Throughout the centuries, travellers in the northern hemisphere have used the altitude of the Pole Star, along with certain corrections, as a reliable guide to their latitude. That the same star might also help to provide an accurate longitude may seem an odd claim, but it is a true one. With the aid of a "Polarum", a home-made device related to the ancient nocturnal or star clock, your students can test out this unusual approach to navigation. Furthermore, the possession of a current nautical almanac is unnecessary, just an accurate time check and a few simple calculations.
The nocturnal Prior to the invention of the chronometer, navigators in the northern hemisphere used the apparent rotation of certain stars around the Pole Star to tell the time at night. The stars generally used were Kochab in the Little Bear or Dubhe in the Great Bear (figure 1). In the northern night sky, these circumpolar stars appear to rotate in an anti-clockwise direction about 361x every 24 hours.
The time was obtained from the nocturnal by sighting the Pole Star through the centre of the device and lining up the selected star on the edge of an arm that rotated over a time scale.
The star's additional daily rotation of almost 1x resulted in a "clock" that gained a little each day. For example, early in May, Kochab appears directly above the Pole Star at midnight local time. Early in November, however, it is located directly below the Pole Star.
This problem was overcome by the addition of a second dial which adjusted the time scale according to the date of the observation.
Around 1770, with the advent of John Harrison's chronometer, nocturnals were discarded and few have survived. Had the early navigators redesigned and used them in conjunction with the chronometer, they might have added a useful aid to their repertoire of methods for measuring longitude, but this simple method, outlined below, has been overlooked throughout the intervening centuries.
Polarum When a star lies directly above the Pole Star, it is almost directly above the observer's longitude. The difference arises because the Pole Star does not coincide exactly with the celestial pole. As the Earth turns, the star appears to move in a circular anti-clockwise direction. Polarum measures the angle through which the star has rotated since it appeared directly above the Pole Star and therefore almost directly over the observer's longitude (figure 2).
This angle is simply the difference between the observer's longitude and the longitude now under the star. With some fairly simple calculations, it is possible to work out the star's new longitude and use the angle measured by Polarum to calculate our own longitude.
Making a simple version of Polarum An inexpensive Helix 360x angle measure with its rotating pointer makes an ideal basis for a simple Polarum. Drill a small hole through the centre of the protractor. Close one eye and hold the device up in the night sky so that Polaris, the Pole Star, is visible through the hole. Rotate the pointer until it coincides with a nearby star such as Kochab in the Little Bear. Using the 0-360x scale that is printed on the angle measure in an anti-clockwise direction, you can obtain an indication of the star's local hour angle, that is the number of degrees it is West of your longitude. By comparing the angle on Polarum to the star's calculated "longitude", you may obtain your longitude.
The obtained angle is a crude measure because not only is the protractor's scale rather small, but it must be held at right angles to the direction of the Pole Star, and vertical. These requirements are not easily achieved, particularly when the Pole Star is at a high altitude (as is the case in northern Europe). With a little ingenuity, each of these problems may be overcome. The smaller angle measure can be mounted on a larger 360x protractor with 0.5x graduations. Mount both on a wooden support and extend the small rotating arm to the outer scale.
A short piece of 15mm plastic piping from a DIY store fits neatly over the front of the rotating pointer and provides a sighting aid for holding the plane of the instrument at right angles to the direction of the Pole Star.
The scale is kept vertical by means of an illuminated spirit level. Wrap a spirit level bubble unit in black tape but leave a small gap where one side of the bubble appears when the unit is level. Attach and conceal a red LED light to that end of the unit and a sharp red light will reflect from the bubble and blink through the gap when the unit is level (figure 3).
Attach the illuminated spirit level to the device so that the gap is just visible through the centre hole without obstructing the view of the Pole Star. Adjust the spirit level setting with the help of a plumb line so that the red light is visible when the line coincides with 0x on the scale (at different levels of elevation).
The calculations Here is the procedure to find one's longitude using the star Kochab. It is followed by a worked example.
1. Measure the star's angle on Polarum by lining up the Pole Star through the centre hole (with the red light visible) and record the Greenwich Mean Time of the observation. Deduct 0.25x from the observed angle to allow for Pole Star error.
2. Calculate the days, minutes, hours, and seconds that have elapsed between the date and time below and the observation.
0958 22 April 1997 The positions of stars (a constant) are related to the First Point of Aries. The date above is when Aries was in transit with the Greenwich Meridian. Any Greenwich transit of Aries will do. The one above is as good as any and can be used for years.
3. Convert the period that has elapsed between the date above and the time of the observation into days and decimal parts of a day. Multiply this figure by 1. 0027378 (a constant).
4. Ignore any whole figures in the result and multiply the remainder by 360x. You now have the "longitude" or Greenwich Hour Angle of Aries at the time of the observation.
5. Add 137.35x and the result is the "longitude" or Greenwich Hour Angle of Kochab at the time of the observation.
6. The difference between the observed and calculated angles is equal to your longitude East or West of Greenwich.
A worked example 1. Date and time of observation: 20h 31m 00s on 28.4. 97 Polarum Angle of Kochab: 292.5x - 0.25x = 292.25x 2. Time elapsed since 0958 on 22. 4.97 = 6d l0h 33m or 6.44 days 3. 6.44 x 1. 0027378 = 6. 4572 4. Ignore 6 and and multiply remainder by 360x = 164.6x 5. Add 137.35x = 301.95x . This is the "longitude" or GHA of Kochab at the time of the observation.
6. The observed Polarum angle of Kochab was 292.25x which is a difference of 9.7x.
This is the observer's longitude but is it East or West? The answer lies in a simple rhyme: "Greenwich Best Longitude West, Greenwich Least Longitude East."
In the example, the Greenwich Hour Angle of the star was the larger (best) so the observer is in longitude 9.2x West.
The same procedure may be used with Dubhe in the Great Bear but the observed angle must be increased by 1x and the addition at stage 5 is 194.15x. The accuracy of the scale reading could be improved by inclusion of a vernier scale on the rotating arm.
With certain alterations, Polarum may also be used to obtain one's latitude by measuring the altitude of the Pole Star. Although its description in this article is limited to use in the northern hemisphere, it has applications in the southern hemisphere as well. The device has the potential to form the basis of an interesting cross-curricular project involving science (astronomy), maths (calculations), history (developments in timepieces, and navigation), geography (latitude and longitude), and even CDT in technology through the development of a more permanent (and more accurate) device.
For further information, please write to Tony Crowley co The Extras Editor at The TES, Admiral House, 66 East Smithfield, London E1 9EY Tony Crowley is a former Merchant Navy officer and lecturer in vocational guidance at the University of Hertfordshire