Whoops, a typo. It is /Cos Dec, not x Cos Dec.
The correct equations for the when, where and how of sunrise and set are:
Time: Cos t = -Tan Lat x Tan Dec
Azimuth: Cos Az = Sin Dec / Cos Lat
Path: Cos Phi = Sin Lat / Cos Dec
Celebrate the return of the Sun
Roger Bailey
www.walkingshadow.info
----- Original Message -----
From: Roger Bailey
To: Frank King ; Jack Aubert ; 'Sundial List'
Sent: Monday, December 24, 2007 10:00 AM
Subject: Re: Azimuth of Sunrise - Sunset
Hi Jack, Frank and al,
Your question "What is the path of the sun at sunrise or set?" is a good one.
The answer given by Frank is simple and subtle. Actually it is a Smart
question, posed by W. M. Smart in his "Textbook on Spherical Trigonometry".
Prof Smart leaves it as a question to be solved by students as a homework
exercise, Question 2, Chapter 2 "If Phi is the angle which a star makes at
rising with the horizon, prove that Cos Phi = Sin Lat x Cos Dec".
I enjoy sunrise and sunset. These are the only times of the day when you can
almost get away with staring at the sun. I summarized the derivation and
application of the sunrise equations in a presentation at the NASS conference
in Hartford in 1999. Have a look at "Sunset Phenomenon: The Use of Spherical
Trigonometry to Determine when, where and how the sun rises and sets" This is
available at my website www.walkingshadow.info as a small, 200 kb pdf file, #
14 on the list of sundial publications. It is a relatively small (200 kb) pdf
file as I deleted all the sunset photos.The spherical trigonometry is all there
except for the Prof Smart's proof. I will leave that as a homework exercise for
students. It remains a great question for the final exam.
Regards,
Roger Bailey
----- Original Message -----
From: Jack Aubert
To: 'Sundial List'
Sent: Monday, December 24, 2007 7:21 AM
Subject: Azimuth of Sunrise - Sunset
Thank you all for the answer(s) to this little problem and for the
bread-crumb trail to try to retrace the derivation.
My original question was sparked by wondering about the maximum deviation
from east-west at the solstice(s) so I could display my erudition and bore
people with comments like: "The sun rises in the East and sets in the West,
right? Well, not really. Actually ...."
What I had meant about reading the answer directly from the sundial,
assuming that you already know the time of sunrise, is that if you were to lay
the straight edge of a protractor between the root of the gnomon and the time
of sunrise on the dial face, you could read the deviation from east-west where
the six o'clock line intersects the protractor scale. Of course, if my
sunrise time were taken from the daily newspaper, I would have to adjust for
longitude.
At my latitude (38.88 north) the formula tells me that on the solstice,
sunrise/sunset deviates from east-west by 30.74 degrees, which looks like it
agrees with the protractor method (allowing for the fact that I can't actually
lay it on the dial without removing the gnomon).
Jack
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