Apollo 13 Snapshot Navigation

 

            There is a popular Mark Twain book  called A Connecticut Yankee in King Arthur's Court. The fictional story’s appeal is a “time machine” type journey back to the days of Yore.   A 20th Century citizen of the United States is planted in the society of  500 A.D. England.   The gimmick making the book enjoyable is how the American Yankee used his modern  knowledge to advantage in King Arthur’s day.   What is now  common technical and scientific knowledge  appears mystical, magical, and supernatural to the knights of the legendary Round Table.  

            The following activity is sort of an opposite approach.  What if the Wright Brothers were transported 7 decades in time and thousands of miles above  to that April week in 1970 aboard the crippled Apollo 13 spacecraft?  How might they use knowledge of their time to advantage?  In lieu of Apollo 13’s powered down computer and star sightings obscured by debris, how could they establish the location of their spacecraft between the Earth and the Moon?  The challenge would be to use simple arithmetic and a process much like Columbus’s dead reckoning navigation.  The exercises and discussions which follow deal with their dilemma.

 

Materials:

 

digital camera and photo printer,  quadrant template, string, a globe, paper punch,  and ruler.

 

 

Figure 1.

 

 

Common knowledge found in 1903  grade school science books:

 

1.      The distance between the Earth and the moon is approximately 240,000 miles.

2.      The diameter of the Earth is approximately 8,000 miles (30 Earth diameters is the distance between the Earth and the Moon), the moon’s diameter is  2160 miles (111 Moon diameters is the Moon to Earth distance), or about one-fourth the diameter of Earth.

3.      The Earth and Moon are spheres.  The Earth  rotates on its axis once every 24 hours. (i.e., any point on Earth rotates 360 degrees about the Earth’s axis every 24 hours. )

4.      On Earth, the elevation angle of the North Star (Polaris) above the horizon is the latitude.

5.      The time recorded on a clock when the Sun is directly overhead, i.e., Noon, is a linear  relationship to longitude as one travels west or east of the prime meridian.

 

      When the Wright brothers hear the explosion of  Apollo 13’s Oxygen Tank 2, they peer into space from their spacecraft’s windows.   How can they determine their distance from the Earth and the Moon?

      All that would be required would be to make a measurement of the apparent diameter of both the Earth and the Moon from their spacecraft’s viewing window.  To make this measurement precisely, an onboard digital camera would be ideal.   The camera would capture the images of the Earth and  the Moon.  The ratio of their diameters or radii would be used to determine the location of Apollo 13 in space.  Here’s how:

 

 

Figure 2.

     

      NASA’s Tom Pendleton used symmetry and the concept of a camera’s focal length to construct similar triangles.  Both the Earth and Moon are imaged on the camera’s focal plane to determine their radii ratio. The above sketch was provided by Mr. Pendleton to explain his derivation.  Tom’s solution transmitted by e-mail is:

 

      Apollo 13 exploded at about 208,000 miles from Earth which is approximately 38,000 miles from the Moon.  Substituting the  Moon and Earth’s radii from the 1903 science book along with the location of Apollo 13 as 208,000 miles from Earth and 38,000 miles from the Moon results in a ratio of the observed size of the Moon to the observed size of the Earth of 1.755 to one.  The appearance of the Moon to Apollo 13’s crew was almost twice as large as that of Earth at the moment of the explosion.  

      The Wright brothers could have taken a photo of the Earth and a simultaneous photo of the Moon.  Next, they could have measured the diameters of the Moon and Earth on the respective photos with a ruler.  Lastly, the brothers need only divide the measured diameters of the Moon by the Earth’s measured diameter to calculate the needed ratio. If they had substituted that ratio of 1.755 in the formula above,  their solution would have revealed that  Apollo 13 was 208,000 miles from Earth and 32, 000 miles from the Moon when the craft exploded.

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