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Finding North for Sundial Construction

Slawek K. Grzechnik

In spite of the title the methods presented here are general and not restricted to the needs of sundials only. The accuracy required for home-made sundials is not that high and half of a degree would be sufficient. But some of the methods presented may yield accuracy of about one tenth of a degree or better provided that the accurate equipment is used during the observations.

• Using compass
• Using Sun for few hours
• Using Sun for a moment
• Using the Polar Star
• Litearure and Links

Using Compass

This is the easiest but we have to realize that the magnetic compass shows magnetic North rather than the true North which we need. The true North of the place is the direction towards the North geographical pole and not the local direction of the magnetic field shown by the compass. The difference between the two was noticed a few centuries ago and for a time authorities and science were quiet about it not knowing what to say. The difference between true and magnetic direction is called the magnetic variation and must not be neglected when using a compass. The variation changes from place to place so it is a function of position. It also changes with time due to continuous travel of Earth's magnetic poles. Geographical poles also wander (nothing is stable in our world) but these changes are very small. The relation between true direction T and magnetic direction M may be noted down

    T   =    M    +    var

The sign of var has to be checked. Because we traditionally measure directions clockwise from the North then the Eastern variation should have the sign (+) and western should have the sign (-). The information about variation of the place may be found on most (good) maps and on all charts. The value for the date of the chart publication is printed and the yearly change of the absolute value is given enabling us to find the variation for the current year. Many charts have also the compass rose, the figure showing true and magnetic roses so that direct magnetic readings can be taken off the chart. The above equation is very simple so it may seem that devoting to it so many lines of text may be wasteful but I have seen many errors committed in navigation because of wrong application of this simple formula. So if the variation is say 14 E then the compass reading of 0 shows really 014 true.

Using magnetic compass properly we may find direction of true North with the accuracy of about 0.5 of a degree which is sufficient for a backyard sundial. Such accuracy corresponds to about 2^m of time. Sometimes reading of the compass, especially in the cities, may be seriously disturbed by metal construction and underground metal pipes. Unfortunately we do not know about it until we use astronomical methods for determining directions.

Using Sun for Few Hours

Most manuals and papers on sundials recommend the straightforward method of finding the true North. Place a vertical post in the backyard at point some point P. At some time before noon mark the end of the shade on the ground. This is the point A. Using a cord draw an arc centered at the post (point P) with the radius PA. Then patiently wait till in the afternoon the shade reaches the drawn arc again and mark it as point B. The bisection of the angle APB gives the direction of the North (the error because of slight change of Sun's declination during these few hours may be neglected). Of course the post should be really vertical.

This method has some disadvantages:

• takes a lot of time (few hours at least)
• relies on fair weather during these few hours
• determination of shade end points is difficult because the taller the post the more fuzzy its shade because of diffraction of Sun's rays on sharp edges
• most people are at work during the  hours around noon
• the error because of the change of Sun's declination during the observation period may be neglected

To apply the method you need patience and time.

Using the Sun just for a Moment

This method is the best. We just take one shot of the Sun and calculate the rest. In order to do this properly follow the steps:

• Place a vertical post in your garden in good place. You can use some otherwise useful construction for it. I used team-ball post I set for my kids making sure it was perfectly vertical
• Place horizontally a flat piece of plywood or some other hard surface material. Level it very carefully, this is your horizon plane for marking the shadow
• At some moment when the sky is clear mark the shade direction on the plywood and note the time with accuracy of 1 second. This is too big accuracy for sundials but the cost is low :-). Note that marking of the shadow direction is a bit easier than marking the shade of the end point of the post
• Convert observation time to GMT
• Using the Nautical Almanach and calculator find the azimuth of the Sun for the time of observation
• Basing on the calculated azimuth and using good quality protractor lay the direction of true North in your garden
• Put permanent markers for the N-S line, may be useful for other applications

Finding the azimuth of the Sun for given time and geographical position may be a challenge for the beginners. However these days you may find the calculators on the web. If you don't you may purchase the the Nautical Almanac for the given year. Why the Nautical Almanac? For many reasons:

• It is extremely easy to use. It was designed for navigators and they need to find data instantaneously when at sea. Of  course you may take data from Astronomer's Almanacs as well however they are more complex and take more time while using them.
• The Nautical Almanac gives positions of the Sun, 5 planets, Aries point and 57 stars for every hour of the year. Interpolation needed to obtain positions for any second of the year is linear and easy to use interpolation tables at the end of the Almanac. Most Almanacs for astronomers give positions at daily intervals and calculations for any second of the year require parabolic interpolation
• Accuracy. Accuracy of the Almanac enables calculations with accuracy of 0.1' (one tenth of minute of arc)
• Other data. There is lots of other interesting data in Nautical Almanacs, like Polar Star tables, Greenwich Hour Angle of Aries, data for positions of few hundred stars (not that accurate as the principal 57 navigation stars), Equation of Time for every day of the year, Meridian Passage of the Sun, the Moon and the Aries Point for every day of the year, tables for sunrise, sunset, moonrise, moonset, eclipses and many other.
• Availability and the price. You can buy it in most ship chandler stores and yacht marinas or order it directly. The price is about $15 and you have the data for the whole year

Using the Almanac requires some investment at the beginning but it really pays off. The explanations in the Almanac are very clear, leave no doubt about the procedure. Even simplest operations are illustrated with examples.

Now calculations. The equation used to find the azimuth is

ctg( A ) = cos( lat )tg( dec ) / sin( LHA ) - sin( lat )tg( LHA )              (**)

Where

• A   - azimuth
• lat - latitude of the place
• dec - declination of the Sun at the time of observation
• LHA - Local Hour Angle of the Sun at the time of observation

It has to be solved carefully and with full understanding of sign rules. The detailed discussion how to do it you may find in the Calculations, equation (**) in the paper on Sundials. At first it may be a bit defying, especially with the naming convention for directions but common sense and a bit of imagination help. In order to solve this equation we may use anything, calculator, logarithmic tables or special tables in the Almanac. At least you may say that you know a bit if you are able to solve correctly  this equation basing on the data extracted from the Almanac.

The last remark. Actually we could use any celestial body whose azimuth we are able to determine but it is a bit cumbersome especially for objects at high altitudes above the horizon. The Sun, nevertheless its altitude, casts shadows on the ground and it makes the whole process much easier.

Using the Polar Star

Unfortunately using the Polar Star could be hardly recommended in the Southern Hemisphere. But in the Northern it is so useful that special tables for determining the exact azimuth of the star are published in the Nautical Almanac. These tables are necessary because the Star is not exactly above the pole. Using tables is easy and is described in the Almanac. This method may be useful because most of working people are at home during night hours.

The higher the latitude of the place the bigger the altitude of the Polar Star and marking its azimuth on the ground may be harder. Some inventiveness may be necessary to do it correctly.

Literature and Links

• Any manual on Celestial Navigation
• H. Robert Mills, "Practical Astronomy", Albion Publishing, Chichester U.K. - excellent book on many practical issues of backyard astronomy
• My paper on Sundials
• The source of the Nautical Almanac
• back to home page

Jul 29, 1996