Re: simultaneous sunset
Dear Frank et al, An intriguing side issue to your puzzle is that it relates to the discussion about the Hawkeshead dial and the notion of a Plane's Longitude and, implicitly, the notion of a Plane's Latitude. Once you have taken on board these notions, the simplest way of expressing the solution to your new puzzle is to say: 1. Consider the plane defined by the great circle through London and Paris. 2. The required declination is that Plane's Latitude. I am fairly confident that John Good of The Art of Shadows could have come up with a solution 200 years ago. Since it was a 1950s problem I took a 1950s approach rather than an 18th century one! I have enjoyed reading the comments by Roger Bailey, Warren Thom, Werner Riegler and Fred Sawyer. I have a comment on one of Warren's notes: two locations with too great of a difference of longitude, would be too far apart to ever have the same moment of sunset. This is not quite the whole story... The great circle which separates light from dark necessarily encompasses EVERY longitude, so there will be points almost 180 degrees apart which have the same moment of sunset. Half the points on this circle will correspond to sunset and half to sunrise. The most northerly and most southerly points will, respectively, be points where sunset is immediately followed by sunrise and sunrise is immediately followed by sunset. My formula doesn't distinguish between sunrise and sunset and the leading minus sign could equally be a plus sign: tan(dec) = [-]sin(d)/sqrt(t1^2 - 2.t1.t2.cos(d) + t2^2) In the case of London and Paris a declination of -18.7 degrees is the condition for common sunset and +18.7 degrees is the condition for common sunrise. I usually like to sleep on any Mathematics before writing about it and I worried a little about whether the argument of the square root function could be negative. Happily it cannot be negative for real t1, t2 and d. I'll leave it as an exercise for you all to demonstrate this! It can be zero and this corresponds to a declination of 90 degrees which doesn't apply to the real sun but I am not bothered about trifles like that! Best wishes Frank --- https://lists.uni-koeln.de/mailman/listinfo/sundial
simultaneous sunset
Greetings, fellow dialists, So many clever solutions! I am no mathematician but I suppose I am in duty bound to add my two pennyworth to the sunset problem. If 90 deg is added to the angle Pole-Paris-London (obtainable from given) then this angle is the internal angle of Pole-Paris-Sun. The sides Pole-Paris and Paris-Sun are known and Pole-Sun may be calculated to give the sun's declination. The 90 deg angle is because the arc Paris-London is part of the great circle whose centre is the sun and Paris-Sun is a radius and so forms a right angle to the circle. OK, where did I go wrong (don't all rush). I may say I think it an unfair question for pilots being examined in navigation but it happened to my brother-in-law who was flying with KLM in 1949. Frank 55N 1W -- No virus found in this outgoing message. Checked by AVG Free Edition. Version: 7.5.472 / Virus Database: 269.9.1/854 - Release Date: 19/06/2007 13:12 --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: simultaneous sunset
Hello Franks and all, I assume we are to neglect altitude differences and use the fictional spherical earth model, but just what do we mean by 'the same time'? I would assume that when two or more persons look at their watches, corrected for standard zone time, that they read the same numbers? If so, this would give a larger number of answers as the sun sets in each time zone, it will probably occur some place within it at the same clock time as in fixed places in other time zones. Something like the New Year always starts at midnight, so each time zone has it's own New Year, 24 in total. Since the time zones are so non-uniform, the problem could be quite difficult to solve. But you probably mean at the same GMT or Universal time. Then we have our troubles with synchronizing. Given that light travels at about 11.8 inches in a nanosecond, someone 11.8 feet away would be off by 12 nanoseconds. Still, the problem is a neat one! Having fun with Sundials! Edley. Greetings, fellow dialists, A little while ago there was a dialling discussion about pairs of places where the sun rose or set at the same moment. I recall that to solve the problem a terminator programme was called into play. There are earlier examples of this question. Around the year 1950 airline pilots (who still used sextants) sat two successive navigation examinations, the higher of which was called the First N. In the written paper a question appeared of the form: Find a day on which the sun sets (altitude 0 deg.) at the same moment in London and Paris (positions given). This problem caused the pilots much head scratching until it was realised that this type of question occurred in every paper that was set. The solution is presumably a simultaneous equation in spherical trigonometry to discover the sun's declination. Would any dialist care to try it? Frank 55N 1W -- No virus found in this outgoing message. Checked by AVG Free Edition. Version: 7.5.472 / Virus Database: 269.9.0/852 - Release Date: 17/06/2007 08:23 --- https://lists.uni-koeln.de/mailman/listinfo/sundial --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: simultaneous sunset
Hello Edley and listeners everywhere, Edley, I don't quite know how to answer you. I explained in an earlier message that this was a real question set for airline pilots but it was fifty eight years ago and the exact wording unfortunately eludes me. Did they have nanoseconds then? And what about refraction, parallax and height of eye? Is Paris higher than London? It looks lower on the map. Which reminds me that the south east of England is running out of water. We have plenty here in the north and the good folks of London are under the impression that if they built a pipe the water would just run down to them. After all, we are at the top of our English maps and they are at the bottom. Frank 55N 1W Edley McKnight wrote: Hello Franks and all, I assume we are to neglect altitude differences and use the fictional spherical earth model, but just what do we mean by 'the same time'? I would assume that when two or more persons look at their watches, corrected for standard zone time, that they read the same numbers? If so, this would give a larger number of answers as the sun sets in each time zone, it will probably occur some place within it at the same clock time as in fixed places in other time zones. Something like the New Year always starts at midnight, so each time zone has it's own New Year, 24 in total. Since the time zones are so non-uniform, the problem could be quite difficult to solve. But you probably mean at the same GMT or Universal time. Then we have our troubles with synchronizing. Given that light travels at about 11.8 inches in a nanosecond, someone 11.8 feet away would be off by 12 nanoseconds. Still, the problem is a neat one! Having fun with Sundials! Edley. Greetings, fellow dialists, A little while ago there was a dialling discussion about pairs of places where the sun rose or set at the same moment. I recall that to solve the problem a terminator programme was called into play. There are earlier examples of this question. Around the year 1950 airline pilots (who still used sextants) sat two successive navigation examinations, the higher of which was called the First N. In the written paper a question appeared of the form: Find a day on which the sun sets (altitude 0 deg.) at the same moment in London and Paris (positions given). This problem caused the pilots much head scratching until it was realised that this type of question occurred in every paper that was set. The solution is presumably a simultaneous equation in spherical trigonometry to discover the sun's declination. Would any dialist care to try it? Frank 55N 1W -- No virus found in this outgoing message. Checked by AVG Free Edition. Version: 7.5.472 / Virus Database: 269.9.0/852 - Release Date: 17/06/2007 08:23 --- https://lists.uni-koeln.de/mailman/listinfo/sundial No virus found in this incoming message. Checked by AVG Free Edition. Version: 7.5.472 / Virus Database: 269.9.1/854 - Release Date: 19/06/2007 13:12 -- No virus found in this outgoing message. Checked by AVG Free Edition. Version: 7.5.472 / Virus Database: 269.9.1/854 - Release Date: 19/06/2007 13:12 --- https://lists.uni-koeln.de/mailman/listinfo/sundial
