Hello Fred, Thanks for the equation. It provides a direct answer in one step*. Your solution to the quiz confirms my point on the time variation with azimuth angle. In the winter the azimuth change of 45º is a time angle of 50.12º or 200'29" and for summer 24.6º or 98'24, about half the time. Both are quite different from the uniform 15º per hour or 180 ' for 45º.
The teaching tool for Oscar's next lesson should be a circle on a stick, a gnomon that can be pointed at the north star and and an equatorial disc with a tangent line. I used on for my "Scratch" presentation at Banff. John Hoy offered a version using a pencil. The most graphic example is Dilbert's donut problem. I have the cartoon posted at http://www3.telus.net/public/rtbailey/images/Dilbert.gif . It is a small 55 kb file worth a chuckle. Regards, Roger My first attempt was an iteration using the the standard equations for azimuth, first the Cosine Rule for altitude and then the Sine Rule for azimuth. This was tedious. My second attempt was a two step solution, first the Sine Rule for the third angle and then Napier's analog for the time angle. This work but your equation looks better. Find B by Sine Law: Sin B= Sin CoLat *Sin Az / Sin CoDec Find t by Napier's Analog: Cot t/2=Tan(B+C)/2 *Cos(b+c)/2 / Cos(b-c)/2 ----- Original Message ----- From: "Fred Sawyer" <[EMAIL PROTECTED]> To: "Roger Bailey" <[EMAIL PROTECTED]>; "Sundial Mailing List" <[email protected]>; "Frank King" <[EMAIL PROTECTED]> Sent: Wednesday, July 11, 2007 10:42 AM Subject: Re: Help Oscar! > Roger, > > There's a direct solution to this problem - but it seldom if ever shows up > in dialing texts. Since it's so seldom presented, I thought list members > might be interested in seeing the equation. I've attached a very small > pdf > that gives the equation (in my preferred form - there are actually several > ways to attack this) and solution for this particular instance. > > Fred Sawyer > > > ----- Original Message ----- > From: "Roger Bailey" <[EMAIL PROTECTED]> > To: "Sundial Mailing List" <[email protected]>; "Frank King" > <[EMAIL PROTECTED]> > Sent: Wednesday, July 11, 2007 12:26 PM > Subject: Re: Help Oscar! > > > This advice is good as far as it goes but if the class repeats the > experiment with a vertical gnomon six months later they will find that the > hour lines are very much different. Will this destroy their trust in > science? I hope not and their teachers will use this as an opportunity to > introduce another topic, the sun and seasonal change. The ancients > building > megaliths etc learned a lot about their world by observing the sun. > Unfortunately a lot of fires were burned and virgins sacrificed to get the > sun to come back. The important thing for teachers today is to get the > kids > to keep asking why and not to take things on faith from all knowing > teachers. > > A little quiz is appropriate here. Calculate the time for the sun to be at > a > azimuth of 45º for the summer and winter solstices for a given latitude, > say > 50º. The difference is remarkable. I found solving this particular > solution > of spherical trigonometry somewhat more difficult than the usual solutions > where the time is known and altitude and azimuth are calculated. > > Regards, > Roger Bailey > --------------------------------------------------- https://lists.uni-koeln.de/mailman/listinfo/sundial
