Fix to NOAA Sunrise/Sunset Algorithm

The Zmanim API was developed from the ground up as an API which allows for easy plugging in of different algorithms. The Zmanim API ships with 3 “Calculator” implementations. Two calculators implement the US Naval Observatory’s algorithm, the SunTimesCalculator and the ZmanimCalculator (no longer available). Both produce identical zmanim using slightly different code and are included for comparison. There is also the JSuntimeCalculator (no longer available – see below information on the NOAACalculator), an implementation of the NOAA algorithm by Jonathan Stott. I was recently contacted by Eliezer Bulka who wanted to know why sunrise/sunset times generated by the NOAA algorithm were about 2 minutes off of the sunrise/sunset times generated by the NOAA JavaScript implementation that is the source of the JSuntimeCalculator. To compare apples and apples required modification of the NOAA JavaScript to allow entry of decimal latitude/longitude and changing the output to display seconds. No change was made to the algorithm itself. I then ported the JavaScript directly to Java. This involved nothing more than slight syntax changes between the languages. Once this was done, I noticed that the sun rise/set output from the Java port exactly matched the output of the NOAA JavaScript. Analysis of Jonathan’s code showed (or at least my interpretation of it did) that there were two areas that caused the difference. Once is that he used a slightly different method of computing the Julian date, a key part of the algorithm. His change includes the time of day as part of the calculation. The net result of this change is that solar time generated using his algorithm varies based on the time of day the calculation is run, something that is incorrect. This means that there can be a discrepancy of up to one calendar day. If the user calculated sunrise at 11:59 PM, sunrise would be calculated for the following day even if the user attempted to calculate it for today. In addition, the other calculations do not match the output of the matching NOAA code. I have deprecated the JSuntimeCalculator and in its place added the NOAACalculator that was the result of the direct port of the NOAA code, shoehorned into the Zmanim API Calculator interface. I ran some tests to compare the maximum and minimum discrepancy between the 2 implementations, and calculations for Lakewood, NJ, latitude 40.0828, longitude -74.2094 show a discrepancy of between a minute and 34 seconds to a minute and 37 seconds for sunrise and sunset across an entire year of sunrise and sunset calculations. I also compared the USNO algorithm to the new NOAA implementation and ended up with a maximum deviation of less than 30 seconds, something that had been about 1.5 minutes apart previously. While I do believe that the Julian date calculation is a bug, I do not know that this is a case as far as the rest of the calculation, but it is clear that it does not match the NOAA implementation that is was based on, and I recommend that you download the latest version that has the new NOAACalculator that fixes this issue. In addition to this fix, an additional patch will be released later this week that will address issues with calculations in the arctic circle. Stay posted for the next post.

Technical Information about the Bearing to Yerushalayim Map

Vintage Map with CompassI have been asked a number of questions recently about the directional accuracy of the Bearing to Yerushalayim and Zmanim Map calculations. The rhumb line calculations are very straight forward, and I will not spend much time on it in this article. As a side note, for those interested in the subject, I would like to mention that Rabbi Gavriel Goetz recently published a very comprehensive pamphlet Gevuras Moishe on the subject and it is well worth reading.
My original calculations and implementation of the great circle route (geodesic line) used simple spherical trigonometry to calculate the initial bearing (and distance) based on a sphere using an authalic mean radius of 6371 km. These calculations were very similar to the method used by Rabbi Yehuda Herskowitz’s article in Yeshurun volume III page 586. This model of the earth is more than accurate enough for these calculations (you would need a very accurate “nose alignment” and the total lack of shuckling to even get within a degree of being correct). Google Maps API, Yahoo Maps and Microsoft’s Live Maps for simplicity sake, all use very similar calculations as mentioned in the cfis blog quoting Morten Nielsen

Google Maps / Virtual Earth / Yahoo Maps(?) all use a spherical datum based on WGS84. That is, it has the same center, orientation and scale as WGS84, but has no flattening. The radius of the sphere is the same as the semi-major axis of WGS84 (6378137 meters).

Using a sphere, all the above-mentioned mapping APIs use spherical law of cosines formula for the calculations, that yield an identical result to the more complex Haversine formula.

Curious how accurate these calculations really are, being based on a perfect sphere and not the oblate ellipsoid/spheroid that the earth actually is, I started looking into the subject a little more. The most accurate current geoid model of the earth is the WGS84 model. The geodesist/mathematician Thaddeus Vincenty published the Vincenty formulae for this type of calculation that is said to be accurate to about one-half millimeter, more than adequate for our calculation. Chris Veness implemented this formula in JavaScript and released it under the LGPL, making it very simple to use. I slightly modified his work to allow easier interaction with the Google Maps API. The implementation that I used when I finally published the map on December 30, 2007, uses the Vincenty formula with the WGS84 geoid model (the Vincenty formula can easily be used with different goeiod models). The table below shows a comparison of the results for the initial bearing and distances of the different calculations based an a Lakewood, NJ (latitude: 40.095965°, longitude: -74.22213°) to Har Habayis (latitude: 31.77805°, longitude: 35.235149°) example.

Calculation MethodInitial BearingDistance
6,371 km (authalic mean radius) sphere using spherical trigonometry53.86555°9224.67442 km
Google Maps API 6,378.137km (WGS84 equatorial mean radius) sphere using spherical trigonometryN/A9235.00819 km
Vincenty formula using a WGS84 geoid53.81786°9244.61686 km
rhumb line95.37152°9891.16074 km

Zmanim Via Instant Messaging Bot

As hinted at in my previous post , there is a new project underway that uses the Zmanim API in a way that I had never really imagined. Using the Smack API, Michael Kopinsky created the ZmanimBot that allows getting zmanim by instant messaging the ZmanimBot, an internet bot. It currently supports the Google Talk IM system, but support for other systems might follow. Please be aware that the system is under development and is not always up. Additional information can be found on the ZmanimBot page.

Update (ג׳ אייר תשע״ג April 13, 2008): The ZmanimBot is now available via AIM

Zmanim Clock Applet Alpha Release

Java Logo Sepia
Note: Java Applets are obsolete, and the applet is no longer available
A while ago, Dr. Irv Bromberg mentioned to me, that my Zmanim API was in use in the Astronomical Clock Applet created by Ali Adams. The applet was a modification to Antony Pranata’s clock applet that displayed Islamic prayer times. Apparently the Zmanim API was so flexible that without any modification it was able to be used for generating “zmanim” well beyond anything I had envisioned. At Dr. Bromberg’s suggestion I modified the clock applet to display zmanim. Using the same Google Map API used in the Calendar Generator, I created the Zmanim Clock page (no longer available). The clock on the Clock page is very configurable, and allows selection of the location, zmanim etc. I contacted Antony Pranata, the original author of the clock who allowed my to release it under the GPL. An easy to use downloadable version will be made available in the near future. For those who think that the clock is upside down, there are more than enough configuration options to tweak the clock to your heart’s content.

Calculation of Zmanim VS Other Sites

A 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 Naval Observatory), and the other from the NOAA (National Oceanic and Atmospheric Administration). I have 2 different implementations of the USNO algorithm (one has since been removed as a duplicate), 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):

SiteAlgorithmElevation
in Meters
Sunrise
Time
MyZmanim?06:12:16
MyZmanim?206:11:25
KosherJavaUSNO06:12:18
KosherJavaUSNO206:11:31
KosherJavaNOAA06:10:11
KosherJavaNOAA206:09:24

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.