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 (1/10th 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
Location7.199° 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)
Lat: 31.7°
2 sec7 sec15 sec26 sec
Lat: 40.1°
2 sec7 sec20 sec38 sec
Lat: 45.5°
3 sec10 sec30 sec90 sec
Lat: 50.05°
3 sec13 sec93 secN/A
Lat: 51.5°
4 sec16 secN/AN/A
Lat: 58.68°
13 sec47 secN/AN/A
Lat: 61.2°
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.
    30 Min7.203°7.199°7.208°7.205°
    50 Min11.432°11.424°11.449°11.442°
    72 Min16.055°16.04°16.092°16.08°
    90 Min19.804°19.784°19.865°19.848°


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 Manat 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 20th back to March 16th (or from September 22nd to the 26th).

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 Manat in his Zmanei Halacha Lema’aseh (4th 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 placesDegreesDistanceNotes
01111 kmA state or small country
10.111.1 kmCity
20.011.11 kmNeighborhood
30.001111 mA specific cul-de-sac
40.000111 mA corner of a house
50.000011.1 mA person in a room
60.00000111 cmA small siddur
70.00000011 cmThe size of Waldo on a page
80.000000011 mmA grain of sand
90.000000001111 μmThe width of a hair
100.000000000111 μmA grain of pollen
110.000000000011 μmA smoke particle
120.000000000001111 nmThe width of a COVID virus
130.000000000000111 nmA red blood cell
140.000000000000011 nmThe length your nails grow every second
150.000000000000001100 pmAn atom. If you need this precision, you probably belong in Lawrence Livermore

Decimal Versus Sexagesimal Based Zmanim Location Errors

Decimal Analog Clock
Converting from one number system to another can be tricky. What time would 6:19 am/pm on this decimal based analog clock be on a regular duodecimal (12 based) clock? See the end of the article for the answer.
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

99557 ZIP code area (the largest in the USA)
99557 ZIP code area (the largest in the USA)
As mentioned in the ZIP Codes and Zmanim – Use With Care article, using ZIP codes to geolocate your position for zmanim can be problematic when the zip code is large. With large zip codes, 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
        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])

The Definition of Chatzos

Arctic Sun
This article was written לז״נ my close friend ר׳ מנחם מענדל בן יחיאל מיכל Menachem Halpert ז״ל. He was never without a smile, was always ready to listen and help, and was a true עניו. Menachem ז״ל was a regular reader of my posts, and in his self-deprecating way always claimed that the contents were over his head, but discussing the subject with him clearly showed a deep understanding. יהא זכרו ברוך.

Possible Definitions of Chatzos

The zmanim of chatzos / חצות are the solar midday and midnight points. The Radak in the ספר השרשים, שרש חצה defines the word chatzos / חצות as splitting the morning from the afternoon (or first half of the night from the second half).
I am aware of three possible definitions of chatzos.

  • When the sun is directly south (or north) of an observer. This is known astronomically as the solar meridian transit. This method is the only one that was measurable (not perfectly) in the time of chazal (and until relatively modern times) using a sundial or the shadow of a perfectly vertical pole with knowledge of the exact point of the cardinal directions. This is the generally accepted definition of solar noon in the astronomical world.
  • When the sun reaches its highest altitude of the day. This is known astronomically as the sun’s upper culmination.
  • Halfway between sunrise and sunset (or variations on this, such as halfway between alos and tzais).

Although these events happen in close proximity to each other, they do not happen at the same time as we will explain in this article. While they generally occur within the span of half of a minute and the difference is therefore halachically almost insignificant, the point of this article is to understand the exact definition of chatzos.

Is Calculating Chatzos As Half of the Day Close Enough?

Currently the KosherJava Zmanim library calculates chatzos as halfway between sunrise and sunset. This was done for simplicity and technical reasons. Rabbi Yehuda Burstein in his זמנים כהלכתם / Zmanim Kehilchasam 7th edition vol. 2 פ״ו ס״ד defines chatzos:

חצות היום הוא הרגע בו השמש נמצאת באמצע הרקיע ממש – בראש גובה קשת מהלכה היומי ממזרח למערב. … חישוב זמן זה הוא פשוט – דהוא במחצית הזמן שבין רגע הנץ המישורי לבין רגע השקיעה המישורית באותו מקום באותו יום.

Chatzos is the moment that the sun is positioned exactly in the center of the sky – at the highest point of its daily east/west path … Calculating this time is simple since it is halfway between the moment of sea-level sunrise and sea-level sunset.

He continues to explain that this half-of-the-day chatzos is not exactly accurate since the time from sunrise to astronomical midday is not equal to the time from astronomical midday to sunset. The difference is minor, and for this reason, most zmanim calendars do not bother calculating the astronomical chatzos.
In an effort to show the actual difference between meridian transit and half of the day chatzos, I graphed the difference between the two chatzos calculations at various latitudes.

Meridian Transit Versus Half of the Day
A chart showing the difference in seconds between meridian transit chatzos and half of the day at different latitudes.

This discrepancy is caused by changes in the length of the day. There are two factors that cause the length of the day to change, the EoT (we covered the equation of time in detail in the earlier FAQ: Chatzos Hayom Versus Chatzos Halayla article) and the change in the amount of daylight time from day-to-day. At the equator, the only change in the length of the day is the EoT, while as you move north of the equator (all examples in this article focus on the spring in the northern hemisphere, though you can mirror it to the other times of the year and southern hemisphere), the length of the day is impacted by the sun’s apparent lengthening path through the day as well (the sun sets farther north), and is above the horizon for more of its day arc. As we move away from the equinox and the day lengthens, the second half of the day is longer than the first half of the day, pushing the halfway point between sunrise and sunset past solar midday. However, even in Gateshead, England (55° N), the difference between the two calculations ranges from -47 to + 36 seconds, not a significant difference. As you get farther north into the Arctic Circle, things start to change more significantly, and that leads to a practical reason to use astronomical chatzos. In the Arctic Circle when the sun does not rise / set, chatzos can’t be calculated as half of the day. However calculating chatzos as the meridian transit can be done without issue because the meridian transit remains the same anywhere along a line of longitude (the algorithm does not factor in the latitude at all). While the impact of the accurate time of chatzos is not something that usually impacts people, in the Arctic Circle with periods of time where there is no day or night on a daily basis, chatzos may be used to define the boundaries of day and night. See the ביאור הגר״א או״ח ס׳ רס״א ב׳ where in the northern regions (including Vilna) chatzos is alos hashachar. In a future post I will discuss changes to the KosherJava Zmanim API to allow calculation of astronomical (local meridian transit) chatzos, and possibly more on the halachic impact of chatzos in the Arctic Circle.

Sun–Meridian Transit Time Versus The Sun’s highest Point (Transit Versus Culmination)

It is interesting to note that the sun is not exactly at its highest point (culmination) at its local meridian transit (when it is at an azimuth of 180° – directly south or 0° – directly north, depending on your location) on most days of the year. This was mentioned over 120 years ago in a paper by D.A. Pio published in the Monthly Notices of the Royal Astronomical Society, Vol. 59, May 1899; p. 513:

The Sun, the Moon, and all the planets culminate out of the meridian.

It is at its maximum difference near the equinoxes when the day arc is changing significantly from day to day. For example at Yerushalayim’s latitude of 31.778°, the sun reaches its highest point on February 20th 11.14 seconds after crossing the meridian. At 60° the difference on March 4th increases to 27.13 seconds.

Culmination Versus Transit
A chart showing how many seconds the culmination happens after the sun’s transit. Courtesy of J. Giesen.
While this seems like a large gap, the actual difference in the sun’s altitude between the transit and the culmination at latitude 31.778° is minimal, about 0.00002° (0.0826 arcseconds), something that can’t be measured without an observatory sized telescope. At 60° latitude, the altitude difference increases to 0.00006° (0.2148 arcseconds), again something almost imperceptible. In Longyearbyen, Norway, latitude 78.22°, the difference between transit and culmination peaks at 73 seconds on March 13. The difference increases significantly farther north. These calculations were done with the help of J. Giesen whose Transit and Culmination article discusses the phenomena in detail. The effect can also be seen using Stellarium.

The cause of the time difference between the two “flavors” of astronomical midday is due to the sun’s increasing declination that causes the sun’s altitude to increase faster than its movement along the day arc moves it down. The farther north you go, the flatter the day arc appears, meaning that it travels much farther horizontally for a small change in altitude. Around midday when the arc is flattest, the change in declination is increasing the altitude immediately after the meridian transit more than the drop caused by the day arc.
At this point you may wonder why chatzos is called chatzos if we define it as one of the astronomical midday definitions that is not exactly half of the day (a case can be made that neither is “half of the day” chatzos, since at the point of splitting it, you have not reached the lengthening full day…). The Radak’s definition of chatzos / chatzi quoted above concludes with:

… כלם ענין חלוקה, בין שיהיו החלקים בשוה או בלא שוה.

… they are all under the category of splitting, whether they are of equal parts or not.

The Radak on Shmuel II, 19:41 restates that chatzi is not always half as is clear from the “half” that is referred to there:

… ולשון חצי ומחצה ומחצית אינו אלא חלוק החלק מהכלל פעמים הוא חלק כחלק בשוה וזהו ברוב ופעמים אינו בשוה.

… the language of half (in its various forms) are only splits of the whole, sometimes it is an equal division, and that is the case most of the time, but sometimes it is not equal.

Note that this concept of chatzi not being half of the day would also seem to counter objections that some have about calculating chatzos from alos to tzais Geonim as an uneven split (regardless, it is very hard to fit such an uneven chatzos into any of the rishonim or achronim we mentioned earlier), though Rabbi Mordechai Kuber pointed out that it is half of the halachic day timewise, so it still fits the definition of chatzos.

Halachic Definition of Chatzos

Rashi in פסחים נח. ד״ה בין הערבים defines the afternoon (starting immediately after the 6th hour or chatzos) as:

משש שעות ולמעלה שהצל נוטה קרינן בין הערבים

From six hours and on, when the shadow leans [to the east], it is called the afternoon.

Rashi has the same definition in Brachos ט. ד״ה שם תזבח את הפסח and כו: ד״ה מנחה גדולה and Shabbos ט: ד״ה מנחה גדולה. Since Rashi defines after chatzos as when the shadows begin leaning, it would indicate that at chatzos shadows do not lean east or west, but point directly north. This seems to be a clear indication that chatzos is defined as when the sun is transiting its local meridian and is directly south at an azimuth of 180° (or directly north or an azimuth of 0° for locations south of the sun, or directly overhead on certain days for locations between the Tropic of Cancer and the Tropic of Capricorn), when there are no east or west shadows. The Tosafos Yom Tov in פסחים פ״ה מ״א expands and clarifies this:

כתב הר״ב דזמן שחיטת תמיד מכי ינטו צללי ערב וכו׳ ואין צל נוטה אלא צל כל אדם תחתיו. רוצה לומר ואין לו נטייה לא למזרח ולא למערב אבל שיהיה תחתיו ממש זהו דבר שהחוש מכחישו בכל אלו הארצות ואף בארץ ישראל. לפי שאין צל כל אדם תחתיו אף באמצע היום. אלא להשוכנים בין עגול סרטן לעגול גדי. ואחת בשנה *) תבא השמש נוכח הראש לקצתם קצתם איש איש ביומו לפי מספר עגולי השמש שתעשה מראשית השנה עד אחרית השנה אבל לשוכנים חוצה לעגולים האמורים לעולם השמש דרומית או צפונית להם ועושה צל בהכרח הנוטה לצפון או לדרום אפי׳ בצהרים והנה ארץ ישראל ידענו כי היא כמו שמנה מעלות לצפון עגול סרטן:

כתב המגיה *) הלשון אחת בשנה אינה מדויקת, כי רק לאלה השוכנים במקום קצות עגולי ההפך סרטן וגדי הרחוקות כל אחת מן קו המשוה כ״ג מעלות וחצי אם לצפון או לדרום רק לאלה תבא השמש נוכח הראש אחת בשנה; לשכנים בקצה גבול עגול ההפך סרטן יהיה זה ביום תקופת תמוז ולשוכנים בקצה גבול עגול ההפך גדי, ביום תקופת טבת. אך לאלה השוכנים בין שתי קצות העגולים האלה תבא השמש נכחם שתים בשנה:

The Raav wrote “The time of the Korban Tamid is from when the afternoon shadows lean etc. and there is no shadow, rather everyone’s shadow is directly underneath him.” Meaning, that the shadow does not project to the east or west, rather that it is directly under him. However, this is something that is not the case in any of our lands, including Eretz Yisrael. This is because a person’s shadow is not under him even in the middle of the day besides for those residing between the Tropic of Cancer and the Tropic of Capricorn … for those residing outside of the Tropics, the sun is always north or south of them [at solar noon] and casts a shadow to the north or south even in the afternoon. We know that Eretz Yisrael is about 8° north of the Tropic of Cancer.

Unlike the Tosafos Yom Tov, Rabbi Avraham Bar Chiyya Hanasi רבי אברהם בר חייא הנשיא in the Sefer Ha’ibur ספר העבור, מאמר אּ׳ שער ט׳ defines chatzos as:

והרגע השני הוא עת היות בחצי השמים מפני שהיא מגעת במקום ההוא אל סוף גבהה ועליותה על המקום אשר זרחה עליו, ומכאן ואילך היא נוטה לערוב

The second moment is the time that the sun is in the center of the sky since at that point it reaches its highest point and its climb from its rising point, and from here forward it begins to dip to the evening.

This definition refers to the sun’s highest point / culmination. Though it does mention the “center of the sky” which can be understood to mean meridian transit, from the ראב״ח’s general context it appears clear that he is referring to the culmination.

I would like to express my thanks to my son Shai for his insights, ideas and calculations, Pinny Markowitz for his debugging work in contrasting the old NOAA implementation of solar noon calculations versus the new implementation, and to members of the Frum software developers #zmanim Slack channel for reviewing and making suggestions that improved this article.