## 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. - These were calculated for the year 2021. There will be slight differences from year to year.

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° |

96 Min | 21.045° | 21.023° | 21.116° | 21.097° |

##### Notes:

^{^}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.^{^}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.^{^}A term coined by astronomers in the 1980s and in “popular” use since ~2006.^{^}It is calculated as if the world had no atmosphere and the radius of the sun is above the horizon.^{^}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}).^{^}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.^{^}Rabbi Meir Pozen in his Kuntres Haneshef and Ohr Meir, 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 170, and Rabbi Aryeh Leib Lipkin in his Ohr Hayom, summary section, no. 9 (page 76).^{^}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.^{^}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.

Has it been conclusively shown that the NOAA algorithm is more accurate than the USNO one (as you mention in this article)?

Hudi,

Gutten Moed. The USNO one is not considered accurate even by the USNO. It is a fixed algorithm year after year. Even with adjustments for leap years, it is not considered accurate. They have since changed their algorithms to use code that is not public since they do not consider their old code to be accurate enough. The NOOA implementation is based on Jean Meeus’s algorithm that is known for accuracy. It accounts for changes from year to year (not just leap-year adjustments). It’s accuracy is considered very good from 1800 – 2100 and sufficiently accurate through the year 3000. The code use by the NOAA implementation and my code uses the less accurate version of the Meeus algorithm code that should be accurate to within 3 seconds. This code is much faster and less processor intensive. Given atmospheric / weather changes from day to day that can cause a minute+ time difference, 3 seconds is more than enough. At some point in the future I may implement the more accurate version to account for the 3 second accuracy, but in reality it would be a waste of time.

Ok, that’s pretty clear then. Thanks a lot, and gut moed!