I have two separate references, listed below, that gives the solution to this problem in terms of the cosine of the hour angle as follows:
cos H =(sin (altitude) - sin (lat) sin (dec))/(cos(lat) cos(dec)) where H is the hour angle. Since the tangent of H is undefined when H approaches +/- 90 degrees, perhaps this form is of more general use. When considering the double-valued azimuth examples being discussed, isn't it true that there can be two values for the hour angle given the same input values, but these two events do not occur at the same time? (The location of the shadow of the tip of a gnomon can only be one place at a time.) I think Fer de Vries' picture he posted on the list makes the explanation of this double-valued solution quite clear. Cheers, Jeff Adkins Astronomical Photometry by Henden and Kaitchuck, (c) 1982 p.121 Spherical Trigonometry by W.M. Smart, (c) 1932 p.35 Gordon Uber wrote: > Steve in an email to me raised the possibility of an error in the equation > I posted for hour angle. It is probable that the corrected equation should > read: > > > > From Jean Meeus's "Astronomical Algorithms," 1951, p. 89: > > > > > > tan H = sin A / (cos A * sin f + tan h * cos f) [corrected] > > > > > > where H is hour angle, A is azimuth, f is latitude and h is altitude, > > > and H is obtained via the arctangent. > > Gordon > > Steve. > > The equation in my edition of Meeus is as I gave it, but the repeated cos f > certainly does look suspicious since it would be a common factor. There is > a Second Edition, which I do not have, which may have a corrected equation. > His equation for tan A does have sin f and cos f factors in the > denominator, which is a hint that it may be incorrect. > > Gordon > > At 10:31 AM 12/14/00 -0400, you wrote: > >Gordon, thanks for the message. > > > >I would like to check whether there is a transcription error in your > >message. > > > >Does "cos f" really appear twice? I got information from Richard Koolish > >which gave the answer in another form - as two equations. When I divided > >them through to convert them to Meeus' form, there was a mismatch which > >implies that the first cos f should perhaps be a sin f. > > > >Thanks, Steve > > Gordon Uber [EMAIL PROTECTED] San Diego, California USA > Webmaster: Clocks and Time: http://www.ubr.com/clocks Content-Type: text/x-vcard; charset=us-ascii; name="Astronomer.vcf" Content-Transfer-Encoding: 7bit Content-Description: Card for Jeff Adkins Content-Disposition: attachment; filename="Astronomer.vcf" Attachment converted: Macintosh HD:Astronomer.vcf (TEXT/ttxt) (0001FECB)
