There has been a lot of discussion about setting sundials by establishing
the north south axis at solar noon. See the previous notes on
www.solar-noon.com . Personally I have been frustrated by trying to use
solar noon for laying out and setting sundials. The first problem is
weather. Whenever I am on site at around solar noon, a big cloud rolls in
and the sun disappears until well after noon. The second problem is
altitude. Even if the sun is out, the sun is so high these days as we
approach the solstice that the shadow is very short and it is difficult to
get an accurate fix on the shadow of a plumb line. A lens as a shadow
sharpener helps some but the vertical distance is still so much greater than
horizontal that the accuracy is poor. It may work for small dials but not
for the layout of a typical analemmatic dial. I have two modest proposals to
work around this problem.
1. Set the east west axis: For the good half of the year, spring and summer,
when the solar declination is positive, the sun is due east in the early
morning and west in the late afternoon, at times when you can get good long
shadows for setting the east west axis. The complicated looking trig
functions simplify to a relatively simple formula solvable with a pocket
scientific calculator. If you found solar noon, which you need to do
anyways, you can solve for the time when the sun's azimuth is due east and
west.
Cos (t) = Tan (Dec) / Tan (Lat)
You needed to know declination and latitude so just plug these angles in
this formula and solve for t. This is time from solar noon as an angle. This
you will have to divide by 15 to convert to hours and then convert the
decimal part to minutes and seconds in the usual way. For example, here is
the calculation that I did recently for Elliston Park in Calgary where I am
laying out an analemmatic dial. The site is at Latitude 51.03632 and
Longitude 113.9438. Local solar noon on 5 June was at 13:34:20 MDST when you
make the usual longitude and EQT corrections. The solar declination was
22.61739. All this information was calculated from the date and site data
using the NASS Dialists Companion. Plugging the dec and lat into the above
formula gave me a time angle t of 70.31 degrees which converts to 4:41:14.
Adding this to local noon gives me 18:15:34. At a quarter after six that
evening the sun was due west. Subtracting from solar noon tells me that the
sun was due east at 8:53:06. This gave me two more shots at specific times,
before and after normal business hours, to get the dial axis set.
For the record, it rained all day on 5 June in Calgary and I missed the
opportunity to set the dial. Oh well, we need the rain. There has been a
drought in Alberta.
2. Use Offsets: The above method is fairly simple but the clouds can still
roll in at the wrong time. What we really need is a way to calculate the
offset from the north south axis for any convenient time. One formula for
azimuth adapted from the one in the BSS Glossary is:
Tan Az = Sin t / (Sin Lat x Cos t - Cos Lat x Tan Dec)
This looks pretty bad but the declination and latitude can be stored as
constants in your calculator leaving only one variable to be dealt with, the
time angle t. A simple program in a pocket calculator will allow you to
input the current time, as an angle from solar noon, and solve for Tan Az.
This is all you need as the tangent is the offset. From 1 meter down the
shadow line from your centre point, measure out at a right angle the offset
(Tan Az) to establish a point on the north south axis. This formula is now
programmed into my Radio Shack / Sharp calculator so I am all set for the
next sunny day when I am in Calgary.
Anyone who stayed with this note this far will see that the first equation
is simply derived from the second for case of the east or west azimuth of 90
degrees. Tan Az is infinite at 90 degrees so the denominator must be zero.
Then Sin Lat x Cos t must equal Cos Lat x Tan Dec. QED
I hope this will help us get past our solar noon fixation.
Roger Bailey
Walking Shadow Designs
N 51.0842 W 115.3775
