|1.1 Eclipse Seasons|
|1.2 Interval Between Two Successive Lunar Eclipses|
|1.3 Lunar Eclipse Repetition|
|1.5 Gamma and Saros Series|
|1.6 Saros Series Statistics|
|1.7 Saros and Other Periods|
|1.8 Saros and Inex|
|1.9 Saros-Inex Panorama|
|1.10 Secular Variations in the Saros and Inex|
The orbit of the Moon around Earth is inclined about 5.1° to Earth's orbit around the Sun. As a consequence, the Moon's orbit crosses the ecliptic at two points or nodes. If Full Moon takes place within about 17° of a node , then a lunar eclipse will be visible from a portion of Earth.
The Sun makes one complete circuit of the ecliptic in 365.24 days, so its average angular velocity is 0.99° per day. At this rate, it takes 345 days for the Sun - and at the opposite node, Earth's umbral and penumbral shadows - to cross the 34° wide eclipse zone centered on each node. Because the Moon's orbit with respect to the Sun has a mean duration of 29.53 days, there will always be one, and possibly two, lunar eclipses during each 345-day interval when the Sun (and Earth's shadows) pass through the nodal eclipse zones. These time periods are called eclipse seasons.
The mid-point of each eclipse season is separated by 173.3 days which is the mean time for the Sun to travel from one node to the next. The period is a little less that half a calendar year because the lunar nodes slowly regress westward by 19.3° per year.
 The actual value ranges from 15.3° to 17.1° because of the eccentricity of the Moon's (and Earth's) orbit.
The time interval between any two successive lunar eclipses can be either 1, 5, or 6 lunations (synodic months).
Earth will experience 12,064 eclipses of the Moon during the 5000-year period from -1999 to +3000 (2000 BCE to 3000 CE ). As published in in both the Five Millennium Canon of Lunar Eclipses and Five Millennium Catalog of Lunar Eclipses, the distribution of the 12,063 intervals between these eclipses is found in Table 1.
|Table 1 - Interval Between Successive Lunar Eclipses|
Lunar eclipses separated by 1, 5, or 6 lunations are usually quite dissimilar. They are frequently of unlike types (i.e., penumbral, partial, or total) with diverse Sun-Moon-Earth alignment geometries, and with different lunar orbital characteristics (i.e., longitude of perigee and longitude of ascending node). More importantly, these short periods are of no value as predictors of future eclipses because they do not repeat in a recognizable pattern.
A simple lunar eclipse repetition cycle can be found by requiring that certain orbital parameters be repeated. The Moon must be in the full phase with the same longitude of perigee and same longitude of the ascending node. These conditions are met by searching for an integral multiple in the Moon's three major periods-the synodic, anomalistic, and draconic months. A fourth condition might require that an eclipse occur at approximately the same time of year to preserve the axial tilt of Earth and thus, the same season, as well as the distance from the Sun. This last factor controls the apparent diameter of Earth's umbral and penumbral shadows.
The Saros arises from a harmonic between three of the Moon's orbital cycles. All three periods are subject to slow variations over long time scales, but their values as of 2000 CE are:
Synodic Month (New Moon to New Moon) = 29.530589 days = 29d 12h 44m 03s Anomalistic Month (perigee to perigee) = 27.554550 days = 27d 13h 18m 33s Draconic Month (node to node) = 27.212221 days = 27d 05h 05m 36s
One Saros is equal to 223 synodic months, however, 239 anomalistic months and 242 draconic months are also equal (within a few hours) to this same period:
223 Synodic Months = 6585.3223 days = 6585d 07h 43m 239 Anomalistic Months = 6585.5375 days = 6585d 12h 54m 242 Draconic Months = 6585.3575 days = 6585d 08h 35m
With a period of approximately 6,585.32 days (~18 years 11 days 8 hours), the Saros is a valuable tool in investigating the periodicity and recurrence of eclipses. It was first known to the Chaldeans as an interval when lunar eclipses repeat, but the Saros is applicable to solar eclipses as well.
Any two eclipses separated by one Saros cycle share similar characteristics. They occur at the same node with the Moon at nearly the same distance from Earth and the same time of year. Because the Saros period is not equal to a whole number of days, its biggest drawback as an eclipse predictor is that subsequent eclipses are visible from different parts of the globe. The extra 1/3 day displacement means that Earth must rotate an additional ~8 hours or ~120° with each cycle. For lunar eclipses, this results in a shift ~120° west in the visibility zones of each succeeding eclipse. Thus, a Saros series returns to approximately the same geographic region every three Saros periods (~54 years and 34 days). This triple Saros cycle is known as the Exeligmos.
Figure 1 shows the path of the Moon through Earth's shadows and the geographic regions of visibility for six lunar eclipses belonging to Saros 136 from 1932 through 2022. The 1932 Sep 14 eclipse was a large magnitude partial eclipse with a visibility zone centered on eastern Europe and Africa. One Saros period later the eclipse was total (1950 Sep 26) with visibility centered on the Americas. After another Saros interval, the eclipse was a larger magnitude event centered on the Pacific Ocean (1968 Oct 06). Finally, after the lapse of one more Saros period (3 Saros = Exeligmos = ~54.1 years), the zone of visibility returned to eastern Europe and Africa with a deeper non-central total lunar eclipse (1986 Oct 17). The same 54.1-year time interval in geographic visibility repeats for the Americas (1950 and 2004) and for the Pacific (1968 and 2022). The westward migration in the zone of eclipse visibility illustrates the effect of the extra 1/3-day in the Saros period. The southward shift of the Moon's path with respect to the shadow axis is due to the progressive decrease in gamma from 0.4664 (1932) to 0.2570 (2022). During this interval, lunar eclipses in the series change from partial (1932) to non-central total (1950) to central total (2022). Although the Moon's path through Earth's shadows is similar from one eclipse to the next, it is not exact. The Moon-shadow geometry changes slowly as do the characteristics of each lunar eclipse in a Saros series.
Saros series do not last indefinitely because the synodic, draconic, and anomalistic months are not perfectly commensurate with one another. In particular, the Moon's node shifts eastward by about 0.48° with each eclipse in a series. The following narrative describes the life cycle of a typical Saros series at the Moon's ascending node. The series begins when the Full Moon occurs approximately 17° east of the node.
A small fraction of the Moon's disk passes through the northern edge of the penumbra and a small penumbral eclipse occurs. One Saros period later, the lunar trajectory shifts a little further south, the Moon passes deeper into the penumbra (gamma decreases) and a penumbral eclipse of slightly larger magnitude results. After ~10 penumbral eclipses, the first partial lunar eclipse occurs as the Moon's southern limb passes through the umbra. Around 20 more partial eclipses occur as the Moon swings increasingly deeper into the umbra after each Saros and the magnitude of each event grows larger. Finally, the Moon passes completely into the umbra and a shallow total eclipse occurs. Over the course of the next 2 centuries, a total lunar eclipse occurs every 18.031 years (= Saros), as the Moon is displaced progressively southward through the umbral shadow with each eclipse. Halfway through this period, the Moon passes through the center of the umbra producing long total eclipses. The last of ~13 total eclipses in the series takes place just inside the southern edge of the umbra. The Saros series winds down now with another string of ~20 partial eclipses following by a final set of ~10 penumbral eclipses. The last eclipse of the series is a small magnitude penumbral event just inside the southern edge of the penumbral shadow. In all, this typical Saros series produces 73 eclipses spanning nearly 13 centuries.
The scenario for a Saros series occurring at the Moon's descending node is similar except that gamma increases as each successive eclipse shifts the Moon's trajectory farther north of the previous one. The magnitude of the shift in gamma is again tied to aphelion and perihelion.
Because of the ellipticity of the orbits of Earth and the Moon, the exact duration and number of eclipses in a complete Saros series is not constant. A series may last 1,226 to 1,587 years and is composed of 69 to 89 eclipses. A series can begin with 6 to 25 penumbral eclipses, followed by 6 to 24 partial eclipses. During the mid-life of a Saros series, there are 11 to 29 total eclipses. Finally, a series ends with 6 to 24 partial eclipses followed by 7 to 25 penumbral eclipses. At present (2008), there are 40 active Saros series numbered 110 to 149. The number of eclipses in each of these series ranges from 70 to 83, however, the majority of them (80%) are composed of 70 to 73 eclipses.
Historically speaking, the word Saros derives from the Babylonian term "sar" which is an interval of 3600 years. It was never used as an eclipse period until English astronomer Edmund Halley adopted it in 1691. According to R. H. van Gent, Halley "...extracted it from the lexicon of the 11th-century Byzantine scholar Suidas who in turn erroneously linked it to an (unnamed) 223-month Babylonian eclipse period mentioned by Pliny the Elder (Naturalis Historia II.10)."
Gamma changes monotonically throughout any single Saros series. The change in gamma is larger when Earth is near its aphelion (June to July) than when it is near perihelion (December to January). For odd numbered series (descending node), gamma increases, while for even numbered series (ascending node), gamma decreases. This simple rule describes the current behavior of gamma, but this has not always been the case. The eccentricity of Earth's orbit is presently 0.0167, and is slowly decreasing. It was 0.0181 in the year -2000 and will be 0.0163 in +3000. In the past, when the eccentricity was larger, there were Saros series in which the trend in gamma reversed for one or more Saros cycles before resuming its original direction. These instances occur near perihelion when the Sun's apparent motion is highest and may, in fact, overtake the eastward shift of the node. The resulting effect is a relative shift west of the node after one Saros cycle instead of the usual eastward shift. Consequently, gamma reverses direction.
The most unusual case of this occurs in Saros series 13. It began in -2313 with 11 penumbral eclipses, followed by 7 partial eclipses. The series then reverted back to 3 more penumbral eclipse followed by another 11 partial eclipses. The series then produced 13 total, 20 partial, and 8 penumbral eclipses. To understand this odd behavior we must look to gamma. The value of gamma increased positively for the first 15 eclipses of the series. It then reversed direction after the fifth partial and decreased for 5 sequential eclipses while changing from partial back to penumbral. Finally resuming its northward motion, gamma began to increase again and continued to do so for the remainder of the series. Saros 13 produced 73 ending in the year -1015.
Among 204 Saros series examined (-20 to 183), there are many other examples of temporary shifts in the monotonic nature of gamma, although none as unusual as Saros 13. In fact, the first 38 Saros series (series -20 to 17) with members represented in the Five Millennium Canon and Five Millennium Catalog, experience short reversals in gamma. The reversals are short and rarely last for more than four or five eclipses in a series. Some series even have two separate reversals in gamma (e.g., series -19, -15, -14, -12, -11, -5, -1, 4, 7, and 25). The most recent eclipse with a gamma reversal was on 1648 Jan 10 (Saros 138). The next and last in the Canon and Catalog will occur on 2353 Jan 20 (Saros 120). In past millennia, the gamma reversals were more frequent because Earth's orbital eccentricity was larger.
Lunar eclipses belonging to 204 different Saros series fall within the five millennium span of the Canon and Catalog. One series (183) has only two eclipses represented, while two series (-20 and 182) have three eclipses each. Another 81 have a larger, but incomplete, subset of their members included (-18 to 19, 12, 25, 139, 142, and 144 to 181). Finally, 120 complete Saros series are contained within the Canon and Catalog (20 to 23, 26 to 138, 140, 141, and 143).
The number of lunar eclipses in each of these series ranges from 69 to 89. Almost a quarter (22.6%) of the series contain 72 eclipses, another quarter (24.0%) has 73 eclipses, and a sixth (15.7%) consists of 71 eclipses. In other words, nearly 2/3 (62.3%) of all Saros series are composed of 71 to 73 eclipses. If all Saros series with 70 to 74 eclipses are considered, then the percentage jumps to 75.0%. The remaining quarter have either 69 eclipses (Saros 181) or 75 to 89 eclipses.
Table 2 presents the statistical distribution of the number of eclipses in each Saros series. The approximate duration (years) as a function of the number of eclipses, is listed along with the first five Saros series containing the corresponding number of eclipses.
All Saros series begin and end with a number of penumbral eclipses. Among the 204 Saros series with members falling within the scope of the Five Millennium Canon and Five Millennium Catalog, the number of penumbral eclipses in the initial phase ranges from 6 to 25. Similarly, the number of penumbral eclipses in the final phase varies from 7 to 25. The initial penumbral eclipse sequence is followed by a series of 6 to 24 partial eclipses, while the final penumbral eclipses are preceded by a sequence of 6 to 24 partial eclipses. The middle life of a Saros series is composed of total (umbral) eclipses, which range in number from 11 to 29.
Saros 13 is an exception to the normal progression of eclipse types through a series. After beginning with 11 penumbral eclipses followed by 7 partial eclipses, the series reverts back to produce 3 more penumbral eclipses. It then resumes the normal pattern to produce 11 partial eclipses followed by 13 total, 20 partial, and 8 penumbral eclipses. This odd behavior is caused by a reversal in gamma (Sect. 1.5).
Figure 2 presents data on the number of penumbral, partial, and total eclipses in Saros series 1 through 150. The information is plotted in four separate histograms showing the numbers of all eclipses (Fig. 2a), penumbral eclipses (Fig. 2b), partial eclipses (Fig. 2c), and total eclipses (Fig. 2d). Several interesting relationships are revealed in these diagrams. Figure 2a shows that Saros series with large numbers of lunar eclipses tend to cluster into tight groups or 3 or 4 series separated by 18 to 19 series. However, most series are composed of 72 to 73 eclipses as already shown in Table 2.
Figure 2b displays the numbers of both leading (solid) and following (dashed) penumbral eclipses in a series. Note how the two histograms are displaced from each other, but their overlapping sections coincide with Saros series with large numbers of eclipses. Figure 2c shows the numbers of leading (solid) and following (dashed) partial eclipses in a series, which are nearly in phase with each other and peak during periods when the number of all lunar eclipses is near the minimum. Finally, Figure 2d presents the number of total eclipses in each series, which is inversely correlated with the number of partial lunar eclipses.
|Table 2 - Number of Lunar Eclipses in Saros Series|
|Saros Series Numbers||Saros Series|
|70||1244.0||10||126, 147, 148, 163, 166, ...|
|71||1262.1||32||59, 92, 93, 94, 95, ...|
|72||1280.1||46||-4, -1, 17, 20, 34, ...|
|73||1298.1||49||-20, -19, -17, -9, -6, ...|
|74||1316.2||16||-8, -7, 0, 10, 11, ...|
|75||1334.2||4||-18, -16, 9, 48|
|76||1352.2||4||3, 44, 46, 85|
|79||1406.3||4||100, 139, 174, 176|
|82||1460.4||5||63, 103, 119, 121, 138|
|83||1478.4||4||-10, 29, 101, 120|
|84||1496.5||7||-12, 64, 66, 82, 83, 84, 120|
|85||1514.5||6||4, 24, 26, 27, 43, 45|
|86||1532.5||4||6, 8, 47, 65|
|87||1550.5||3||-15, -13, 25|
To generalize these relationships, it appears that Saros series that are rich in the number of total eclipses are also rich in the number of penumbral eclipses, but poor in the number of partial eclipses. Conversely, Saros series poor in total eclipses are also poor in penumbral eclipses, but rich in partial eclipses. Finally, Saros series with large numbers of eclipses are centered on series with large numbers of total eclipses. Figure 2a also shows that the extremes in the minimum and maximum numbers of lunar eclipses in a Saros series is gradually decreasing.
A concise summary of all 204 Saros series (-20 to 183) is presented in:
The number of lunar eclipses in each series is listed followed by the calendar dates of the first and last eclipses in the Saros. Finally, the chronological sequence of lunar eclipse types in the series is tabulated. The number and type of eclipses varies from one Saros series to the next as reflected in the sequence diversity. Note that the tables make no distinction between central and non-central total lunar eclipses. The following abbreviations are used in the eclipse sequences:
T = Total Lunar Eclipse P = Partial Lunar Eclipse N = Partial Penumbral Lunar Eclipse N* = Total Penumbral Lunar Eclipse
The Catalog of Lunar Eclipse Saros Series contains links to 180 web pages, each one listing the details of all eclipses in a particular Saros series:
The numbering system used for the Saros series was introduced by van den Bergh in his book Periodicity and Variation of Solar (and Lunar) Eclipses (1955). He assigned the number 1 to a pair of solar and lunar eclipse series that were in progress during the second millennium BCE based on an extrapolation from von Oppolzer's Canon der Finsternisse (1887).
There is an interval of 1, 5, or 6 synodic months between any sequential pair of lunar eclipses. Interestingly, the number of lunations between two eclipses permits the determination of the Saros series number of the second eclipse when the Saros series number of the first eclipse is known. Let the Saros series number of the first eclipse in a pair be "s". The Saros series number of the second eclipse can be found from the relationships in Table 4 (Meeus, Grosjean, and Vanderleen, 1966).
|Table 4 - Some Eclipse Periods and Their Relationships to the Saros Number|
|Length of Time||Saros Series Number||Period Name|
|1||~1 month||s + 38||Lunation|
|5||~5 months||s - 33||Short Semester|
|6||~6 months||s + 5||Semester|
|135||~11 years - 1 month||s + 1||Tritos|
|223||~18 years + 11 days||s||Saros|
|235||~19 years||s + 10||Metonic Cycle|
|358||~29 years - 20 days||s + 1||Inex|
|669||~54 years + 33 days||s||Exeligmos (Triple Saros)|
A number of different eclipse cycles were investigated by van den Bergh, but the most useful were the Saros and the Inex. The Inex is equal to 358 synodic months (~29 years less 20 days), which is very nearly 388.5 draconic months.
358 Synodic Months = 10,571.9509 days = 10,571d 22h 49m 388.5 Draconic Months = 10,571.9479 days = 10,571d 22h 55m
The extra 0.5 in the number of draconic months means that eclipses separated by one Inex period occur at opposite nodes. Consequently, an eclipse occurring in the northern half of Earth's shadows will be followed one Inex later by an eclipse occurring in the southern half of Earth's shadows, and vice versa.
The mean time difference between 358 synodic months and 388.5 draconic months making up an Inex is only 6 min. In comparison, the mean difference between these two cycles in the Saros is 52 min. This means that after one Inex, the shift of the Moon with respect to the node (+0.04°) is much smaller than for the Saros (-0.48°). While a Saros series lasts 12 to 15 centuries, an Inex series typically lasts 225 centuries and contains about 780 eclipses.
Van den Bergh placed all 5,200 lunar eclipses in von Oppolzer's Canon der Finsternisse (1887) into a large two-dimensional matrix. (Von Oppolzer did not include penumbral lunar eclipses in his Canon der Finsternisse.) Each Saros series was arranged as a separate column containing every eclipse in chronological order. The individual Saros columns were then staggered so that the horizontal rows each corresponded to a different Inex series. This "Saros-Inex Panorama" proved useful in organizing eclipses. For instance, one step down in the panorama is a change of one Saros period (6585.32 days) later, while one step to the right is a change of one Inex period (10571.95 days) later. The rows and columns were then numbered with the Saros and Inex numbers.
The panorama also made it possible to predict the approximate circumstances of lunar (and solar) eclipses occurring before or after the period spanned by von Oppolzer's Canon. The time interval "t" between any two lunar eclipses can be found through an integer combination of Saros and Inex periods via the following relationship:
t = a * i + b * s (1) where t = interval in days, i = Inex period of 10571.95 days (358 synodic months), s = Saros period of 6585.32 days (223 synodic months), and a, b = integers (negative, zero, or positive).
From this equation, a number of useful combinations of Inex and Saros periods can be employed to extend von Oppolzer's Canon from -1207 back to -1600 using nothing more than simple arithmetic (van den Bergh, 1954). The ultimate goal of the effort was to a produce an eclipse canon for dating historical events prior to -1207. Periods formed by various combinations of Inex and Saros were evaluated in order to satisfy one or more of the following conditions:
No single Inex-Saros combination meets all three criteria, but there are periods that do a reasonably good job for any one of them. Note that secular changes in the Moon's elements cause a particular period to be of high accuracy for a limited number of centuries. The direct application of the Saros-Inex panorama allows for the determination of eclipse dates in the past (or future); however, the application of the longer Saros-Inex combinations permit the rapid estimation of a number of eclipse characteristics without lengthy calculations. Table 5 lists several of the most useful periods.
|Table 5 - Some Useful Eclipse Periods|
(Inex + Saros)
|Heliotrope||58i + 6s||1,787||Geographic longitude of eclipse|
|Accuratissima||58i + 9s||1,841||Geographic latitude of eclipse|
|Horologia||110i + 7s||3,310||Time of ecliptic conjunction|
Modern digital computers using high precision solar and lunar ephemerides can directly predict the dates and circumstances of eclipses. Nevertheless, the Saros and Inex cycles remain useful tools in understanding the periodicity and frequency of eclipses.
Because of long secular variations in the average ellipticity of the Moon's and Earth's orbits, the mean lengths of the synodic, draconic, and anomalistic months are slowly changing. The mean synodic and draconic months are increasing by approximately 0.2 and 0.4 s per millennium, respectively. Meanwhile, the anomalistic month is decreasing by about 0.8 s per millennium.
Although small, the cumulative effects of such changes has an impact on both the Saros and Inex. Table 6 shows how the number of draconic and anomalistic months change with respect to 223 synodic months (Saros period) over an interval of 7000 years. Of particular interest is the last column, which shows the mean shift of the Moon's node after a period of 1 Saros. It is gradually increasing, which means that the average number of eclipses in a typical Saros series is decreasing. This explains the trend in the number of lunar eclipses seen in Figure 2a.
|Table 6 - Number of Anomalistic and Draconic Months in 1 Saros|
(after 1 Saros)
Table 7 shows how the number of draconic months is changing with respect to 358 synodic months (Inex period) over a 7000-year interval. The mean shift in the lunar node after 1 Inex is much smaller than the Saros and is gradually decreasing. This explains why the lifetime of the Inex is so much longer than the Saros and is still increasing.
|Table 7 - Number of Draconic Months in 1 Inex|
(after 1 Inex)
Although the Inex possesses a long lifespan, its mean duration is not easily characterized because of the decreasing nodal shift seen in Table 7.
If the instantaneous mean durations of the synodic and draconic months for the years -2000, +2000, and +4000 are used to calculate the mean duration of the Inex, the resulting lengths are about 14,500, 26,600, and 51,000 years, respectively (Meeus, 2004a).
The information presented on this web page is based on material originally published in Five Millennium Canon of Lunar Eclipses: -1999 to +3000 and Five Millennium Catalog of Lunar Eclipses: -1999 to +3000.
Permission is freely granted to reproduce this data when accompanied by an acknowledgment:
"Eclipse Predictions by Fred Espenak (NASA's GSFC) and Jean Meeus"
Espenak, F., and Meeus, J., Five Millennium Canon of Lunar Eclipses: -1999 to +3000 (2000 BCE to 3000 CE), NASA Tech. Pub. 2006-214172, NASA Goddard Space Flight Center, Greenbelt, Maryland (2009).
Espenak, F., and Meeus, J., Five Millennium Catalog of Lunar Eclipses: -1999 to +3000 (2000 BCE to 3000 CE), NASA Tech. Pub. 2008-214173, NASA Goddard Space Flight Center, Greenbelt, Maryland (2009).
Gingerich, O., (Translator), Canon of Eclipses, Dover Publications, New York (1962) (from the original T.R. von Oppolzer, book, Canon der Finsternisse, Wien, ).
Meeus, J., Mathematical Astronomy Morsels III, Willmann-Bell, pp. 109-111, (2004).
Meeus, J., Grosjean, C.C., and Vanderleen, W., Canon of Solar Eclipses, Pergamon Press, Oxford, United Kingdom (1966).
van den Bergh, G., Periodicity and Variation of Solar (and Lunar) Eclipses, Tjeenk Willink, and Haarlem, Netherlands (1955).
von Oppolzer, T.R., Canon der Finsternisse, Wien, (1887).