When comparing the results of the KosherJava NOAA algorithm to the output of NOAA’s new Solar Calculator almost two years ago, a discrepancy was encountered between the two. There was no discrepancy compared to the output of the old NOAA calculator. The NOAA code is an implementation of the accurate Jean Meeus algorithm for solar time calculations, and the KosherJava code is a Java port of this algorithm. While attempting to debug the issue, I turned to Pinny Markowitz who ported the KosherJava library to both Ruby and Python. He was able to trace the issue to what seemed to be a small accuracy adjustment missing in the noon calculation on the new NOAA implementation. Based on Pinny’s analysis, the old implementation seemed correct, but without confirmation from the NOAA developers this was not a certainty. We reported the issue to NOAA for clarification, and after an almost two-year delay, the NOAA development team confirmed and corrected the bug. After NOAA’s fix there is no longer any discrepancy. The fix can be seen in line 342 of the NOAA JavaScript file, where a half day adjustment is made in the noon time calculation. This bug was never present in the KosherJava library, or other language ports of the KosherJava code, since our code was based on the original NOAA code.

# Tag: Zmanim Accuracy

## Equinox VS Equilux in Zmanim Calculations

## Degree based *Zmanim*

Degree-based *zmanim* are considered most accurate by many *poskim* since *zmanim* calculated using degrees for *alos* and *tzais* have a consistent level of light at all dates and locations. The alternatives of using fixed minutes (for example 72 minutes) or percentage of the day-based calculations (1/10^{th} of the day) result in *alos* and *tzais zmanim* having different levels of light at different dates and locations. The number of degrees for a given *zman* is calculated based on how many degrees the sun is below the horizon on an equal day in Yerushalayim^{[1]}. For example, the sun is 16.1° below the horizon 72 minutes before sunrise (or after sunset)^{[2]} on an equal day (defined below) in Yerushalayim. The subject of degree-based *zmanim* is extensive and deserves its own detailed article, עוד חזון למועד.

## Equinox VS Equilux in Halacha

A question explored by *poskim* and *luach* authors is; how we define an equal day to use for degree-based *zmanim* calculations. Should it be calculated at the astronomically equal day of the equinox or the *halachic* equal day of the equilux^{[3]}. At the equinox, the 12-hour duration of the day is calculated astronomically without accounting for refraction or solar radius^{[4]}. At the equilux there are exactly 12 hours of daylight from sunrise to sunset. Due to these two factors, the *halachic* length of the day from sunrise to sunset at the equinox is longer than 12 hours. In Yerushalayim on March 20, 2021, the day of the March equinox, sea level sunrise is at 5:42:51 AM and sunset is at 5:50:33 PM, or a day length of 12 hours, 7 minutes and 42 seconds. You would have to go back four days^{[5]} to March 16, the equilux, for a 12-hour day^{[6]}. There are *halachic* opinions supporting both the equinox and the equilux as the equal day for *zmanim* calculations^{[7]}.

## Practical Differences Between Equinox and Equilux Calculations

Many calendars and *seforim* list the 72-minute *alos / tzais* as 16.1° and 90 minutes as 19.8° (using the global refraction average + solar radius of 0.8333). Calculations using the KosherJava *Zmanim* API (utilizing the Jean Meeus / NOAA algorithms) show that the actual figures are 16.08° and 19.848° at the equilux, and 16.04° and 19.784° at the equinox. The table below shows the difference between these numbers at the summer solstice when twilight is the longest (the most extreme expected gap between two different degree-based times).

Difference Between the Equilux and Equinox Calculations at the Summer Solstice | ||||
---|---|---|---|---|

Location | 7.199° VS 7.205° (30 Min)^{[8]} | 11.424° VS 11.442° (50 Min) | 16.04° VS 16.08° (72 Min) | 19.784° VS 19.848° (90 Min) |

Jerusalem Lat: 31.7° | 2 sec | 7 sec | 15 sec | 26 sec |

Lakewood Lat: 40.1° | 2 sec | 7 sec | 20 sec | 38 sec |

Montreal Lat: 45.5° | 3 sec | 10 sec | 30 sec | 90 sec |

Krakow Lat: 50.05° | 3 sec | 13 sec | 93 sec | N/A |

London Lat: 51.5° | 4 sec | 16 sec | N/A | N/A |

Vilnius Lat: 58.68° | 13 sec | 47 sec | N/A | N/A |

Anchorage Lat: 61.2° | N/A | N/A | N/A | N/A |

N/A indicates that the sun does not get this far below the horizon at this time of the year due to the high latitude of the location. See Why Some Zmanim Never Occur for more details. |

While the above question is interesting from an academic perspective, the measurements above show a negligible difference between calculating at the equinox VS the equilux for most locations and *zmanim*. The difference in calculating *zmanim* up to 16.1° *alos / tzais* on the equinox vs the equilux isn’t significant until the 30 second difference at the 72-minute *zman*. Since this is typically calculated as 16.1° *lechumra*, there is no difference at all for this *zman*. The less commonly used 19.8°, has an up to 90 second difference (at the latitude of Montreal). The ~11.5° *misheyakir* times start showing a difference at high latitudes. This is not significant even as far north as London but becomes significant at the 58.68° latitude of Vilna (Vilnius) since it reaches 47 seconds.

## Observations on Degree Based Calculations

- The commonly used 16.1° time is a slightly rounded
*chumra*for both the equinox and equilux. The actual numbers are 16.04° and 16.08°. - The 19.8°
*zman*mentioned by many calendars and*seforim*is calculated at the equinox where it is 19.784° and not equilux where it is 19.848°. It should possibly be rounded up to 19.9°*lechumra*to account for the equilux calculation^{[9]}. - The
*misheyakir*11.5° times are a slight*kula*since both the equilux (11.442°) and the equinox (11.424°) calculations show a sightly later time. - As noted above, the degree-based calculations were done using the more accurate Jean Meeus / NOAA algorithms.
*Seforim*printed in the past did not have access to the newer algorithms and typically used the USNO algorithm, but as seen below, there is only a trivial difference between the algorithms.*Zman*Equinox Equilux USNO NOAA USNO NOAA 30 Min 7.203° 7.199° 7.208° 7.205° 50 Min 11.432° 11.424° 11.449° 11.442° 72 Min 16.055° 16.04° 16.092° 16.08° 90 Min 19.804° 19.784° 19.865° 19.848°

##### Notes:

1. ^{^} The assumed location for these calculations in most calendars (and the KosherJava *zmanim* library) is Yerushalayim, something that is debatable. See Hazmanim Bahalacha 19:2, pages 169-170.

2. ^{^} As an example, *alos hashachar* according to some opinions is 72 minutes before sunrise (the time it takes to walk 4 *mil* at a speed of 18 minutes per *mil*). The time of twilight from *alos* / dawn to sunrise and sunset to *tzais* / night is known as *neshef* נשף in Hebrew. The time of twilight differs by location and time of year with the longest duration during the summer solstice, shortest by the equinoxes and somewhere in between in the winter. According to many opinions this *zman* should be calculated by measuring the sun’s degrees below the horizon at the equal day and applying the same number of degrees to any location and date.

3. ^{^} A term coined by astronomers in the 1980s and in “popular” use since ~2006.

4. ^{^} It is calculated as if the world had no atmosphere and the radius of the sun is above the horizon.

5. ^{^} Rabbi Yedidya Manet mentions that there are 5 to 6 days separating the equinox and equilux, while Rabbi Yonah Merzbach in a letter to Rabbi Manat mentioned a week or two. Calculations show the difference between the equinox and equilux to be 4 days in Yerushalayim, moving the calculation date from March 20^{th} back to March 16^{th} (or from September 22^{nd} to the 26^{th}).

6. ^{^} In March 2021 it is 8 seconds off from a true 12-hour day due to the location where the equinox occurs for that season (it is at a single point and time globally), but it is more than close enough for our purposes. The figure varies from year to year. Calculations on the September equinox show similar results.

7. ^{^} Rabbi Meir Pozen in his Kuntres Haneshef and Or Hameir, is of the opinion that the equilux should be used. Opinions that the equinox should be used are brought down by Rabbi Yedidya Manet in his Zmanei Halacha Lema’aseh (4^{th} edition part 2, pages 22 and 24), Rabbi Yonah Merzbach (in a letter published by Rabbi Manat) and Rabbi David Yehuda Burstein in his Zmanim Kehilchasam, 1:8 (pages 56 – 61). This is also the opinion of Rabbi Chaim Pinchas Banish in Hazmanim Bahalacha vol 1, 19:3, page 270, and Rabbi Aryeh Leib Lipkin in his Ohr Hayom, summary section, no. 9 (page 76).

8. ^{^} This is close to the 7.083° *tzais zman* and used for comparison. The 7.083° *zman* was first brought down by Dr. Baruch (Berthold) Cohn in his *luach* Tabellen enthaltend die Zeitangaben für den Beginn der Nacht und des Tages für die Breitengrade + 66 bis -38, published in Strasbourg, France in 1899. It was based on actual observation of star visibility. Some list the 7.083° *zman* as based on the 30-minute calculation, but as seen in the chart, it is not an exact match. In Yerushalayim at the equinox (when there is the smallest difference), 7.083° is 33 seconds earlier than the 30-minute time of 7.199° and in Vilna it is 49 seconds earlier. At the solstice in Yerushalayim 7.083° is 39 seconds earlier than 7.199°, and in Vilna it is 97 seconds earlier.

9. ^{^} At this point the KosherJava *Zmanim* API will continue using the 16.1° (a minor *chumra*), and 19.8° (a minor *kulah* at the equilux) used by the Yisrael Vehazmanim and many others.

## FAQ: Location Precision for Zmanim Calculations

While overly broad ZIP code based *zmanim* geolocation can be an issue in calculating *zmanim* accurately, going overboard in geolocation precision and accuracy for *zmanim* is a (harmless) waste of time.

Let’s start with the basics. Asking what the *zmanim* are for the USA is too broad of a location. Narrowing it down to a state is also too broad since *zmanim* at one side of the state are likely to be different than the other side. How small (or precise) does an area have to be for the zmanim calculated to be considered accurate? The location of *zmanim* are calculated based on degrees of longitude (east to west) and latitude (north to south).

The earth’s circumference at the equator is about 40,000 km (about 25,000 mi). There are 360 degrees of longitude around the world (The 0° line is centered on the Royal Greenwich Observatory in England, and longitude lines extend 180° to the west and -180° to the east). For simplicity we will deal with longitude degrees at the equator. If we divide the earth’s circumference by 360°, each degree of longitude will be 111 km (69 mi) apart. The sun’s path travels 1° of longitude in 4 minutes, so calculating *zmanim* with one degree accuracy (no decimal points such as the latitude of 40° and longitude of -74° for Lakewood, NJ, a point in the Atlantic about 3 mi off the coast of Toms River, NJ) results in *zmanim* accurate to 4 minutes in each direction or an 8 minute spread, not quite accurate enough to rely on. Moving to one decimal point will pinpoint the location for *zmanim* calculation to an accuracy of 11 km or 48 second accuracy. That is close to being accurate enough, especially given the inaccuracy of solar time calculations resulting from hard to predict refraction caused by varying atmospheric conditions. However, this should be avoided. Adding a second decimal point (such as the latitude of 40.09° and longitude of -74.22° for Lakewood, NJ – a spot at the edge of Lake Carasaljo in Lakewood) would have a precision of about 4 seconds, more than enough accuracy for *zmanim*.

A concrete example of how *zmanim* differ from place to place in a small area would be the difference between Beth Medrash Govoha (BMG) and the Westgate Bais Medrash in Lakewood. They are 2.7 km (1.69 mi) or a drop more than 0.01° apart and calculations show that there is about a 6 second difference in sunrise and sunset times between these two locations.

From time to time I am contacted by developers with *zmanim* related technical questions. Debugging their issues often requires information on the latitude and longitude that they are using to try and replicate the issue. Often the latitude and longitude are sent with multiple decimal points. The most extreme was 14 decimal points. To understand the ridiculousness of this level of precision, see the table below. To read more on the subject, see the Stack Exchange page Measuring accuracy of latitude and longitude? and the xkcd cartoon on the subject.

Decimal places | Degrees | Distance | Notes |
---|---|---|---|

0 | 1 | 111 km | A state or small country |

1 | 0.1 | 11.1 km | City |

2 | 0.01 | 1.11 km | Neighborhood |

3 | 0.001 | 111 m | A specific cul-de-sac |

4 | 0.0001 | 11 m | A corner of a house |

5 | 0.00001 | 1.1 m | A person in a room |

6 | 0.000001 | 11 cm | A small siddur |

7 | 0.0000001 | 1 cm | The size of Waldo on a page |

8 | 0.00000001 | 1 mm | A grain of sand |

9 | 0.000000001 | 111 μm | The width of a hair |

10 | 0.0000000001 | 11 μm | A grain of pollen |

11 | 0.00000000001 | 1 μm | A smoke particle |

12 | 0.000000000001 | 111 nm | The width of a COVID virus |

13 | 0.0000000000001 | 11 nm | A red blood cell |

14 | 0.00000000000001 | 1 nm | The length your nails grow every second |

15 | 0.000000000000001 | 100 pm | An atom. If you need this precision, you probably belong in Lawrence Livermore |

## Decimal Versus Sexagesimal Based Zmanim Location Errors

There are two different ways to reference latitude and longitude. One uses a sexagesimal (60 based) system of degrees indicated by a ° symbol, minutes indicated by a ' and seconds indicated by ". Think 60 minutes in an hour, 60 seconds in a minute and apply it to latitude numbers. The other system uses the more familiar decimal based format. For example The main BMG

*beis hamedrash*is located at latitude 40.096, longitude -74.222 in degree/decimal. In degrees, minutes and seconds this would be latitude 40° 5′ 46″ N, longitude 74° 13′ 19″ W.

I was recently shown a *zmanim* calendar that seemed to be slightly inaccurate. Researching the issue showed that the intention was to generate the calendar for the location XX° 46′ N XX° 15′ W (latitude and longitude degrees are masked), but was mistakenly calculated for XX.46° -XX.15°. This confusion of the sexagesimal based system with the decimal based system is not uncommon. The discrepancy in sunrise and sunset in the calendar versus what it should have been was about 80 seconds in the summer. If someone were to confuse XX° 9′ with XX.9° (for both latitude and longitude) you have a much more significant relative error of 0.75°. The impact of this type of mistake is mostly caused by longitude, but latitude changes impact *zmanim* calculations as well. This 0.75° mistake can result in a *zmanim* discrepancy of up to five and a half minutes at the latitude of Lakewood, NJ. As confirmed by Dr. Noson Yanofsky, this scenario has the most extreme error, while 10′ confused with 0.10° has the least significant error of 0.066°.

An interesting variant of such a mistake is calculating a *zman* for a depression angle (how far the sun is below the horizon) that is based on degrees and minutes using degree/decimal. An example is mistakenly calculating *tzais* of 7° 5′ , or 7.083° as 7.5°. See Hazmanim Bahalacha vol II p. 520 footnote 21 for a case where this mistake happened. It should be noted that many are of the opinion that a depression angle of 7.5° is the proper time of *tzais*. This was used in the first ever known printed calendar calculated based on depression angles. It was published in תקכ״ו / 1766 by Raphael Levi Hannover. See Hazmanim Bahalacha p. 524 for a picture of the *luach* and a list of other calendars that calculate *tzais* as 7.5°.

To answer the question in the image caption above, the time in a regular 12 hour / duodecimal based clock would be 7:40. With 10 hours instead of 12, each decimal hour on this clock is 72 minutes of regular time. Therefore 6 hours = 432 minutes. Add ~19/50 decimal minutes that are equivalent to ~28/72 regular clock minutes and you end up with 460 minutes after noon/midnight, or about 7:40 🙂.

## ZIP Codes and Zmanim – A Practical Approach

*zmanim*on the west side of the zip code can be quite a bit later than zmanim on the east side of the zip. Recently, Lazer Guttman created an SMS based zmanim service at (914) 409-9394 that provides a warning when

*zmanim*are requested for large zip codes. This approach is probably the best that can be done. I would recommend that any zmanim service that is zip code based (and does not have a map to allow zeroing in to a precise location), use this data to provide a warning whenever the zip codes is wider than 0.5° of longitude. A degree of longitude spans 4 minutes (regardless of the latitude), so half of a zip code with half of a degree would span 2 minutes (one minute east or west of the center). It should be noted that Canadian postal codes are much smaller than zip codes (usually covering one side of a city block), and most likely do not face the same issue. A spreadsheet listing all zip codes with the maximum longitude and latitude distances (in degrees), was generated by Avraham David Gelbfish from OpenDataDE that is based on US Census data. His Python source code is below.

import json import csv jsonfile = open("tl_2019_us_zcta510/out2.geojson") zipcodes = json.load(jsonfile) def getop(geolist, operation, longitude = None, latitude = None): if isinstance(geolist[0], list): answers = [getop(geo, operation) for geo in geolist] for answer in answers: lat, lng = answer if latitude is None: latitude = lat if longitude is None: longitude = lng latitude = operation(latitude, lat) longitude = operation(longitude, lng) return latitude, longitude else: return geolist with open("out2.csv", "w") as csvfile: zwriter = csv.writer(csvfile) zwriter.writerow(["Zip", "Latitude max distance", "Longitude max distance"]) for zipcode in zipcodes["features"]: zip = zipcode["properties"]["ZCTA5CE10"] geometry = zipcode["geometry"]["coordinates"] maxlat, maxlng = getop(geometry, lambda x, y: x if x > y else y) minlat, minlng = getop(geometry, lambda x, y: x if x < y else y) dlat = abs(maxlat - minlat) dlng = abs(maxlng - minlng) zwriter.writerow([zip, dlat, dlng])