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Math in the News: 10/3/11
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Math in the News: 10/3/11


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In this issue of Math in the News, we look at the UARS satellite, which recently crashed after over a decade of service. What were the factors that made it so difficult to predict where it would land?

In this issue of Math in the News, we look at the UARS satellite, which recently crashed after over a decade of service. What were the factors that made it so difficult to predict where it would land?

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  • 1. 10/3/11
  • 2. The UARS Satellite
    The Upper Atmosphere Research Satellite (UARS) was launched in 1991. Its mission was to study what at the time was termed the “ozone hole” in the upper atmosphere.
  • 3. The UARS Satellite
    This video shows the deployment of this satellite from the Space Shuttle Discovery back in 1991.
    The UARS looks stationary, but it is moving at the same speed as the Space Shuttle, around 17,000 mph.
    Go to
    Click on “NASA Archives: About UARS.”
  • 4. The UARS Satellite
    This video gives an overview of the role that UARS played in the management of the ozone hole problem and how it has been alleviated. (Note: This video is 17 minutes in length.)
    Go to
    Click on “UARS and Discovery.”
  • 5. The UARS Satellite
    As a result of the data obtained from the UARS, global levels of CFCs have reached a plateau since the satellite’s launch.
    Launch of UARS
  • 6. The UARS Satellite
    The UARS was decommissioned in 2005. Since it was originally scheduled for a three-year mission, it outlived its mission by over a decade.
    When decommissioned a satellite is allowed to continue its orbit, but with no further intervention.
  • 7. The UARS Satellite
    We think of space travel as movement through a vacuum. If this were really the case, then given enough energy, a satellite would continue orbiting the Earth indefinitely.
  • 8. The UARS Satellite
    But a satellite, like any other spacecraft, is subject to solar wind, which is made up of subatomic particles and photons. Like the wind that can slow down an airplane, solar wind acts as a drag on the motion of a satellite, slowing it down.
  • 9. The UARS Satellite
    As the satellite slows down, its orbit around the Earth shrinks. The is an example of a decaying orbit.
  • 10. The UARS Satellite
    If solar wind was a drag on the satellite, even more dramatic changes occur when the satellite enters the Earth’s atmosphere. Here, a much more dense collection of gases and other particles not only slow down the satellite but eventually cause it to break apart.
  • 11. The UARS Satellite
    This is where it becomes difficult to predict where the satellite will ultimately land. There are two main reasons for this. Let’s look at the first.
  • 12. The UARS Satellite
    There are a number of detectors around the Earth. As the satellite passes each detector, its expected path past the next detector is determined. When the satellite is not detected, that means it descended somewhere between the two detectors. So, there is a margin of error introduced.
  • 13. The UARS Satellite
    The second reason that it becomes difficult to predict the satellite’s descent is the Earth’s atmosphere, which changes the path of the satellite. Imagine throwing a paper airplane past a fan. You cannot accurately predict where the airplane will land because of the deflection from the fan.
  • 14. The UARS Satellite
    As a result, there is a degree of uncertainty about where exactly a satellite’s begins its descent. Let’s estimate that this uncertainty is 15 minutes in duration. Let’s also assume that the satellite is traveling at a speed of 17,000 mph.
  • 15. The UARS Satellite
    Now let’s assume that the satellite is at an altitude of 100 miles when it begins its descent. We can use this freefall function to determine the range of where the the satellite debris will land. (All distance units shown are in feet.)
  • 16. The UARS Satellite
    Solve this quadratic equation for tto find out the time the satellite will be in the air before it comes to a crashing halt.
  • 17. The UARS Satellite
    Use this value for t to find the horizontal displacement of the satellite. One it enters the atmosphere its speed is no longer 17,000 mph. Assume a horizontal speed of 600 mph.
  • 18. The UARS Satellite
    So the total range of where the satellite can land has increased to 4,280 miles.