A question for the mathematically inclined
Dear dialists, Does anyone know a formula for calculating the hour angle given the azimuth, declination, and latitude? I’d like to know the time of day, throughout the year, when the sun will be positioned at a particular angle. This will allow me to determine when sunshine will stream squarely through a window on any (sunny) day. I’ve seen several formulae for calculating azimuth. I suspect that one of them could be rewritten to solve for the hour angle given the azimuth instead of the finding the azimuth using the hour angle (plus the declination and latitude). Unfortunately, I don’t have the math skills for this conversion. Thanks for any suggestions. --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: A question for the mathematically inclined
If you know the zenith distance, z, of the sun (90° - elevation angle) as well as the azimuth (A) then you could use: sin(h) = -sin(z)*sin(A)/cos(delta) where delta is the sun's declination. The latitude of the site, phi, is not needed. Computing the hour angle when the zenith distance is not known is a little trickier. In principle, this equation could be used: sin(h) = tan(A)*(sin(phi)*cos(h) - cos(phi)*tan(delta)) but you'll notice that h appears on both sides of the equation. Possibly this can be solved in an iterative fashion by selecting an approximate trial value for h and using it on the r.h.s. to compute a new value of h. You would then use this new value on the r.h.s. and continue the iterative procedure until the new value does not change significantly from the previous value. I've not actually tried this myself so proceed with caution. -- Richard Langley On Saturday, January 31, 2015, 31, at 11:05 AM, John Goodman wrote: > Dear dialists, > > Does anyone know a formula for calculating the hour angle given the azimuth, > declination, and latitude? > > I’d like to know the time of day, throughout the year, when the sun will be > positioned at a particular angle. This will allow me to determine when > sunshine will stream squarely through a window on any (sunny) day. > > I’ve seen several formulae for calculating azimuth. I suspect that one of > them could be rewritten to solve for the hour angle given the azimuth instead > of the finding the azimuth using the hour angle (plus the declination and > latitude). Unfortunately, I don’t have the math skills for this conversion. > > Thanks for any suggestions. > --- > https://lists.uni-koeln.de/mailman/listinfo/sundial > - | Richard B. LangleyE-mail: l...@unb.ca | | Geodetic Research Laboratory Web: http://gge.unb.ca/ | | Dept. of Geodesy and Geomatics EngineeringPhone:+1 506 453-5142 | | University of New Brunswick Fax: +1 506 453-4943 | | Fredericton, N.B., Canada E3B 5A3| |Fredericton? Where's that? See: http://www.fredericton.ca/ | - --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: A question for the mathematically inclined
You can download a free excel spreadsheet, sunpositioncalculator at http://precisionsundials.com/sunpositioncalculator.xls. The Azimuth page allows you to input date, latitude, longitude, and azimuth, and it gives you the civil time, eot, declination, and altitude. When opening, you must allow macros to run if the computer asks. -Bill On Sat, Jan 31, 2015 at 10:49 AM, Richard B. Langley wrote: > If you know the zenith distance, z, of the sun (90° - elevation angle) as > well as the azimuth (A) then you could use: > > sin(h) = -sin(z)*sin(A)/cos(delta) > > where delta is the sun's declination. The latitude of the site, phi, is > not needed. > > Computing the hour angle when the zenith distance is not known is a little > trickier. In principle, this equation could be used: > > sin(h) = tan(A)*(sin(phi)*cos(h) - cos(phi)*tan(delta)) > > but you'll notice that h appears on both sides of the equation. Possibly > this can be solved in an iterative fashion by selecting an approximate > trial value for h and using it on the r.h.s. to compute a new value of h. > You would then use this new value on the r.h.s. and continue the iterative > procedure until the new value does not change significantly from the > previous value. I've not actually tried this myself so proceed with caution. > > -- Richard Langley > > On Saturday, January 31, 2015, 31, at 11:05 AM, John Goodman wrote: > > > Dear dialists, > > > > Does anyone know a formula for calculating the hour angle given the > azimuth, declination, and latitude? > > > > I’d like to know the time of day, throughout the year, when the sun will > be positioned at a particular angle. This will allow me to determine when > sunshine will stream squarely through a window on any (sunny) day. > > > > I’ve seen several formulae for calculating azimuth. I suspect that one > of them could be rewritten to solve for the hour angle given the azimuth > instead of the finding the azimuth using the hour angle (plus the > declination and latitude). Unfortunately, I don’t have the math skills for > this conversion. > > > > Thanks for any suggestions. > > --- > > https://lists.uni-koeln.de/mailman/listinfo/sundial > > > > > - > | Richard B. LangleyE-mail: l...@unb.ca >| > | Geodetic Research Laboratory Web: http://gge.unb.ca/ >| > | Dept. of Geodesy and Geomatics EngineeringPhone:+1 506 453-5142 > | > | University of New Brunswick Fax: +1 506 453-4943 > | > | Fredericton, N.B., Canada E3B 5A3 > | > |Fredericton? Where's that? See: http://www.fredericton.ca/ >| > > - > > --- > https://lists.uni-koeln.de/mailman/listinfo/sundial > > --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: A question for the mathematically inclined
The USNO Webpage http://aa.usno.navy.mil/data/docs/AltAz.php will also compute elevation angle (altitude) and azimuth of the sun for a given date and location at specified intervals. On Saturday, January 31, 2015, 31, at 12:31 PM, Bill Gottesman wrote: > You can download a free excel spreadsheet, sunpositioncalculator at > http://precisionsundials.com/sunpositioncalculator.xls. The Azimuth page > allows you to input date, latitude, longitude, and azimuth, and it gives you > the civil time, eot, declination, and altitude. When opening, you must allow > macros to run if the computer asks. > > -Bill > > On Sat, Jan 31, 2015 at 10:49 AM, Richard B. Langley wrote: > If you know the zenith distance, z, of the sun (90° - elevation angle) as > well as the azimuth (A) then you could use: > > sin(h) = -sin(z)*sin(A)/cos(delta) > > where delta is the sun's declination. The latitude of the site, phi, is not > needed. > > Computing the hour angle when the zenith distance is not known is a little > trickier. In principle, this equation could be used: > > sin(h) = tan(A)*(sin(phi)*cos(h) - cos(phi)*tan(delta)) > > but you'll notice that h appears on both sides of the equation. Possibly this > can be solved in an iterative fashion by selecting an approximate trial value > for h and using it on the r.h.s. to compute a new value of h. You would then > use this new value on the r.h.s. and continue the iterative procedure until > the new value does not change significantly from the previous value. I've not > actually tried this myself so proceed with caution. > > -- Richard Langley > > On Saturday, January 31, 2015, 31, at 11:05 AM, John Goodman wrote: > > > Dear dialists, > > > > Does anyone know a formula for calculating the hour angle given the > > azimuth, declination, and latitude? > > > > I’d like to know the time of day, throughout the year, when the sun will be > > positioned at a particular angle. This will allow me to determine when > > sunshine will stream squarely through a window on any (sunny) day. > > > > I’ve seen several formulae for calculating azimuth. I suspect that one of > > them could be rewritten to solve for the hour angle given the azimuth > > instead of the finding the azimuth using the hour angle (plus the > > declination and latitude). Unfortunately, I don’t have the math skills for > > this conversion. > > > > Thanks for any suggestions. > > --- > > https://lists.uni-koeln.de/mailman/listinfo/sundial > > > > - > | Richard B. LangleyE-mail: l...@unb.ca | > | Geodetic Research Laboratory Web: http://gge.unb.ca/ | > | Dept. of Geodesy and Geomatics EngineeringPhone:+1 506 453-5142 | > | University of New Brunswick Fax: +1 506 453-4943 | > | Fredericton, N.B., Canada E3B 5A3| > |Fredericton? Where's that? See: http://www.fredericton.ca/ | > - > > --- > https://lists.uni-koeln.de/mailman/listinfo/sundial > > - | Richard B. LangleyE-mail: l...@unb.ca | | Geodetic Research Laboratory Web: http://gge.unb.ca/ | | Dept. of Geodesy and Geomatics EngineeringPhone:+1 506 453-5142 | | University of New Brunswick Fax: +1 506 453-4943 | | Fredericton, N.B., Canada E3B 5A3| |Fredericton? Where's that? See: http://www.fredericton.ca/ | - --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: A question for the mathematically inclined
Hello John, I routinely use Napier's Analogue as suggested by Fred Sawyer when I asked this question several years ago. This involves an intermediate step involving an angle B. Here are the formulae. Napier's Analogues: Knowing Latitude, Declination and Azimuth, Solve for Altitude and TimeFindangle B : Sin B =Sin Az Cos Lat/ Cos Dec. Then Tan .5(90-Alt)=Tan .5(Lat-Dec)Cos.5(B-Az)/Cos .5(B+Az), Then the Sine Rule for t: Sin Az=CosDec Sint/CosAlt or Sint=SinAzCosAlt/CosDec These are fairly easy to program into a spreadsheet. Regards, Roger Bailey -- From: "John Goodman" Sent: Saturday, January 31, 2015 7:05 AM To: "Sundial List" Subject: A question for the mathematically inclined Dear dialists, Does anyone know a formula for calculating the hour angle given the azimuth, declination, and latitude? I’d like to know the time of day, throughout the year, when the sun will be positioned at a particular angle. This will allow me to determine when sunshine will stream squarely through a window on any (sunny) day. I’ve seen several formulae for calculating azimuth. I suspect that one of them could be rewritten to solve for the hour angle given the azimuth instead of the finding the azimuth using the hour angle (plus the declination and latitude). Unfortunately, I don’t have the math skills for this conversion. Thanks for any suggestions. --- https://lists.uni-koeln.de/mailman/listinfo/sundial - No virus found in this message. Checked by AVG - www.avg.com Version: 2015.0.5646 / Virus Database: 4273/9032 - Release Date: 01/31/15 --- https://lists.uni-koeln.de/mailman/listinfo/sundial
