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Note that this web page has nothing to do with the Hebrew calendar, it only concerns specific Jewish ritual events that are considered to have some astronomical relevance.
For reviews of the traditional rabbinic sources for the Jewish laws pertaining to the request for rain, please see the following references:
None of the above sources dealt adequately with the relevant astronomy, so my intention here is to focus specifically on the astronomical aspects.
As a Canadian living in a cool climate with rain (or snow) all year round, the timing of all prayers with regard to rain and dew seemed meaningless to me until I spent a year living in Israel in 1977, during which I realized that those prayers perfectly suit the climate in Israel. When Israel prays for rain, the rains come to Israel. When Israel prays for dew, the rains stop in Israel, or so it seemed at the time. In the present era during the summer in Israel rain essentially never falls.
Therefore one might expect that it would be appropriate or at least permissible for those in the diaspora to say the prayer for rain starting on the same date as those living in Israel, specifically having the Land of Israel in mind, but the traditional sources have soundly rejected that idea (see the references cited above).
One might expect that Sh'ela in Israel should begin in the first regular weekday service after the official prayer for rain during the Musaf service on Shemini Atzeret. However, the ancient sages delayed Sh'ela for 11 days until the 3rd of Cheshvan, to allow time for pilgrims to return home after Sukkot. Subsequently Rabban Gamaliel extended the delay to the 7th of Cheshvan, to allow 15 days "for even the tardiest Israelite to reach the Euphrates". Their concern was that rain would turn the roads to mud before the pilgrims could reach their homes.
That traditional delay continues today in Israel, even though the Temple was destroyed and there are no pilgrims anymore. When the Temple is re-built there will be no risk of roads turning to mud because today almost all roads are paved, and anyhow probably all pilgrims will return home safely and rapidly in some sort of vehicle.
Later, for those living in lands outside Israel in Mesopotamia, Syria, Egypt, and "nearby or similar places", the delay was extended to 60 days after Tekufat Tishrei, a term that is loosely translated as the southward equinox.
The Talmud Bavli tractate Eruvin page 56a, in the context of surveying a city to determine its orientation with respect to the 4 cardinal directions prior to setting up an eruv, very briefly mentions that on Tekufat Nisan (northward or spring equinox, near the beginning of the Hebrew month of Nisan) and on Tekufat Tishrei (southward or autumn equinox, in the Hebrew month of Tishrei, near Yom Kippur), Sun rises at the middle of the range of sunrise points during the year and sets at the middle of the range of sunset points. This definition is astronomically quite valid for tropical and mid latitudes on planet Earth, within plus or minus a day or so of the actual equinox, but it's an observational method that doesn't yield a way to compute the moment of any equinox. This method is also rather inconvenient because it requires a very long period of observation (at least from one solstice to the opposite solstice). It isn't mentioned in calendrical contexts in tractate Rosh HaShanah nor in tractate Sanhedrin, which suggests that it may never have been directly used to support the Sanhedrin Calendar Committee's decision to declare a leap year. The insertion of the leap month had to be announced well in advance of the spring equinox, so that pilgrims would know when to begin their journey to arrive in Jerusalem before Passover.
The Talmud didn't specify whether the observer is to note the direction to the upper limb, center, or bottom limb of the solar disk. Atmospheric refraction near both horizons severely limits the accuracy of this observational method, especially on a cold morning with high humidity pressure inversion over the eastern horizon, which can make Sun appear to rise several minutes prematurely (4 minutes earlier per degree of refraction, and as much as +2° of refraction is possible). Observation from an elevated point also introduces errors, making the apparent sunrise earlier and the apparent sunset later. To minimize the error due to atmospheric refraction, the recorded direction should be that of the center of the solar disk at the moment that the bottom limb of Sun is just touching the horizon, and to minimize the error due to elevation, the observation should be made as close to sea level as possible. Refraction variability is less when observing sunrise and sunset at sea rather than on land, because the atmospheric conditions are less variable, but it would be difficult to permanently mark the observed directions at sea!
A comparable method, not mentioned in the Talmud, but requiring only a brief observation period (a few days near the date of an equinox), is to draw a line from the direction of sunrise to the observer, and another line from the direction of sunset to the observer. That day was an equinox day if those two lines connect as one straight line, or if they come closer to doing so than on the prior or next day.
More precisely, the Sun would rise exactly due East only if Earth had no atmosphere, Sun were a point of light instead of a disk, and the equinox moment coincided with the moment of sunrise for the observer's sea-level locale, likewise it would set exactly due West only if the equinox moment coincided with the moment of sunset at the observer's locale. Obviously Sun can never do both on the same day for any given locale, it is rare for the equinox moment to coincide with either sunrise or sunset for any given locale, although for every equinox there is always exactly one meridian of longitude somewhere on Earth that sees sunrise at the moment of the equinox, and at the same moment the antipodal meridian sees sunset. In modern astronomical terminology the date of an equinox could be described as the day when the sunrise azimuth is 90° east of north or when the sunset azimuth is 90° west of north.
If the sunrise and sunset directions are observed at Earth's equator, then the errors due to refraction and elevation will be negligible, because at the equator Sun rises and sets nearly perpendicularly to the horizon. Also, at the equator when it is local apparent noon on the day of an equinox, sunlight will shine straight down any vertical shaft, and no object will cast a shadow, especially when the equinox moment is close to noon. Requiring that observations be made at the equator, however, is not very convenient for inhabitants of the Holy Land!
The direction or azimuth of sunrise or sunset must be distinguished from the angle made by the path of the rising or setting Sun with respect to the horizon, known as the solar parallactic angle. Although Sun rises at the true east direction and sets at the true west direction on the day of an equinox, on that day the solar parallactic angle would be 90° only at the equator.
At non-equatorial latitudes, due to atmospheric refraction near the horizon and the solar semi-diameter, especially at elevations above sea level, the upper solar limb is seen to rise before it reaches the direction observed at the equator (Jean Meeus, Mathematical Astronomy Morsels V, chapter 63, "Where does the Sun rise at the equinoxes?", pages 343-4), and for the same reasons the upper solar limb remains visible as Sun sets until after it passes the direction observed at the equator. The range of variation of sunrise and sunset directions are least at the equator, where they are equal to the Earth equatorial obliquity (axial tilt relative to the plane of Earth's orbit around Sun). At higher latitudes the variations are progressively greater, as shown in the following charts:
At sunrise or sunset only, the solar parallactic angle is given by:
solar parallactic angle at sunrise or sunset = arccosine[ sine( geographic latitude ) / cosine( solar declination ) ]
where all angles are in degrees (Jean Meeus, Astronomical Algorithms, second edition, chapter 14, "The Parallactic Angle", on page 99). The parallactic angle can only be 90° on days when the solar declination equals the observer's geographic latitude, which occurs twice per year for locales between the Tropic of Cancer and Tropic of Capricorn, but never occurs at latitudes further north or south of the tropics. Locales that are on the Tropic of Cancer see a 90° parallactic angle only when sunset or sunrise occurs near the moment of the north solstice, and locales that are on the Tropic of Capricorn see a 90° parallactic angle only when sunset or sunrise occurs near the moment of the south solstice.
The English word "equinox" comes from the Latin for "equal night", because the duration of daytime and night time are approximately equal on the date of the equinox. At both equinoxes the daytime length is actually slightly longer than the length of night, due to atmospheric refraction making Sun appear higher at sunrise and sunset, and due to the approximately 1/2° diameter of the solar disk. Nevertheless, a properly set up sundial will show sunrise at 6 am and sunset at 6 pm on equinox days at equatorial latitudes, which explains why prior to the invention of mechanical clocks the length of day and night were believed to be equal on those days.
At the north solstice the daytime length is maximal in the northern hemisphere, and minimal in the southern hemisphere, and the converse applies to the south solstice.
Saying that the day and night are equal is the same as saying that the daytime temporal hour (sha'ah zmanit) is exactly 60 minutes, when calculated by the method of the Vilna Gaon as (sunset sunrise) / 12.
At Jerusalem on the day of the southward equinox in the Hebrew year 5765 the daytime was 12 hours and 17 minutes, whereas the night time was 11 hours and 42 minutes (this doesn't add up to exactly 24 hours because the seconds were rounded and because it is already changing for the next day). That day was 35 minutes longer than the following night. For Jerusalem the day that actually has equal day and night in the autumn is 8 days later than the southward equinox, and in the spring it is 8 days earlier than the northward equinox.
The number of days that separate the equinoxes from the dates of equal day and night varies with latitude, with the least separation at higher latitudes where the length of the day changes more rapidly in the vicinity of the equinoxes. For example, in Toronto the equal day and night occurs 4 to 5 days after the southward equinox and about 5 days before the northward equinox.
Click here to see charts of the variation of daytime length with latitude throughout the year 133KB. This chart shows that in both the north and south hemisphere the dates of equal length daytime and nightime (where the daytime lengths cross the 12-hour line) are several days prior to the spring equinox and several days after the autumn equinox. The gaps separating the 12-hour days from the equinoxes are unequal and differ between hemispheres, because of differences in Earth's orbital velocity, and are shorter at higher latitudes (further away from the equator), because the length of day changes more rapidly near the equinoxes at latitudes further away from the equator, as shown in the chart.
The calculation of the date to start Sh'ela is based on Amora Shmuel, who took the year as exactly 365+1/4 = 365.25 days or 365 days 6 hours long, which is the same as the mean length of the Julian calendar year. According to astronomical calculations, however, such a long year length was last appropriate more than 100,000 years ago, when Earth was rotating slightly faster. Shmuel who assumed that the intervals between the four Tekufot (equinoxes and solstices) are equal in length, each 1/4 of 365+1/4 days = 91+5/16 days = 91 days 7+1/2 hours. For more information about Tekufat Shmuel, see "Method #1" on my "Rambam and the Seasons" web page.
Astronomically, the actual intervals between the equinoxes and solstices are not equal, and they vary over the centuries, as shown on my web page "The Lengths of the Seasons" at <http://www.sym454.org/seasons/>.
The actual southward equinoctial year length is considerably shorter than 365.25 days, currently around 365.242 days (365 days 5 hours 48 minutes 30 seconds), which is 42 seconds shorter than the mean Gregorian calendar year length of 365+97/400 = 365.2425 days (365 days 5 hours 49 minutes 12 seconds), and it is continuing to get progressively shorter as aphelion slowly advances towards the southward equinox, as explained on the "Lengths of the Seasons" web page at <http://www.sym454.org/seasons/>.
The Tekufat Tishrei according to the calculation of Shmuel drifts about 3 days later on the Gregorian calendar for each elapsed 400 years. Today it is about 13 days later than its original proleptic Gregorian date, for essentially the same reason that the Julian calendar is 13 days behind the Gregorian calendar. The Gregorian calendar itself, however, has drifted almost 3/2 days later with respect to the autumnal equinox, so today Tekufat Tishrei is almost 14+1/2 days late with respect to the actual astronomical southward equinox, and during the 21st century it will continue to be fall behind at a fairly steady rate averaging 11 minutes and 30 seconds later per year, as shown in the chart below (click here or on the chart to open a higher-resolution PDF version 18KB).
The drift of the seasons according to Tekufat Shmuel is contrasted with the more accurate method of Rav Adda bar Ahavah and the solar longitude method of Rambam on my web page entitled "Rambam and the Seasons".
My freeware Windows program, Kalendis, has the built-in ability to generate a Tekufot report for any Hebrew year. Click here to see an example of such a report for Hebrew year 5769, showing astronomical clock times calculated for Israel civil time, along with the traditional Tekufot moments reckoned according to the methods of Amorah Shmuel, Rav Adda bar Ahavah, and Rambam.
Relative to the actual solar cycle, the traditional Hebrew calendar periodically varies over a range of more than 30 days, as shown in the following chart depicting the actual timing of the southward equinox relative to certain events in the month of Tishrei 192 KB:
In the era of Hillel ben Yehudah (Hebrew year 4116, near the left edge of the chart) the southward equinox could land as late as the 27th of Tishrei, but in the present era the latest equinox lands on the 19th of Tishrei. The day below the Hoshanah Rabbah line shown on the chart above is Shemini Atzeret, which falls on the 22nd of Tishrei. In those years in which the autumn equinox falls in the second half of Elul the weather in Israel will already be cold and wet before Sukkot ends.
To establish which years of the 19-year cycle will be cold and wet before Shemini Atzeret, and which will still be warm and dry even for a while after Shemini Atzeret, it is simplest to evaluate the exact repeating 19-year cyclic relationship between Tekufat Adda and the molad of Tishrei:
The relative timing between years holds for any method of reckoning the southward equinox. The nearer the equinox lands to the top of the chart shown above, the colder and wetter the weather will be in Tishrei, and the nearer the equinox lands to the bottom of that chart, the warmer and dryer the weather will be in Tishrei. Therefore, the three years in which the southward equinox always occurs earliest (actually in Elul) and hence have the coldest and wettest weather at Sukkot are years 9, 1, and 12 of every 19-year cycle, especially so for years in which Rosh HaShanah is postponed by two days (such as Hebrew year 5823, a 9th year starting on Thursday, October 5, 2062). Likewise, the three years in which the southward equinox always occurs latest in Tishrei and hence have the warmest and driest weather at Sukkot are years 17, 6, and 14 of every 19-year cycle, especially so for years in which Rosh HaShanah is not postponed (such as Hebrew year 5774, a 17th year starting on Thursday, September 5, 2013).
There is a Jewish tradition of reciting a special blessing for the Sun, known as Birchat haChamah, once every 28 years (Talmud Bavli, tractate Berachot 59b). This period is known as machzor gadol shel chamah (the Great Solar Cycle). According to Tekufat Shmuel, every 28 years the weekday and time-of-day of the spring equinox are the same as they were at Creation, that is exactly at the mean sunset that begins a Yom Rivii. Because of the 28-year intervals, observant Jews consider themselves fortunate to have the opportunity to recite this blessing multiple times in a lifetime. Like Sh'ela, this 28-year cycle is based on the arithmetic of Tekufat Shmuel.
With the 365+1/4 day year of Shmuel, as each year passes, Tekufat Nisan lands 5/4 day later in the week, so after 4 years, known as machzor katan shel chamah (the Small Solar Cycle), it lands 4 × 5/4 = 5 weekdays later. After 7 such sub-cycles it lands again at the beginning of the original weekday that started the 28-year cycle.
The Julian calendar similarly starts again on the same weekday every 28 years, but that's not the whole story. The Julian New Year Day of year 1 AD was on Saturday, but the Julian year also starts on Saturday in the 7th, 18th, and 24th year of each 28-year group, corresponding to intervals of 6, 11, 6, and 5 years between years that start on Saturday.
A given Hebrew year number hYear is the beginning of a 28-year cycle if ( hYear – 1 ) / 28 leaves a remainder of zero. Note that a year is deducted because the first year was year number one, not year number zero. For example, the next such year is 5769, because 5769 – 1 = 5768, into which 28 divides exactly 206 times. The specific moment will be at the beginning of Yom Rivii on the 14th of Nisan 5769, which will start at sunset on the evening of Gregorian Tuesday, April 8, 2009, although the blessing is generally recited during the Wednesday morning prayers. The actual astronomical northward equinox will be on on Gregorian Friday, March 20, 2009 at about 2pm, based on a clock showing the mean solar time at the midpoint between the Nile River and the end of the Euphrates River (about 16 minutes ahead of Jerusalem mean solar time), so the spring equinox according to Tekufat Shmuel will be about 18 days and 10 hours too late.
The year according to Shmuel is 365 days and 6 hours = 365.25 days in length, whereas the mean year of the traditional Hebrew calendar = molad interval × 235 months / 19 years = 365+24311/98496 days = 365 days 5 hours 55 minutes and (25+25/57 seconds or 7+12/19 chalakim) ≈ 365.2468222 days. In other words, the year of Tekufat Shmuel is exactly 4 minutes 34+32/57 seconds = 313/98496 of a day longer than the Hebrew calendar mean year, so its long-term drift rate to dates that average progressively later in the Hebrew calendar year is the inverse of that fraction = 98496/313 = 314+214/313 ≈ 314.6837 years per day of drift. Tekufat Shmuel will eventually drift all the way through the Hebrew calendar and start the drift cycle over again, always >1 year late. To calculate how long that will take, divide the Hebrew calendar mean year by the Tekufat Shmuel yearly drift rate:
= (365+24311/98496 days) / (313/98496 of a day) = 114937+70/313 years.
The following chart depicts a long-term view of Tekufat Shmuel spring equinox moments, with the Birchat haChamah moments highlighted, for traditional Hebrew calendar years 2000 through 6000 (click here or on the chart to open up a higher-resolution PDF version 133KB):
As expected, the spring equinox according to Tekufat Shmuel falls progressively later in Nisan as the years pass, with the average at the start of Nisan in Hebrew year 2000 and at the 13th of Nisan in the present era. The total drift of Tekufat Nisan according to Shmuel from Creation to the present era was 5768 × 313/98496 days ≈ 18+1/3 days later in the calendar year (if you are checking my arithmetic, keep in mind that there is no day zero at the start of Hebrew months). As a consequence, it is becoming progressively less likely for Tekufat Nisan to land in the month prior to Nisan (which can only be Adar Sheini in the present era). Birchat haChamah will fall in Adar Sheini only one more time, in Hebrew year 5993, and the last time that Tekufat Nisan will fall in Adar Sheini will be in Hebrew year 6563.
In the traditional Hebrew calendar Birchat haChamah never lands on day 1, 3, 6, 8, 10, 13, 15, 17, 20, 22, 24, 27, or 29 in Nisan, because those days can never be Wednesday. For the same reason, Birchat haChamah never falls on day 2, 4, 9, 11, 16, 18, 21, 23, or 25 in the month before Nisan.
The zig-zag pattern in the chart above is due to the Rosh HaShanah postponement rules and makes it hard for the eye to discern the underlying pattern. We can remove that effect to reveal the Birchat haChamah pattern by plotting the difference between Shmuel Tekufat Nisan and the traditional molad of Nisan, connecting every 19th year of the Tekufah and every 28th year for Birchat haChamah (click here or on the chart to open up a higher-resolution PDF version 65KB):
Relative to the molad of Nisan, each Birchat haChamah moment is 28 years 9 days 9 hours 56 minutes and 14 parts later except that after every 3rd Birchat haChamah the pattern jumps earlier by one molad interval (29 days 12 hours 44 minutes and 1 part), and after 5 such cycles the Birchat haChamah moment lands in the 8th year of the 19-year cycle (top diagonal line on the chart) and from there takes 4 steps later before the next time that it jumps one molad interval earlier. Thus there is an 84-year cycle of 3 × 28 years, which repeats 5 times and is completed by a group extended to 4 × 28 years, so the pattern repeat interval is 19 × 28 = 532 years. The Birchat haChamah moment for 5769 falls exactly at the average, as it does every 532 years. The last time that Birchat haChamah falls before the molad of Nisan will be in Hebrew year 5993.
The following chart compares the equinox and solstice moments calculated by SOLEX version 9.1 (a "postcard-ware" computer program based on numerical integration, the "gold standard" in celestial mechanics, written by Professor Aldo Vitagliano of the Chemistry Department at the University of Naples, Italy) with Tekufat Shmuel (thick dashed lines) and with Tekufat Adda (thick solid lines) and the solar longitude method of Rambam (thin solid lines), from the traditional year of Creation for the first 10 millennia of the Hebrew calendar. Click here or on the chart to open up a full-page higher-resolution PDF version 43KB (due to limitations of the image generator, the low resolution version has jagged lines for Tekufat Shmuel instead of the intended dashed lines):
My freeware Windows program, Kalendis, has the built-in ability to generate a Tekufot report for any Hebrew year:
Click here to see an example of such a report for Hebrew year 5769, showing astronomical clock times calculated for Israel civil time, along with the traditional Tekufot moments reckoned according to the methods of Amorah Shmuel, Rav Adda bar Ahavah, and Rambam.
Although the seasons of Tekufat Shmuel pass at fixed intervals, it is impossible to quote a constant error rate with respect to the actual mean astronomy, beause the astronomical changes are non-linear, for reasons that are explained on my "Lengths of the Seasons" web page at <http://www.sym454.org/seasons/>.
The Tekufat Shmuel calculation was probably established near the era when the Shmuel minus SOLEX spring equinox difference was closest to zero. That was around Hebrew year 3400, which was well before the era of Shmuel. The back-calculated Traditional Equinox of Creation according to Tekufat Shmuel was astronomically more than 26 days too early. Today Tekufat Nisan of Shmuel is astronomically about 18 days too late. If the original timing at Creation was correct then today we reckon the tekufah more than 26 + 18 = 44 days too late!
The polynomials posted at the NASA Eclipses web site at <http://eclipse.gsfc.nasa.gov/SEcat5/deltatpoly.html> estimate that for the start of Tishrei at the epoch of the Hebrew calendar the approximate Delta T was 1 day 3 hours and nearly 40 minutes, so if the uncertainty is better than 25% then the SOLEX astronomical start of season moments ought to be accurate to within less than 7 hours at the calendar epoch.
Should Jews employ a more accurate method for reckoning the seasons? For the purposes of the request for rain, this is not a calendrical question, it is a ritual matter. The more pressing needs are the reform of the traditional fixed arithmetic Hebrew calendar and molad. Please see "The Seasonal Drift of the Traditional (Fixed Arithmetic) Hebrew Calendar" at <http://www.sym454.org/hebrew/drift.htm> and "Moon and the Molad of the Hebrew Calendar" at <http://www.sym454.org/hebrew/molad.htm>.
Could it be that the calculation for Birchat haChamah was not intended to arrive at the same ecliptic solar longitude as existed at Creation but rather to arrive at the same sidereal solar longitude, when Sun will be at the same position against the background of stars in the zodiac? (Or alternatively that the zodiac will be in the same position as existed at Creation.)
The so-called First Point of Aries was established around 600 BC, but that is not where Sun was at Creation in 3760 BC (no year zero). At the the northward equinox in 3760 BC, Sun was actually about 10+1/2° west of the eastern border of the astronomical constellation Taurus, or about 44° east of the "true" First Point of Aries.
Relative to the mean sidereal solar year the mean year of Tekufat Shmuel = 365+1/4 days = 365 days 6 hours has a deficiency of more than 9 minutes. When averaged over the past 6 millennia using astronomical algorithms the mean sidereal year was about 365 days 6 hours 9 minutes 17 seconds, which would have accumulated a sidereal solar longitude deficiency of about 6 degrees 15+1/2 arcminutes per millennium and in the present era when we say Birchat haChamah Sun would be more than 35+3/4 degrees west of its original position at Creation (Sun moves slightly less than one degree per 24 hours eastward along its path through the middle of the zodiac constellations). This error is almost as inaccurate as the 11 minute excess that 365 days 6 hours has with respect to the present era mean northward equinoctial year of 365 days 5 hours 49 minutes and 0 seconds.
Today the mean sidereal year is usually quoted as about 365 days 6 hours 9 minutes 9+1/2 seconds. It is worth noting, however, that a mean year of 365+10/39 days = 365 days 6 hours 9 minutes 13+11/13 seconds would have accumulated until today a deficiency of less than 12 arcminutes of sidereal solar longitude error since Creation, and is an attractively short cycle.
The Birchat haChamah blessing itself is very short: Baruch attah hashem, elokeinu melech ha-olam, o'seh ma'aseh b'reishit. Blessed are you, hashem our God, King of the Universe, who makes the work of creation.
Although this blessing is known as Birchat haChamah, blessing the Sun, it doesn't actually mention the Sun.
However, there are a variety of traditions that embellish this with various psalms before and after the blessing, making quite a ceremony out of the occasion. Although one might expect that the evening service would be the appropriate time to recite the blessing, because the calculated moment of the tekufah (ignoring any time zone differences) corresponds to sunset at the start of a Yom Rivii, nevertheless the practise is to carry out the ceremony in the morning, preferably when Sun is visible, with the earliest time being the moment when the entire solar disk is above the horizon after sunrise and the latest time being the solar culmination at mid-day. Even though it might be recited only 2 or 3 times in a human lifetime, the Birchat haChamah blessing is not followed by the she'hechiyanu blessing.
The reader may wonder why Tekufat Adda, long known to be more accurate that Tekufat Shmuel, and having a mean year that is exactly equal to that of the traditional Hebrew calendar, was never used for the ritual purpose of Birchat haChamah. The usual reason given is that the Rav Adda's arithmetic is "more complicated" than Shmuel, but that can't be the real reason, because they employ identical calculations, only differing in the assumed length of the mean year and the starting epoch (Tekufat Adda started 7 days later than Tekufat Shmuel). The real reason is that the repeat interval of Tekufat Adda, or the number of years that it will take for Tekufat Nisan of Rav Adda to again land exactly at mean sunset at the beginning of Yom Rivii, as it did in the traditional year of Creation, is "inconveniently" long, being equal to the full repeat cycle length of the traditional Hebrew calendar = 689472 years! It takes "only" 1/7 of that, or 98496 years, for Tekufat Adda to arrive again exactly at mean sunset but one weekday earlier!
On the other hand, relative to the molad, Tekufat Adda has a permanently fixed relationship which repeats exactly in every 19-year leap cycle. For example, at Nisan of year 1 the tradition is that Tekufat Nisan of Adda was exactly 9 hours 35 minutes and 40 seconds before the molad moment (equal to 9 hours and 642 parts), and that relationship recurs in the first year of every 19-year cycle, as shown below (click here or on the image to open a higher resolution PDF version, 19 KB):
To have the original Tekufat Adda minus molad of Nisan difference recur on the original weekday one "only" needs to wait 7 × 19 = 133 years, but even this would be an inconveniently long interval for the purposes of reciting the Birchat haChamah blessing.
Astronomically there does exist a solar activity / sunspot cycle that lasts approximately 22 years, having two alternating solar magnetic half-cycles of approximately 11 years duration. The length of each cycle is currently unpredictable, in fact each half-cycle is only recognized retrospectively upon review of records of sunspot counts and radiation flux measurements, because as each cycle declines there are frequent resurgences of solar activity, see <http://en.wikipedia.org/wiki/Solar_cycle>. As of February 2008, it was thought that Sun had just passed the cycle minimum and had begun a new cycle, and the web page at <http://antwrp.gsfc.nasa.gov/apod/ap080206.html> shows one of the last sunspots of the old solar cycle. Compared to other stars we see in the heavens, however, Sun is exceptionally stable, an attribute that is of critical importance to life on Earth.
In conclusion, although we have plenty of reasons to be infinitely grateful to HaShem for creating a wonderful and stable Sun to keep our home planet bright, warm. and cozy, the traditional celebration of a 28-year solar anniversary has nothing to do with any physical solar cycle, nor any other astronomical cycle, nor any events purported to have occurred at the time of Creation. It's just arithmetic.
Because of doubt, therefore, it might be appropriate for those who insist on saying the Birchat haChamah blessing to immediately also say Baruch shem kvod malchuto l'olam va'ed, "Blessed be the Name of His glorious kingdom for all eternity".
Many authorities have offered opinions as to when the optimal, earliest and latest time to recite the blessing, ranging from sunset at the beginning of yom rivii all the way to any time of day on yom rivii. A commonly held and usually unexplained opinion is that the optimal time is during the third hour of the morning. According to the traditional reckoning of Tekufat Nisan, when it falls at the beginning of yom rivii that first hour starting at sunset was in ancient times known as the hour of Shabbtai (Saturn). This is just the ancient Hebrew name for that hour, it has nothing to do with the astronomical planet that we call Saturn. The hour of Shabbtai recurs every 7 hours throughout each week. Those who hold that Sun must be fully visible when the blessing is recited often also hold that the blessing should be recited during the first hour of the daytime that is an hour of Shabbtai, which is the third hour of the morning, where each hour is 1/12th of the inteval from sunrise to sunset on that day. Click here to learn about the ancient Hebrew hourly mazalot and weekday names 14KB.
Should Jews employ a more accurate method for reckoning the seasons? For the purposes of the Birchat haChamah blessing, this is not a calendrical question, it is a ritual matter. The more pressing needs are the reform of the traditional fixed arithmetic Hebrew calendar and molad. Please see "The Seasonal Drift of the Traditional (Fixed Arithmetic) Hebrew Calendar" at <http://www.sym454.org/hebrew/drift.htm> and "Moon and the Molad of the Hebrew Calendar" at <http://www.sym454.org/hebrew/molad.htm>.
This page updated 14 Tevet 5772 (Traditional) = 14 Tevet 5772 (Rectified) = Jan 7, 2012 (Symmetry454) = Jan 7, 2012 (Symmetry010) = Jan 8, 2012 (Gregorian)