A Quest for a Civil Time Polar Sundial

 

(Assembled 5 Jan 2003 by Mac Oglesby)

Introduction

Courtesy of Dave Bell, this site is a temporary home for certain materials which relate to the quest for a civil time polar sundial, as discussed during recent weeks on the Sundial Mailing List and in private email messages. These materials are gathered here to make it easier for interested sundial list members to access the photos and drawings whose size exceeds that allowed for attachments. I apologize in advance if I have misrepresented or misunderstood anyone, and I take full responsibility any typos and other errors which occur as a result of my editing.

The Problem

Dialist John Close asked for help in designing a civil time polar sundial that didnít use analemmas, half analemmas, or unfolded analemmas. He wondered if a gnomon could be constructed which would satisfy his requirements.

The dialists he contacted gave differing advice as to the possibility of such a gnomon: some said yes, some said no, and some said maybe. Thatís about where we are still, but the journey has been interesting.

The Journey

To clarify our thinking, letís consider what a polar sundial is, and perhaps mention what the sundial user must do in order to obtain civil time.

A very strict definition of a polar might be that it has a flat dial face which lies parallel to the Earthís axis and perpendicular to the plane of the meridian, has straight hour lines, and has either a pointed post gnomon, or a straight edge gnomon parallel to the Earthís axis. (The edge gnomon may be replaced by a taut cable.)

[NOTE:Commenting on my draft, Fer de Vries writes, ďI call any dial parallel to the earthís axis a polar dial. Also the east and west facing dial, and many more, are polar. But this is arbitrary of course.Ē I agree with Fer, knowing that others will disagree.]

This strict polar dial would commonly be delineated to show solar time, but never as many as 12 hours in a day. Though it may easily be designed to show zonal solar time, to get civil time one must consult a graph or table of EoT values, which are then applied to the dialís reading.

As we alter the strict definition, our polar dial becomes a modified polar. One useful modification would be to allow the dial plate to be rotated around an axis parallel to the Earthís, such as the edge of the gnomon, or an edge of the dial face. [Fer writes, ďIn my definitions it stays a polar dial if you rotate the plate. The equivalent horizontal dial is always at latitude 0 degrees. The pole style always is parallel to the dial.Ē] If the dial is turned 15 degrees, the time on the dial will be 1 hour earlier or later, depending upon which way it is rotated. To change the dial time by 4 minutes, turn it 1 degree, etc. Thus we can easily make allowance for summer time, and/or longitudinal offset, and/or EoT, although, since the EoT value is constantly changing, one would need to reset the dial now and then.

H. Robert Mills, in his book ďPractical Astronomy,Ē details such an arrangement on pages 106-109, where he suggests using a wedge to rotate the dial plate. Also, his polar dial has each end of the dial plate folded up 90 degrees, which shortens the length of the dial and makes it usable for a full 12 hours.

Graphic 1 shows a modified polar dial which, once properly installed, will give either solar time or zonal solar time without further adjustment. The dial may be periodically rotated on its mounting post to show civil time directly. Having hour lines only, the dial doesnít allow for highly precise time readings, so frequent adjustments arenít necessary. This dial, entitled Sun Bather, uses 2 gnomons, color coded to indicate which set of hour lines to look at. If the green gnomon is casting the shadow, look at green hour lines, etc. The time shown is just before 8 AM.


Graphic 1

A different approach is to use hour lines which have the EoT correction factored in. If the hour ďlinesĒ are full analemmas, the dial may be confusing to read, especially if the curves overlap. If half-analemmas (from solstice to solstice, for example) are used, then one needs 2 dial plates and has to swap them twice a year. Another choice is to use ďunfoldedĒ analemmas in conjunction with date lines. Christopher St J.H. Danielís polar sundial at Otley indicates civil time as well as solar time (see Graphic 2). This dial plate is not flat, which allows a wider range of hours than a traditional polar. (Iím not certain about the source of this photo, but I believe it came from John Davis.)


Graphic 2

A very different approach is the cycloid dial of Thijs J. de Vries, designed about 1980 (see Graphic 3). An article by Fred Sawyer on this dial appeared in the December 1998 Compendium. On this dial an adjustment for civil time may be made not only by rotating the entire dial, but, since the hour lines are equally spaced, they may be shifted east or west, relative to the gnomon(s).


Graphic 3

The Journey Continues

Letís return to the problem of designing a shaped gnomon which could be used to solve John Closeís puzzle. Bill Gottesman circulated a preliminary sketch of a possibility (see Graphic 4). The axis of the 3D gnomon is parallel to the Earthís, and civil time is read along a line of hour points perpendicular to the gnomonís axis. We wait for some one to show that Billís design will work, or to prove that it cannot.


Graphic 4

Quite recently Thibaud Taudin-Chabot and Fer J. de Vries sent me some very fascinating material from 20 year-old issues of the Journal of The Dutch Sundial Society. Graphic 5 shows a model by Willem Bits which uses a shaped gnomon (in this case a curved wire in space), has equidistant and straight hour lines, and straight date lines. Solar time is read where the gnomonís shadow intersects the current date. The hour lines may be slid east or west for zonal solar time, or shifted every day or so to show civil time. (Of course, this dial can also be rotated around a polar axis.) I mistakenly thought that this dial could give civil time directly, but such is not the case. Perhaps someone can figure out how to modify the gnomon so this dial will display civil time, or else prove that it cannot be done.


Graphic 5

The final group of exhibits in this collection from The Dutch Sundial Society consists of 5 pages and a diagram. These items are labeled as Graphic 6 through Graphic 11, just for identification purposes. I donít read Dutch, so I asked Fer de Vries for comments. Gracious as always, Fer provided the following text, in two parts.


Graphic 6 Graphic 7 Graphic 8
Graphic 9 Graphic 10 Graphic 11

Part 1 of Text by Fer J. de Vries

The dial you mentioned [see Graphic 5] was an idea of Willem Bits.

He started after he had seen the polar dial of Thijs de Vries, the dial recently spoken about on the list.

A polar dial is the same as an horizontal dial at latitude 0 and then itís rather easy to visualize the process.

The first goal of Bits was to make equidistant hour lines.

Place a vertical gnomon with length g1 on the north-south line.At one oíclock this points to the hour line 1 at distance x1 from the n-s line.

Place a second gnomon on the north-south line with length g2 but shorter than g1.

This second gnomon has to point to hour line 2 at a distance x2 from the n-s line that is twice the distance of the hour line 1.

Do this for all the hours and the hour lines are equidistant and we have a number of gnomons on the n-s- line.†† (Shall we call them hour gnomons?) The distance between all these gnomons is arbitrary.

You may choose any space between the gnomons but keep them on the n-s- line.

If you do this process for all the times and not only for the hours you get a plate with a curved edge on the n-s- line.

But you canít read the time. What gnomon has to be used?

So incorporate the date and also draw date lines.

For each hour line and its related gnomon the points for the dates on that hour line may be calculated.

Connect all the points for a certain date with a curve and you have the date lines also.

Now the dial may be read at the intersection of the date line and the shadow of the edge of the plate on the n-s- line.

See [Graphic 6] for some possible solutions.

Many shapes are possible, depending where you place the gnomons.

Have in mind that the pattern isnít in the east-west direction but in an arbitrary direction.

The hour lines are equidistant.

If you place the hour-gnomons also equidistant then the date line for 0 degrees is a straight line, else this date line is curved.If the distance between the gnomons is the same as the distance between the hour lines the angle of the equinox line is 45 degrees.

This was published in our bulletin in March 1981.

Part 2 of Text by Fer J. de Vries

In the previous part you saw a polar dial with equidistant hour lines and the equinox line as a straight line, but all other date lines as curves.The gnomons for each hour are on a n-s line and all equidistant apart.

Otherwise the equinox isnít straight.

Assume:

n = the hour (6 -18)

q = equidistant distance between hour lines

gn = height of gnomon (g0, g1 .... g6)

We now have:

gn = (12- n) * q / tan (n*15)

For hour 6 and 18 this gives g6 = g18 = 0

For hour = 12 (noon) the equation fails.

Take n = 0.0001. Then g0 is about 3.82 q

The distance between a point on the straight equinox line and a point on another date line is

delta = tan (decl) * (n-12) * q / sin (n*15)

Because of the equidistant hour lines any correction for longitude and EoT can be made by shifting the hour lines or the combination of gnomons and date lines.

But it canít have a built in EoT correction.

----------------------------------

Second problem to solve.

Now we try to make all the date lines straight.

For a normal polar dial the distance of a date point from the equinox line is:

y = g tan (decl) / cos (n*15)

We want to have y as a constant value to get parallel lines to the equinox line and name that value k.

We then have:

g tan (decl) / cos (n*15) = k

and g = k * cos (n*15) / tan (decl)

We choose for k :

k = k1 * tan (decl)

which is allowed because for a certain date decl is constant and also tan (decl) is constant.

and then gn = k1 * cos (n*15) in which k1 = k / tan (decl) is a constant.

Again place a number of such gnomons on the n-s- line.

Doing so we have parallel date lines, but no longer equidistant hour lines.Only the gnomons g0, g6 and g18 are correct and that hour lines stay where they were. The gnomon g6 and g18 have a height = 0.See also the attached example.[see Graphic 7, fig. IV] But also Graphic 12 is such an example. This last one comes from an article by P. Oyen in ďZonnetijdingenĒ, nr. 5, 1997, the bulletin of our Belgian friends.


Graphic 12

The other hour lines are shifted and we want to shift them back as may be seen in fig. VII on page 702[see Graphic 8].This shift is in x and y direction.

Well, do this including the appropriate gnomon for that hour and all the hour lines become equidistant again.

And the final gnomon wire gets its nice curve.

But still the EoT isnít built in and I donít see a possibility for that.

At page 704 [Graphic 10] and 705 [Graphic 11] Bits gave the formulas.

The gnomon g12 is H

For any other gnomon gn = H cos (N*15)

Distance of the date line

AB = .... = H tan (decl)

That is a parallel line to the equinox line.

The shadow length of a gnomon for decl = 0 is:

SL = .... = H sin (N*15)

and the distance AN = H sin (N*15)*tan(alpha)

This alpha may be chosen at will.

At page 705 [Graphic 11] the shift is described.

The not linear hour line _._._. has to be shifted to ........

The final formulas you need for the wire are:

(a) for the x shift

(b) for the y shift

Hn for the height of the hour gnomon

An for the place of that hour gnomon

With g12 the hour lines can be calculated.

And alpha is your own choice.

[end of text by Fer de Vries]

Conclusion

Although we havenít (yet) found a gnomon shape which gives civil time directly on a polar dial, speaking personally, Iíve learned several new things about polar sundials. And there always seem to be new lessons around every bend in the road.

If any reader has new thoughts on this problem of civil time on a polar dial, please post to the sundial mailing list, or write to me at†† <[email protected]>.

After the Conclusion - the Quest Continues!

On 10 January 03, Fer J. de Vries sent some additional information to the Sundial Mailing List. He wrote:

As a contribution to the discussion about the polar dial I made a drawing of the construction of such a dial.

In this picture [see Graphic 13] I try to explain what I did.

At the left from top to bottom you find a series of gnomons for each hour starting at noon with the appropriate hour line with date points for the solstices and equinoxes.

The lengths of the gnomons are as Bits formula (Graphic 11, bottom) gn = g * cos t

Now all the hour lines with date points are equal, only the place and length of the gnomons is different.

In the middle top figure these gnomons and hour lines are placed on a north-south line, using the formula an = g * sin t * tan alpha, with alpha = 45. The distances between the foot points of the gnomons aren't equal.

In that case we get the pattern as by Oyen. The shape of the line through all the endpoints of the hour gnomons is a semi-circle.

The second half of the pattern is added in the figure middle bottom.

The difference with the figure by Bits is that he had all the gnomons equidistant as may be seen at the left in Graphic 8.

Now shift the gnomons and hour lines with date points according the formula by Bits for delta-x and delta-y and the result is in the figure right bottom.


Graphic 13

The foot points of the hour gnomons are drawn in red. The foot points for 6 and 18 are arbitrary because the gnomon length is 0.

Now it's easy to draw a side view of the wire. We know the distance of the foot points from the center point and the lengths of the hour gnomons. The figure at top right shows roughly the result in blue.

Reading the time before about 7 am and after 5 pm is difficult because the length of the hour gnomon is small.

If I didn't make a mistake I conclude that the principle by Bits is all right but his drawings are somewhat misleading.

[end of text by Fer de Vries]