UDC

The Universal Dialist’s Companion

A really useful device

© Mike Shaw

Incorporating many additions, suggestions and graphics from

Tony Moss of

Sundial Equations – there’s a lot of them sin(A) = sin(L) sin(D) + cos(L) cos(D) cos(HA) cot(Z) = {cos(D)sin(HA)}/cos(A)

cos(HA) = - tan(L) tan(D)

But what do they all mean? Can you visualise what they represent?

I couldn’t, but then I found this book by Peter Drinkwater

…and inside was this diagram that explained it all

With a bit of refinement, and a few additions, it can help you find out all sorts of things for any Northern Latitude. But first, let’s recap on what we know …

We know that the earth goes round the sun like this …

… and that it is tilted at about 23.5 degrees from the vertical

We also know that the earth rotates once on its axis each day, with every 15 degrees of longitude representing one hour

So from the top it looks like this … … or this could be a slice representing the equator

Usually the earth is shown tilted at 23.5 degrees like this

Which is OK if you live at the Arctic Circle

But I think that I live at the top, like this

Actually, everyone thinks that they live at the top, that’s what makes designing dials difficult.

So what do things look like from where I live? How does the sun appear to move across the sky? How does the time and direction of sunrise and sunset change with the seasons? What are the dial limits at my location?

We need a Dialist’s Companion

This is how you construct one

Everyone carries a sphere around with them It’s the sky above them, and the earth beneath their feet

On a flat surface, we represent it as a circle.

This is the horizon, you can’t see anything below this line You are looking due east into the screen, with north to your left, south to your right and west behind you.

If you point at the pole star with your left hand, you are pointing along the axis around which the earth rotates

53’22”

In Bebington, where I live, that is 53 degrees, 22 minutes North

This vertical line is the end of a wall that runs due east : west.

So this is a direct south facing wall and this is a direct north facing wall

The line perpendicular to the earth’s axis is in the same plane as the equator

23.5 23.5

Lines drawn 23.5 degrees from the plane of equator mark the northern and southern limits of the sun’s travel, the tropics of Cancer and Capricorn.

If we extend these lines parallel to the equator, and draw a circle within the limits, this then represents the earth’s annual path around the sun We can use this to find the sun’s position throughout the year

We do this by dividing the circle into 12 equal parts, starting at the equator (the equinox)

These now represent the dates covered by the 12 signs of the zodiac

We extend these divisions parallel to the equator, they then represent the sun’s path on the surface of our sphere on the respective date.

Next the equator is rabatted and laid flat at the base of the axis of rotation

We then divide it to show the hour lines, and bearings (in degrees) from North

And use it to determine where the hour lines intercept our sphere

So that we can draw “hour lines in the sky”

Now we’re ready to use our Dialist’s Companion

Remember that you can’t see anything below the horizon

This is the sun’s path on the longest day

At the equinox it’s here

On the shortest day

Longest day

But, of course, it’s really in 3 dimensions, like this

Equinox

Shortest day

On the longest day, the sun rises at 03:20 and sets at 20:40

All times are local apparent time

These are the overall dial limits for this latitude

On the shortest day the sun rises at 08:20 and sets at 15:40

As this is the pivot point, this happens at every latitude

At the equinox, the sun rises at 6am and sets at 6pm.

Limits for a direct north dial are 03:20 to 07:20 then 16:40 to 20:40

What about other Northern latitudes?

What you need is a Universal Dialist’s Companion,

one that can be easily adjusted for any latitude.

It looks like this

The rebatted equator is discarded, and the new circle at the bottom is the rebatted local horizon.

The bottom circle has the azimuth marked in degrees from due south. The circles show altitude; effectively we are looking down on the top of our sphere

Let’s have a look at the top part.

There is a pivot in the centre, so that any latitude can be selected. The latitude scale has moved to the bottom to get it out of the way, here set at 51 degrees

Note that a scale of the sun’s declination has been added.

And the date scale now shows calendar months rather than zodiac signs

The previous latitude scale is now used to find the sun’s altitude … … and there are crepuscular lines showing civil, nautical and astronomical twilight

We are in the South of England at 51º Latitude

It is 9 am on 1st August

So what can we find out?

The sun’s altitude is 42º

9am on 1st August

The sun’s declination is +18º

Sunrise was at 04:20 and sunset will be at 19:40

Civil twilight is at 20:20

Nautical twilight is at 21:15 Astronomical twilight is at 22:45

Note that the sun does not reach astronomical twilight at this latitude near the summer solstice

We use the lower circle to find azimuths

Sunrise is at 121 degrees east of south

Sunset is at 121 degrees west of south

The dial limits for a horizontal dial at this latitude are 03:55 to 20:05

Dial limits for a direct North dial are

03:55 to 07:15 and 16:45 to 20:05

The lower circle can be used to find the dial limits for a declining wall.

A wall declines 14º east of south Draw in the wall

Project up to the upper circle

Dial limits for 51º latitude are 05:15 to 17:10

The lower circle can be used to plot the sun’s path on our sphere, here plotted for the longest day

Drop a vertical wherever the sun crosses an altitude line and mark the position

This can be used to find the azimuth at any time on that day. Drop a vertical down to the sun’s path Here, at 9am the azimuth is 73º east

Then draw a line from the centre through to the azimuth scale.

But the great thing about the Universal Dialist’s Companion is that we can go anywhere we like – lets start at the North Pole, and see what we can discover

At the north pole, the sun is above the horizon from the spring to the autumn equinox.

Set the pointer to 90 degrees

It maintains the same altitude all day. The sun’s altitude is equal to the declination

At the Arctic circle

The sun never sets on the longest day

At the Arctic circle And the sun never rises on the shortest day

At the northern tropic

The sun is directly overhead only on the longest day The sun shines all day on a north facing wall at the summer solstice.

At the equator

The sun shines on a north facing wall for 6 months, and a south facing wall for the other six months

The sun rises at 6am and sets at 6pm every day There is very little twilight

And you can look at any northern latitude So, with this simple device, you can, for any chosen date:

Find the time and azimuth of sunrise and sunset.

Find the time of Civil, Nautical and Astronomical twilight. Find the sun’s altitude and azimuth at any time. Find the sun’s declination.

Find the dial limits for horizontal, declining and direct north dials.

That’s really all there is to it … and not an equation in sight.

Thanks for your attention