A DOUBLE HORIZONTAL SUNDIAL FOR LATITUDE DEGREES NORTH. Design, Construction and Use

Transcription

1 A DOUBLE HORIZONTAL SUNDIAL FOR LATITUDE 0.8 DEGREES NORTH Design, Construction and Use John Hayes 9 Northlands House, Salthill Road Chichester, PO19 3PY July 8, 2004

2 A DESCRIPTION OF THE DIAL The double horizontal dial for Northlands House, Chichester at latitude 0.8 degrees North is based on the 1620 design of English mathematician William Oughtred. In the 17 th century, such dials were hand engraved and expensive. Samuel Pepys had one of the few that were made. Today they can be drawn on a computer CAD program and photo-etched with precision. When etched on brass, bronze or stainless steel they should last for many years, as have the 17 th century dials. My interest in double horizontal dials was inspired by articles written recently in the quarterly journal of the British Sundial Society (BSS) by Michael Lowne, a retired astronomer, and John Davis, who designs, makes and restores dials (see his website ). They and a couple of other sundial enthusiasts have lately revived interest in these early dials. The Northlands dial is one of the first since to be made. Michael and John were both active in helping me draw the dial using TurboCAD software. John photo-etched the drawing on the dial s brass plate and machined its edges. The gnomon was made by Tony Moss, another member of the BSS who has an international reputation for his metalworking and sundial design skills. Tony also advised on setting up the dial. Two styles, two shadows The dial is called double because the gnomon has two shadow-casting styles. The shadow of the sloping style, which forms an angle to the horizon equal to the appropriate latitude, gives the time in the same way as traditional garden sundials. The spacing of the hour lines for such a dial varies with the latitude of where the dial is to be located. As with any horizontal dial the double horizontal dial has to be set up perfectly horizontal, with 12 o clock pointing to true north. Sun or solar time is shown by the Roman numerals. Adjustments have to be made to convert local sun time to standard time as explained below. The vertical style of the double horizontal dial is at the centre of the dial and it has a knife edge or vertex. Its shadow gives the altitude and bearing of the sun. There is a practical limitation on the use of the vertical style. Its shadow is obscured wholly or partially around noon by the shadow of the gnomon, of which it forms a part, and for about an hour either side of noon the shadow is cast not by the vertex, the knife edge, but by one of the adjacent shoulders of the vertical style. Also, because the vertical style is necessarily not very high, one may need visually to extend the shadow to get a reading. The azimuth circle and the double horizontal dial s horizon Inside the ring of Roman numerals are figures used to mark the azimuth of the sun in relation to due south. The bearing at any given time of day varies according to the time of year, and the shadow of the vertical style will give that bearing. At this latitude at 11 am on August 1 the bearing of the sun is 24 degrees east of south. At 11 am on June 21 it is 29 degrees east of south. Inside the azimuth figures is a circle representing the dial s horizon. One must understand that the compass points of the dial s horizon are reversed. Although XII at the top is oriented to true north, the shadow cast there by the sun at noon derives from its southerly position, so that part of the dial represents the southern horizon. Similarly, the eastern horizon, sunrise and the morning hours are on the left; and the western horizon, sunset and afternoon hours are on the right. The map of the sky above the dial Within the horizon circle is what may be called a map of the sky above the dial. The horizon circle and the grid of fine lines within it are based on a stereographic projection, the same method of drawing used for the plate of an astrolabe. The grid is composed of declination and hour angle arcs. It can be used to identify the position of any heavenly body above the dial, but particularly for our purposes the sun. Knowing the sun s declination is necessary to make full use of the information given by the shadow of the vertical style. 1

3 The declination of the sun is its angular distance above or below the celestial equator, the celestial equator being a projection of the plane of the equator into the imaginary celestial sphere. Because the Earth s axis is tilted in relation to its orbit around the sun, the sun appears to move back and forth across the equator during the year, between the Tropic of Capricorn in December (23. degrees south of the equator) and the Tropic of Cancer in June (23. degrees north). The twenty or so transverse arcs above the centre of the dial mark every two degrees of the changing declination of the sun through the year. The arcs close to the centre of the dial show the sun s declination during the summer, when the sun at noon is high in the sky. 2

4 The arcs marked by the inner circle of Arabic numerals show the time given by the shadow of the vertical gnomon, as explained later. Their position relative to the north-south line also indicates the local hour angle, the angular distance from the dial s north-south meridian of the sun, planets or stars. The path of the sun through the year The two ecliptic arcs with dates, drawn across the grid of declination arcs and hour angles, show the path of the sun through the year. The lower arc shows the sun s path through the summer months from the March equinox on the right to the September equinox on the left. The dates enable one to see the declination of the sun for any given day. For example, the declination for August 1 is +14 degrees (north). The ecliptic arcs show also the progression of the sun s Right Ascension through the year, as explained later. HOW TO USE THE DIAL Using the shadow of the sloping style to tell the time The time shown by the dial s Roman numerals and their ten minute divisions is local apparent solar time, the time indicated by the sun. Two corrections are needed to convert to Greenwich Mean Time. The simple correction is for longitude. For every degree of longitude west the sun arrives at its highest point in the sky 4 minutes later than at Greenwich, and 4 minutes earlier for every degree east. The sun arrives at Chichester in West Sussex 3 minutes later than at Greenwich, and at Penzance in Cornwall some 22 minutes later. The other correction is for the equation of time (EOT). The motion of the Earth in its annual orbit around the sun is not uniform, due to its slightly elliptical orbit and the tilt of its axis. Some days are slightly longer than others. EOT is the difference between local apparent time (the time shown by the sun on the dial) and mean solar time, which is an average over the year. The difference is zero on April 1, June 12, September 1 and December 2. In between these dates the difference accumulates. For example, between April 1 and May 13/14 it grows to nearly 4 minutes, and that is deducted from the time shown on the dial to give mean solar time, which is used as a basis for civil time. The difference then reduces daily until it is zero on June 12, and so on. The correction to be made for EOT is highest around November 1 st, when an accumulated 16 minutes has to be deducted from the time shown on the dial, and in February, when 14 minutes has to be added. The graph which accompanies the dial combines the corrections to be made at the location of the dial for longitude and EOT in order to obtain standard time, which in Britain is GMT. During the summer one adds an hour to the corrected dial time to get daylight saving time. The sun is a disc which subtends on the Earth an angle of about one half of a degree. Consequently, the shadow cast by the gnomon is not as sharp as if the sun were a point of light. This, along with any imperfections in the construction or setting up of the dial, and the averaging of the EOT to take account of leap years, limit the accuracy of any dial relying on shadows to within perhaps two minutes. The gap in the Roman numeral XII matches the width of the gnomon. At noon by the sun (not by your watch) the shadow of the gnomon should fill the gap, and the shadow then begins to use the other edge of the gnomon to show the time in the afternoon. Using the stereographic projection As noted above, there is a practical limitation to the use of the shadow because of the gnomon s shape. 3

5 A u g u s t + 20 Decl J u l y A l t i t u d e o f S u n J u n e 12 1 (a) To obtain the bearing of the sun The bearing or azimuth is where the shadow of the vertical style falls on the circle of figures immediately outside the horizon circle. Those figures indicate the azimuth, the bearing east or west of south. (b) To obtain the altitude of the sun The centre of the dial marks the dial s zenith, the point in the sky directly overhead. The altitude of the sun at the horizon is zero and at the zenith it is 90 degrees. It is possible to etch a series of concentric circles around the centre of the dial to show different altitudes, but this would crowd the information shown by the grid of declination and hour angle arcs. The alternative is to use a fixed or revolving alidade. The scale marked from to 80 degrees projecting left from the centre of the dial is a fixed alidade. Dividers are used as in the following example to determine the sun s altitude. Let us assume that the slanting style gives the time by the Roman numerals as.13 am on August 1, and that the straight line in the upper part of the adjacent circular diagram shows the shadow of the vertical style. The date August 1 is on the declination line of + 14 (North). Set one point of the dividers on the centre of the dial and the other on the intersection of the +14 declination line with the shadow, and transfer the measurement to the alidade. The transfer is shown by the arc, and the altitude of the sun on that date and at that time is about 47. degrees. The declination on any day determines the sun s maximum altitude at noon. It is equal to 90 degrees minus the latitude of the dial, plus the declination (minus if the declination is negative). On August 1 the maximum at Latitude 1 is 90 minus 1 plus 14, equals 3 degrees. (c) To tell the time The vertical style provides an alternative, although less precise and convenient, way of telling the time. The intersection of the shadow of the vertical style with the declination line of +14 degrees in the above diagram is just to the right of the hour angle line of. am. So the time is about.13. This time must be corrected for longitude and EOT, in the same way the time indicated by the sloping style is corrected. (d) To tell the date This is simply the reverse of (c) above. If one knows the time one can tell the date. Look first at the time given by the shadow of the sloping style on the Roman numerals. Assume the time is.13 am. Look next at the corresponding hour angle line and see where it intersects the shadow of the vertical style. In the diagram on the previous page the intersection is at declination 14 North. Follow that declination line to the ecliptic curve. The relevant date is August 1. One could follow the declination arc in the other direction, where the date is April 27, but you would generally know which of the two dates is right! It is not possible to tell the precise date at the time of the winter and summer solstices, when the declination of the sun changes slowly, because a given declination will match several dates. The ecliptic curves illustrate this. 4

6 (e) To tell the time and bearing of sunrise and sunset for any day of the year The date on the ecliptic curve gives the declination of the sun for that day. If you follow the declination curve to where it meets the horizon circle you will see the bearing of sunrise or sunset. To get the time of sunrise or sunset you need only identify the time arc that intersects the horizon circle at that bearing. For example, in the next diagram the declination of the sun on April is + 6 (6 North). Follow the declination curve to the right, which is the side of the dial where the afternoon hours are marked, and you get the bearing of the sun at sunset, almost 0 degrees west of south. The declination of + 6 meets the horizon circle just before the time arc for 6.30 pm, so that is the time of sunset. This is local solar time, so it has to be adjusted to give standard or daylight saving time. Similarly, to get the bearing and the time of sunrise for April, follow the + 6 declination curve to the left, to the eastern horizon. The bearing is the same figure of almost 0 degrees but east of south, and the time arc gives the time of sunrise as just after.30 am. By contrast, on January the bearing is 3 degrees and the times are 4 pm and 8 am. At the equinoxes of March 21 and September 23, when the sun s declination is zero (the sun is on the equator), the bearing of sunrise and sunset is 90 degrees east or west of south, and the time is 6 am and 6 pm respectively, day and night being equal in length. (f) To find the Right Ascension of the sun for any day in the year In the same way that latitude and longitude are used to map places on the surface of the globe, declination and Right Ascension (RA) are the two coordinates used to identify the position of the stars, sun and planets in the celestial sphere as viewed from Earth. The declination and Right Ascension of the stars change only imperceptibly over the years, but those of the sun, moon and planets change constantly as they move against the backdrop of the stars. One can read from the dial s stereographic projection both the declination and RA of the sun and therefore its position against the celestial sphere for any day of the year. We have already seen that on August 1 the sun s declination, shown by the lower ecliptic curve, is +14. What is the sun s RA on that date? Right Ascension is measured, usually in hours from 0 to 24, eastward from the vernal equinox, the point where the sun s path as seen from Earth, the ecliptic, crosses the celestial equator. It is the celestial equivalent of the Greenwich meridian. The vernal equinox used to be in the constellation of Aries; it is now in Pisces and tending towards Aquarius. At this point, around March 20/21, the sun s declination is zero (because it is on the equator) and its RA is zero hours. Decl M a y A p r i l Decl On the dial, the March 20/21 equinox is on the horizon circle at the right, at the end of the two ecliptic arcs. To find the sun s RA for any day, first locate the required date on the ecliptic arc. Then count the number of hours towards the left or eastern horizon from the 6 pm time arc of March 21, when the sun s RA is zero, until the time arc for the required date is reached. For example, the sun s RA at noon on August 1 is 9 hours and 40 minutes. M a r c h

7 The sun s RA at the autumnal equinox, at the left hand end of the ecliptic arcs, is 12 hours, so for dates between September 23 and March 21 count the hours towards the right or western horizon from that point and add 12 hours. For example, the RA at noon on January 20 is 20 hours minutes. (g) to see what stars will be overhead at night Some of the 17 th century double horizontal dials give the RA of several bright stars. For example the RA of Sirius is 6 hours 4 minutes and that of Spica is From the dial, the sun s RA at noon on April 13 is 01.2, so at midnight it will be 13.2 (or perhaps add two minutes because of the 4 minute difference between a solar and a sidereal day see below). On April 13 at midnight Spica will be on the dial s north-south meridian. (h) to tell the time at night As a mental exercise and not because it is the most convenient way of telling the time at night! one can use the sun s RA, from the dial, and the RA of bright stars, from an Almanac, to estimate solar time from sidereal time (star time). First a word about the stars and sidereal time. At any given latitude the stars, unlike the sun, rise and set at the same point on the horizon every night of the year. Some stars with high declinations (i.e. remote from the celestial equator) never set, they are circumpolar, and these include at latitude 1 north the stars in the Plough and others such as Capella. Vega, declination 38 46, like other stars whose declination is almost exactly 90 degrees minus our latitude, skims the horizon. But the stars that set on the western horizon do so about 4 minutes earlier each night, so the constellations seem to progress westwards through the sky during the year. After a year they are more or less back where they started. This difference of about 4 minutes is the difference between a solar day and a sidereal day. Not surprisingly, sidereal or star time is related to Right Ascension, the longitude coordinate of the stars. The sidereal time at Northlands, our local sidereal time (LST), is the same as the RA of any star on Northlands meridian. Our LST at noon is also the sun s RA on the day in question. It is by comparing these two ways of determining LST and by relating them to solar time that we can estimate the time at night. For example, on April 30 the sun s RA from the dial is 0230 at noon, which equals our LST. At midnight on April 30 it will be 12 hours later, 1430 (but to be precise add two minutes, half of the daily difference, so let s make it 1432). Assume that on the night of April 30 we observe that Spica is 30 degrees of arc past (west of) our meridian. Every 1 degrees of arc equals one hour so it is two hours past the meridian. We know that Spica s RA is 132 and that when it was on our meridian that was our LST. Our LST must now be 132 plus 2 hours, equals 12. We can convert this LST, the LST at the moment we observe Spica, to standard time because we know from what the dial tells us about the position of the sun on April 30 that our LST at midnight standard time was Clock time now is 12 minus 1432, which is 003. This is evidently a laborious and imprecise way of telling the time at night, but to understand it is to learn something about the stars and sidereal time. THE STEROGRAPHIC PROJECTION AT DIFFERENT LATITUDES At the end of this paper the projection is shown for four different latitudes including the equator, but at the equator and at low latitudes a double horizontal dial would not be the best kind of dial to use. At the equator the sun is in the northern sky between March 21 and September 23 and to the south between September 23 and March 21. The stereographic projection for the equator shows the bearing of the sun at sunrise and sunset it is where the declination line for the date in question meets the horizon circle. On the equator it is at degrees north (of east or west) on June 21 and south on December 21. The projection shows also that the horizon circle is coterminous with the 6 o'clock time 6

8 line, which illustrates that at the equator the sun rises and sets at the same time through the year, at 6 am and 6 pm. Although the stereographic projection shows useful information, the styles of a double horizontal dial at the equator would not be very useful. The sloping polar style, used by traditional garden dials, forms an angle with the horizontal dial plate equal to the latitude. At the equator the latitude is zero, so that style would be horizontal and cast no shadow, unless it were raised above the dial plate to become what is called a polar dial, but that is a different kind of dial. A vertical style would cast a shadow, but it would not serve very well. Using the shadow to tell the time and the altitude of the sun depends on being able to get a good reading of the shadow s intersection with the declination arc on the date in question. At the equator the shadow would most often form an acute angle with the declination lines, making a reading difficult. At latitudes near the equator the problems would be similar. One practical difficulty at low latitudes is that the low angle of the polar style constrains the height of the vertical style. The double horizontal dial happens to be well suited to the mid-latitudes, such as the one for which this dial is made. 7