is skepticism in science? How skeptical should you be when you hear something from (a) a textbook; (b) a famous scientist on television; (c) a scientist on television identified as a member of a "public interest" group; (d) a television reporter? Would a good scientist be more or less skeptical than you of each of these sources? How do scientists finally decide whether what they have been told is true? D,R
11. Compare the audio recording of the discovery, the paper describing the results in Nature, and (if you can find one) a newspaper article announcing the discovery. Which comes closest to describing what was in fact discovered? Which is most objective? Is anything entirely objective? I,R
12. The official communication of the discovery in Nature is written in a highly stylized form. Why is this form used?

Take a long, hard look at some everyday phenomenon (for example, water emptying from a sink, or a sunset) and imagine that it has never been observed before. Write (a) a paper in scientific style, and (b) a letter to a friend about your "discovery." What purposes are served by each style?
13. The audio recording that you have listened to was pieced together from interviews with three people and from the tape made at the time of the discovery. In what ways is the tape made at the time likely to be more truthful than the accounts recorded a decade later? In what ways might the tape made at the time be less useful to understanding what happened? Some historians think it is a waste of time to interview people about what they did many years ago; these historians spend their time studying old correspondence and other writings recorded near the time of the events they study. Other historians try to interview as much as possible. Discuss the advantages and disadvantages of each approach. Use some examples from the optical pulsar story. D,R
14. If you could stand on a neutron star, how far away would the horizon appear? C
15. Name some objects that rotate at about the same speed as a pulsar. Which have the most stable rate of rotation? Why? C
16. Review the description of the environment as Disney saw it on his way to the observatory. Is this your impression of what a scientist would notice? Is it different from what a non-scientist would notice? D
17. We know that ice skaters can spin faster by pulling their arms in: the angular momentum is constant, but the decreased radius produces an increased rotation rate. Calculate the angular momentum of our sun (radius 6 X 108 meters, period of rotation 25.3 days), making an arbitrary assumption about the distribution of mass. Using conservation of angular momentum, calculate the new rotation rate if the sun shrank to the size of a neutron star (radius 10 kilometers = 104 meters). How does this compare with the rotation rates of pulsars? C
18. If all the mass of the sun were to become a neutron star, what would be its density? (Solar mass = 2 X 1030 kilograms.) How does this compare with the density of the nucleus of an atom? If you could get a piece of a neutron star the size of a pea, how much would it weigh on earth? C
19. On the earth, the gravitational force at the equator is partly balanced by a centrifugal force which tends to throw things off this spinning planet (that is why the earth bulges slightly at the equator). Find the gravitational and centrifugal forces at the equator of a pulsar 10 km in diameter with a pulse interval of 33 milliseconds. Compare these with the same forces on earth. Would you expect much of an equatorial bulge on a pulsar? C
20. Calculate an example of Coriolis forces on the surface of a neutron star, assuming a plausible position and velocity for a moving object. If creatures somehow lived on the surface of the neutron star, would Coriolis forces severely affect their movements? C
21. The sun has a magnetic field at the surface averaging about 1 gauss. If the sun were to shrink to the size of a neutron star, the field lines would not escape but would become more tightly wrapped. What magnetic field strength would the neutron star have? Does your result seem reasonable? If creatures lived on the surface of the neutron star, and if their bodies had magnetic properties like our own, would they have to take magnetic field lines into account in their movements?

(Note for the teacher: There is a science-fiction novel by the astrophysicist Robert L. Forward, Dragon's Egg, Ballantine: 1980, which imagines life on a neutron star. The "biochemistry" is made of combinations of nuclei, which can exist in a very shallow surface zone. The creatures are blobs a few mm across. While gravity is too great for them to throw objects, anything they roll is noticeably affected by Coriolis forces. Magnetic field lines are far more important: at the magnetic equator, a creature is pulled out into a cigar shape; it can move along field lines easily, but across them only with great difficulty.)
22. Find the Crab Nebula on a star map. Can you find it in the sky during XXX? (Note for the teacher: fill in month when exercise is assigned.) I
23. In November 1982 a new pulsar was discovered, PSR 1937 + 214, better known as "the millisecond pulsar." Its period is 1.56 ms, that is, a frequency of 642 pulses per second, twenty times the rate of the Crab pulsar. Calculate the gravitational and centrifugal forces for this pulsar, assuming a diameter of 10 km. What is the velocity at the equator? Is this close enough to the speed of light so that useful calculations would have to call upon the equations of special relativity? C
24. The "South Preceding" star in the Crab Nebula (the Crab pulsar) was recorded on a photographic plate made in 1899, and preserved since then. State some questions that could be answered by comparing this old plate with a recent one, and that could not be answered by a recent plate alone.

Old astronomical photographic plates are worth preserving for scientific use, whereas old laboratory measurements in physics are of little scientific value—because a physicist can always measure a physical quantity anew, and usually more precisely than the first time. Astronomy, but not physics, is in part a "historical science." Name some other "historical sciences." How are they like the study of human history? How are they unlike it?
25. Following the audio part of the exhibit, outline the steps in the process of the optical pulsar observation. Include "null" observations, checks of various kinds, changes in the data accumulated, repetitions of earlier trials. From this outline and your general ideas about scientific process, answer the following:

(a) How skeptical are Disney and Cocke?

(b) Why does Taylor tell them to sit on their results?

(c) Does this case agree with your ideas about "good" scientific method? Explain your answer.

(d) Examine the sample from Cocke's notebook. Does this agree with your ideas about how "good" scientists record data?
26. Inspect the page from the proceedings of a pulsar conference. Using this as evidence, estimate about how many people were working on pulsars at that time? State your reasoning.

Some scientific subjects are studied only occasionally by people who soon move on to other subjects. Other subjects may become institutionalized fields, first with conferences, then perhaps with textbooks and university professorships, finally with entire journals and scientific societies dedicated to the field. How far along this series of steps would you expect the study of pulsars to go, and why? Name some things that might influence whether a subject of inquiry will attract the full-time attention of many hundreds of researchers?
27. Define the Doppler effect in general. What effect did it have on the search for an optical pulsar? R
28. Suppose a pulsar is about equally hard to detect at radio and at optical light frequencies. Atmospheric absorption of radio and optical frequencies is slight, and instruments to detect their energy are of roughly equal sensitivity. A typical major radio telescope has 50 times the diameter of a major optical telescope.

(a) Roughly what is the ratio of the energy the pulsar emits in radio frequencies to the energy it emits in optical frequencies?

(b) Suppose the pulsar emits the same amount of energy in X-ray frequencies as in optical frequencies, should this be very easy or very hard to detect? Give two reasons for your answer.
(Note for the teacher: X-rays from pulsars have been detected by satellites above the atmosphere.)

(c) Why can a radio telescope be built much larger than an optical one? What sets the limit on how big a radio telescope can be?
(Note for the teacher: The largest radio telescope is at Arecibo, Puerto Rico, and fills a natural hemispherical depression 305 meters in diameter.)
A larger project for an advanced student or student team can be made from four Web exhibits which all describe "moments of discovery:" nuclear fission, an optical pulsar, the electron, and the transistor. The student(s) should study all four exhibits and discuss similarities and differences -- socially in terms of individuals, scientific institutions, and communication, and scientifically in terms of technologies and thought processes. Students can review the lists of questions in the Teachers' Guides for ideas on directions to follow (perhaps too many!). The students should conclude with general statements about things that seem necessary for all discoveries, at least in modern physical science.