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Measuring Longitude
An Introduction


Prof. David Bernstein
James Madison University

Computer Science Department
bernstdh@jmu.edu

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Review
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  • Longitude:
    • Specifies the east-west position using an angular measure
    • Ranges from \(0^\circ\) at the prime meridian to \(180^\circ\) at the international date line (with positive value to the east and negative values to the west)
    • Denoted by \(\lambda\)
  • Day:
    • The amount of time it takes for the Earth to rotate around it's axis one time
    • Divided into 24 (equally-sized) hours
Implications
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  • Some Basic Math:
    • Since the Earth rotates \(360^\circ\) in 24 hours, it rotates \(360/24 = 15^\circ\) per hour (i.e., \(1^\circ\) in 4 minutes)
  • An "Obvious" Conclusion:
    • The longitude of a given location can be determined by measuring differences between times
Some History
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  • 150 BC:
    • Hipparchus proposed using a prime meridian through Rhodes and lunar eclipses (visible "everywhere") to determine time differences, unfortunately clocks were unavailable
  • 1500s-1600s:
    • Clocks began to improve but were not reliable enough to use at sea
  • 1700s:
    • The English Parliament offered a prize of £20 000 to anyone who could determine longitude at sea to within 0.5 degrees (i.e., to build a clock that was accurate at sea to two minutes)
    • John Harrison built such a chronograph (on his fourth attempt) around 1760
Determining the Local Time
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  • Local Noon Defined:
    • The time when the sun is at its zenith (i.e., the highest point in the sky)
  • Measuring Local Noon:
    • In principle, a gnomon can be used to determine when the sun is at its zenith
    • In practice, the arc is flat near the zenith so identifying it is difficult and an average is often used (e.g., of the angles at sunrise and sunset)
  • Local Time:
    • Is then defined relative to local noon
Determining Longitude Using Lunar Eclipses
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  • Data Collection:
    • An observer at the undetermined location measures the local time of the lunar eclipse
    • An observer at the prime meridian measures the local time of the same lunar eclipse
  • Communication:
    • The observer at the prime meridian transmits their observation to the observer at the undetermined location
  • Calculation:
    • The observer at the undetermined location uses the difference in local times calculate their longitude
Determining Longitude Using Lunar Eclipses (cont.)
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  • An Example Eclipse (August 21, 2017):
    • Measured Start at the Prime Meridian: 15:46
    • Measured Start in Harrisonburg, VA: 10:31
  • Calculations:
    • Time difference: 5 hours and 15 minutes (i.e., 5.25 hr)
    • Longitude of Harrisonburg: 5.25 hr \(\cdot\) 15 deg/hr = 78.75 deg (which is pretty close to the absolute value of -78.8689, the actual longitude)
Determining Longitude Using an Accurate Clock
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  • Preparation:
    • The clock is set to 12:00 at local noon at the prime meridian (or some other known longitude)
  • Determining Longitude of an Undetermined Location:
    • At local noon at the undetermined location, noon is subtracted from the time on the clock to get the time difference
    • The longitude is calculated from the time difference
Determining Longitude Using an Accurate Clock (cont.)
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  • Example Data:
    • The HMS Beagle sailed with 22 chronometers to provide redundancy and error correction
    • In June of 1837, the chronometers were set to 12:00 at local noon at the prime meridian
    • On November 15, 1837, the average time on the working chronometers was 19:45
  • Calculations:
    • Time Difference: 7 hours and 45 minutes (i.e., 7.75 hr)
    • Longitude: 7.75 hr \(\cdot\) 15 deg/hr = 116.25 (i.e., near Perth Australia)
There's Always More to Learn
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