Measuring Longitude

Measuring Longitude
An Introduction

Prof. David Bernstein
James Madison University

Computer Science Department

bernstdh@jmu.edu

Review

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

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

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

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

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.)

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

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.)

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)