Calculation of Zmanim VS Other Sites

pocket watchA user contacted me with the following note (some details removed).

Could you please explain to me why there seems to be nearly a three minute discrepancy between the zemanim for XXXXX, NJ at 20 meters (exact location) and those listed for 20 meters on the MyZmanim.com site. Your site calculates 4/20 at 6:08:20 and theirs has it at 6:11:25….

I really can’t comment on the MyZmanim.com site since their code is not open source and I do not know the exact algorithm they use. The Zmanim Project code used here has 2 different algorithms available for calculating zmanim, one using the algorithm published by the USNO (US Navel Observatory), and the other from the NOAA (National Oceanic and Atmospheric Administration). I have 2 different implementations of the USNO algorithm, and both return the exact time to the millisecond, so I believe that the code is a correct implementation of the USNO algorithm. You can look through the code at my site and see if you find any issues. Having said that, I did a comparison between MyZmanim.com and my site using the USNO algorithm and both sites are within seconds (all calculations were for April 19 for XXXXX, NJ):

Site Algorithm Elevation
in Meters
Sunrise
Time
MyZmanim.com ? 0 6:12:16
MyZmanim.com ? 20 6:11:25
KosherJava.com USNO 0 6:12:18
KosherJava.com USNO 20 6:11:31
KosherJava.com NOAA 0 6:10:11
KosherJava.com NOAA 20 6:09:24

Based on this chart, the sunrise difference between KosherJava.com using USNO and MyZmanim.com who are probably also using the USNO algorithm is 3 seconds for sea level and 6 seconds at 20 meter elevation (see below for why the elevation calculation has this additional 3 second discrepancy). Keep in mind that while I was able to zoom in to the user’s exact location using the Google Map interface, myZmanim.com is using a zip code level calculation, and they might be calculating using a slightly different location.
I believe that the times the user was referring to in his note were generated using the NOAA algorithm implementation. As mentioned above, At this point I have no information to indicate what algorithm is more accurate, and I will hopefully one day post more information on the accuracy of these algorithms.

As far as the elevation calculation, I use the formula

zenith = zenith + Math.toDegrees(Math.acos(earthRadiusInMeters / (earthRadiusInMeters + elevationMeters)));

found in Calendrical Calcuations, as mentioned on the Zmanim Calendar Generator page. For this time of year and the XXXXX location and a 20 meter elevation returns a time 47 seconds earlier than sea level using my calculation, and 51 seconds earlier using myZmanim.com. I reran the calculation using the more commonly used but less accurate formula of

adding 0.0347 * squareRoot(elevationMeters) to the zenith

that is mentioned in the Maaglay Tzedek written by Rabbi Moshe Kosower (as mentioned in the JavaDocs and Calendar Generator page) and the difference between the 0 and 20 meter elevation was the same 51 seconds that appeared in MyZmanim.com. Besides relying on the more accurate calculation mentioned in Calendrical Calculations (considered the most authoritative book on calendars and astronomical time calculations), I spoke to Professor Moshe Koppel a few years ago about the subject and he validated the Calendrical Calculations algorithm (a seemingly common mathematical calculation). Looking back at the Maaglay Tzedek, it seems that this calculation includes changes to the refraction caused by the elevation, and I will have to research the subject some more. All this elevation difference boils down to an insignificant 3 second difference. Keep in mind that atmospheric refraction will have a much greater impact than this. Even if one were to know the exact atmospheric pressure, temperature and humidity, this varies in different parts of the atmosphere and would be extremely hard to calculate without measuring it at all elevations of the atmosphere. Please make sure to speak to a posek before using the time with elevation.

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