For Educators

Timing An X-ray Pulsar

by James Humphreys


Introduction, Requirements, Lesson Plan, References



Certain X-ray objects in the sky emit X-rays in a very rapid regular periodic pattern. These objects are X-ray pulsars. The pulsar contained in the Crab nebula is the most well known of these objects.

A pulsar is a neutron star with a very strong magnetic field. Neutron stars are the remnants of massive stars which have reached the end of their lives. These stars have an initial mass greater than 10 times the mass of the sun. Throughout the star's life, hydrogen burns into helium through nuclear fusion. Once enough helium has formed and the pressure is high enough in the star's center, the helium fuses into carbon and oxygen. This process of lighter elements fusing into heavier elements continues until an iron core is formed. Iron has the largest binding energy of all the elements, and hence it cannot fuse into any heavier element. Thus iron simply accumulates under the weight of shells of successively lighter elements which continue to burn.

When the iron core can no longer support its own weight and the weight of the rest of the star, it collapses. Within milliseconds, gravity overwhelms the pressure of the remaining material and a full collapse ensues. In this collapse, protons and electrons combine to make neutrons. The energy released is emitted outward and explosively blows away the remaining material of the star. This material glows and is visible as a supernova remnant. The core of the star is left as a neutron star.

The initial magnetic field of the star intensifies during the collapse. Hence, neutron stars can have very strong magnetic fields. Neutron stars can "pulse" because of electrons accelerated near the magnetic poles. These electrons travel outward from the neutron star. They emit x-rays and gamma rays when they must travel faster than the speed of light to continue rotating with the magnetic field. An external observers sees a pulse of radiation whenever the magnetic pole is visible. The pulses come at the same rate as the rotation of the neutron star (30 times a second for the Crab pulsar), and are very regular.


In this segment the student is introduced to the use of X-ray data to identify an object as a rotating neutron star, and determine its period of rotation.


Using XTE observations of the Crab Pulsar, the student will be able to:

  1. Determine the period of the light intensity.
  2. Interpret this period as the period of rotation of the object.
  3. Identify this object as a neutron star.


  • Grade level: 11th or 12th
  • Prerequisites: Algebra, physics, chemistry or physical science (some formal introduction to waves and periodic phenomena)
  • Preparation: The teacher must download and distribute copies of the following plots:
  • Alternatively, the teacher may instruct the students how to get these files.

    This lesson depends heavily on the student's understanding of periodicity, and the Principle of Superposition. A brief explanation of the principle of the Fourier transform is required.

  • Materials (per lab group): One copy of the FFT graph, calculator, pencil, flashlight or other safe, low intensity light source
  • Setup: Be sure to know how the student groups will obtain the necessary graphs.
  • Estimated class time: About 10 minutes of discussion of the Principle of Superposition, and Fourier transforms. Once the graphs are obtained, approximately 25 minutes.

Lesson Plan


Choose a familiar, common periodic phenomenon. Describe some measurable parameter that would show this periodicity, including measurements that could be taken, and devices that could be used to take them. Since you are already familiar with this phenomenon, calculate the period and frequency of the variation in the suggested parameter, and express these in appropriate units.


Distribute the graph with the following directions/questions:

The graph you have is of a computer generated Fast Fourier Transform over a light curve obtained from the Crab Pulsar during 100 seconds. In general, the lowest frequency spike corresponds with the fundamental frequency of the wave form being analyzed.

  1. Using this information, what is the frequency of the variation in X-ray intensity coming from this object? the period?
  2. What explanation might you offer for the presence of the other prominent spikes? (Hint: How do the frequencies compare to that of the fundamental?)
  3. The usual, and generally accepted explanation of this rapid variation in intensity is that the neutron star itself is spinning, and that the "bright spot" is alternately pointed toward and then away from us. No other theoretical process is able to adequately explain the brightness changing so rapidly and over so many wavelengths with such regularity.
  4. To see how this effect is produced, have one lab group member stand, holding the flashlight, in the center of a small circle formed by the rest of the lab group. The center person then points the flashlight with an arbitrary orientation. Then, keeping the flashlight still with respect to her/himself, this student spins at some comfortable rate. The others observe the changing intensity of the light they can observe from the source.

    Use a clock or watch to time the "flashes" of light and calculate the period of the student's rotation. Finally, have the student in the middle spin as fast as s/he can, measure that period and compare it to the period of the Crab Pulsar.


Neutron stars have masses up to about 3 solar masses, and radii of the order of 10 km. Assume such a mass and radius for this object, and use the period you calculated earlier to answer the following questions:

  1. What is the linear speed of a point on the equator of the star?
  2. Express this as a percentage of the speed of light.
  3. What is the centripetal force necessary to keep the material of the surface near the equator in place (in orbit around the center)?
  4. What is the gravitational force at the surface of this object ? How does this answer compare to your answer to number 3 above?


  1. Zeilik, Michael. Astronomy: The Evolving Universe. John Wiley & Sons, Inc., New York (1991, 6th ed.) pp.378-382
  2. Arny, Thomas. Explorations: an Introduction to Astronomy. Mosby, St. Louis (1994) pp.395-401
  3. Graham-Smith, Sir Francis. "Pulsars Today." Sky and Telescope 80 (September 1990): 240