Investigation 5: Variable X-ray Sources - Activity 3

Interpreting the Light Curve of 4U1822: Linear Size

Overview: Students learn to use light curve timing to estimate maximum linear size of a system. They connect a given model (binary system with neutron star and main sequence star with given separation) to these observations to determine the size of the main sequence star.

Physical resources: None

Electronic resources: Images of light bulbs at different brightnesses, X-ray image of 4u1822

• Predict the linear size of object from angular size of blob observed, and linear distance (25 kiloparsecs = 25 x 10³ parsecs).
• Is this reliable? What about angular resolution?
• Show images of same size bulb but different luminosity to show that larger flux makes an object look like a larger angular size, even though the linear size of the bulb is always the same. Images of different brightness bulbs: (bulb brightness 1, bulb brightness 2, bulb brightness 3)

Predicting light curves from models with different parameters:

• Instructor asks students to develop models to explain the difference in light curves observed from two hypothetical lighthouses (one light curve goes to zero, with shorter period, and one dips only to half original height with longer period).
• Discuss what is meant by parameters: descriptions of the model, which can change.
• Outline the parameters of the X-ray binary system
• For relative angular sizes of the two components of an X-ray binary system, have students make predictions of the X-ray light curve for these three systems:
  • Angular radius of the X-ray emitting compact star ~ angular radius of the companion star (light curve touch zero for a short time)
  • Angular radius of the compact star << angular radius of the companion star (light curve drops to zero for a long time)
  • Angular radius of the compact star >> angular radius of the companion star (light curve never drops to zero)
  • Students use bulbs to simulate X-ray emitting source, and balls of different sizes to represent companion star, and use empty tubes to simulate observing the system from a distance using a telescope.
• Discuss the implications for the parameters of 4u1822 (since the light curve does not drop to zero, the angular radius of the X-ray emitting compact object would be larger than the compact star.)

Another way to estimate linear size: Estimate the maximum size of objects from durations of segments in light curve:

• Instructor gives simple example of using eclipse length to determine size of object, for simplest case of a much larger companion (blocking) star (see important note below)
  • Images of notes: (size eclipse 1, size eclipse 2)
• The fastest anything can move or change is the speed of light, so v = delta x / delta t. Thus, the largest delta x could be is c * delta t. Compare this with the size of the object as determined from angular size estimate. Compare this to the size of the solar system or distance to nearest star.
• Measure times between lowest states (eclipses), should be about 20,000 seconds.
• Calculate orbital speed from time between eclipses, given an assumed orbital radius of binary system (~10¹² cm). (Project group will discover how to measure this using Kepler's third law!)
• Calculate size of blocking star: diameter (delta x) = v * delta t
  • Notes from above problem-solving: (blocking material calculations 1, blocking material calculations 2, blocking material calculation 3)
  • Instructor notes for above problem-solving: (XRB size)
• Wrap up discussion: Compare the characteristic size of the system gotten above to the distance to the object, as well as the size of the solar system:
  • From orbital speed: Distance to system ~ 7.5 x 10¹⁹ m / 4u1822 system diameter ~ 3 x 10¹⁰ m = 2 x 10⁹, or 2 billion times farther away than it is large!
  • Diameter of solar system = 2 x radius of pluto's orbit = 80 Astronomical units = 12 x 10¹² m / 4u1822 system diameter ~ 3 x 10¹⁰ = 4 x 10² = 400 The solar system is 400 times wider than this system. In other words, this system is only about 20 times larger than the diameter of our sun.

Teacher Tips/Tricks

• Timing: 3.5 hours in Summer 2008. To minimize time, make binary system demo shorter and clearer (as suggested below), getting to their predictions of light curves faster. However, we went through the last two calculations of the size of the blocking star rather quickly.
• When introducing changing the parameters of the XRB model to alter angular size in the physical model, have the orbital radius fixed (perhaps drawn on a piece of butcher paper), and students simply change the linear diameter of the objects.
• Discussing the implications for 4U1822 beyond simply "the X-ray emitting object is larger than the companion star" may be better left for activity 5 where the students develop and rule out or support their own more detailed models of the system.
• The method of using the entire duration of the eclipse to determine the size of the blocking object only works strictly for the case where the X-ray emitting compact object is much smaller than the companion star. (For instance, when the compact object is the same size as the companion star, the proper time to use for estimating the diameter is the time from when the eclipse first begins to when it reaches its minimum.) However, applying this technique to any eclipse situation gives a reasonable (within a factor of a few) estimate.

Assessment Ideas

• Show other light curves (some dip to zero, some do not) How would you interpret what's happening in this system? (i.e. X-ray object completely blocked if light curve drops to zero.)
• Ask students to list observations, models and predictions they used during this activity.

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