Tropical year
A tropical year (from ancient Greek τρόπος (tropos) = twist, turn) is, put simply, the time between two identical points in time in the course of the seasons , for example from a spring equinox (beginning of spring) to the next or from a summer solstice (beginning of summer) to the next. The term "tropical" is derived from the relationship to the solstice.
On the one hand, however, the angular velocity with which the direction of the connecting line changes from the earth to the sun is periodically variable according to Kepler's second law . Therefore the direction of the true sun is replaced by the direction of the mean sun , the angular velocity of which does not show any periodic fluctuations. On the other hand, the reference direction, namely the vernal equinox , to which the direction of the sun in the ecliptic is related, also moves . This movement is made up of an even part, the precession , with an angular velocity of around 50 ″ per year and a periodic part, the nutation . The mean direction of the vernal equinox, which only changes with precession, is the mean vernal equinox . The related direction of the mean sun is the mean ecliptical longitude of the sun. According to a more precise definition, a tropical year is the period in which the mean ecliptical longitude of the sun increases by 360 °.
Because of the precession movement of the midspring equinox, a tropical year is around 20 minutes shorter than a sidereal year , which is equal to the duration of one orbit of the central sun relative to the star's background at rest. Since the direction of the sun relative to the other stars is not decisive for the seasons, but rather relative to the spring equinox, the length of a calendar year (which always contains a whole number of days) should on average approximate the tropical year.
At the beginning of 2000 ( epoch J2000.0 ) the length of the tropical year was
 365,242,190.52 days = 365 days 5 hours 48 minutes 45.261 seconds or 31,556,925.261 seconds
Since the precession movement of the middle spring equinox does not take place at a constant angular velocity, but rather slightly accelerated, the length of the tropical year is slightly variable. It is currently decreasing by about half a second per century.
introduction
There are two different definitions of the tropical year:
 The older understands the tropical year as the period between two passes of the sun through the spring equinox . This definition is clearer and more directly accessible to observation. However, it is difficult to give it a clear, universally valid numerical value. The period between two such passes fluctuates by a few minutes from year to year due to the gravitational influences of the planets and the moon, so that a reference value would have to be averaged over a sufficient number of years to determine a reference value. Since the length of the year is also subject to a longterm drift, the result depends on the arbitrary choice of the averaging period. For the arithmetical determination of this year length, the mean orbital elements of the earth can be assumed from which the mentioned disturbances were arithmetically removed. The year length of the solar calendar is based on the tropical year according to this older definition .
 The newer definition refers to the instantaneous rate at which the Sun's mean ecliptical longitude changes with respect to the mean vernal equinox. This speed is not directly observable, but results as a mathematical quantity in the planetary theories. The definition is very abstract, but its numerical value is welldefined and precisely determinable.
Depending on the definition, slightly different numerical values result. According to the old definition, the length of the tropical year is currently approx. 365.2424 days (slowly increasing) and according to the new definition it is currently approx. 365.2422 days (slowly decreasing). There is also often confusion about the two definitions in the specialist literature; the old definition is often given, but the numerical value according to the new definition is given for the year length.
Old definition: return to the vernal equinox
Basics
In a solar calendar , a year is traditionally the period after which the seasons repeat, after which the sun returns to the same point on its apparent annual path around the fixed star sky .
After Hipparchus discovered the precession of the earth , it had become necessary to distinguish whether “the same point” should refer to the fixed star background or to the equinox and solstitial points of the ecliptic (equinoxes or solstices). The period of time that the sun needs to return to the same fixed star (imagined infinitely far away) is the sidereal year with a length of 365 ^{d} 6 ^{h} 9 ^{m} 10 ^{s} . The equinox and solstitial points, on the other hand, migrate along the ecliptic as a result of the precession, in the opposite direction to the apparent annual movement of the sun ( retrograde ) . Since these reference points move towards it, the sun needs about 20 minutes less to e.g. B. to return to the vernal equinox , currently about 365 ^{d} 5 ^{h} 48 ^{m} 45 ^{s} . This period of return to the vernal equinox was called the tropical year (from the Greek trope "reversal", "solstice", because originally it referred to the solstice instead of the spring equinox).
Since the seasons depend on the position of the sun in relation to the equinox and solstitial points (e.g. astronomical spring begins when the sun passes through the vernal equinox), the tropical year and not the sidereal year is decisive for the annual rhythm of life.
Railway disruptions
2000 → 2001  365 ^{d} 5 ^{h} 55 ^{m} 28 ^{s} 
2001 → 2002  365 ^{d} 5 ^{h} 45 ^{m} 26 ^{s} 
2002 → 2003  365 ^{d} 5 ^{h} 43 ^{m} 37 ^{s} 
2003 → 2004  365 ^{d} 5 ^{h} 48 ^{m} 52 ^{s} 
2004 → 2005  365 ^{d} 5 ^{h} 44 ^{m} 47 ^{s} 
2005 → 2006  365 ^{d} 5 ^{h} 52 ^{m} 10 ^{s} 
2006 → 2007  365 ^{d} 5 ^{h} 41 ^{m} 51 ^{s} 
The ancient and medieval astronomers had no reason to doubt that the length of the tropical year defined in this way was always constant . For the measurement it was sufficient, e.g. B. measure the time interval between any two spring equinoxes and divide it by the number of years that have passed. The Newtonian theory of gravity , however, showed that the planet their orbits each other slightly influence. Because of these orbital disturbances , the earth does not always follow its orbit in exactly the same period of time. The table gives some examples of the time between two passes of the sun through the vernal equinox.
It would have been necessary to subtract these disturbances or to form an average over a sufficient number of tropical years in order to obtain a clear numerical value for the length of the tropical year.
Elliptical earth orbit
In addition to these orbital disturbances, there is a second complication that affects the period between the beginning of spring. The earth moves through its elliptical orbit with variable speed . It runs fastest in perihelion and slowest in aphelion . Since the vernal equinox moves towards the sun, the sun has not traversed the entire orbit ellipse when it meets it again: The period between two vernal equinox passages is therefore shorter than the period between two perihelion passages, namely by the time span which the sun would have needed for the section of the orbit that was not traversed. Now the section of track saved is always (almost) the same size, but according to Kepler's Second Law it is run through at different speeds, depending on whether it is in the vicinity of the perihelion or aphelion. Correspondingly, the time saved is shorter (and the tropical year defined in this way longer) or longer (and the tropical year shorter) than the mean. It takes about 21,000 years for the vernal equinox to migrate back from the perihelion via the aphelion to the perihelion; accordingly, the duration of the tropical year defined in this way is subject to a fluctuation of 21,000 years. To get a mean tropical year, one would have to average over 21,000 years. In addition, the amplitude of this oscillation is slightly variable, as the eccentricity of the earth's orbit fluctuates a little.
At present, the time interval between two passages through the vernal equinox (after deducting the abovementioned fluctuations due to orbit disturbances) is 365 ^{d} 5 ^{h} 49 ^{m} 1 ^{s} . It increases by almost 0.9 seconds per century as the vernal equinox approaches the perihelion of the apparent path of the sun .
The length of the tropical year defined as the return to the precessing starting point depends on the position of the starting point in relation to the perihelion. It also follows from this that, in particular, the time intervals between two passages through the autumn point, the summer solstice point or the winter solstice point are each different because they have different positions with respect to the perihelion. The table shows the current distances between two passages through the relevant points (after deducting the path disturbances):
Beginning of Spring → Beginning of Spring:  365 ^{d} 5 ^{h} 49 ^{m} 01 ^{s} 
Beginning of summer → beginning of summer:  365 ^{d} 5 ^{h} 47 ^{m} 57 ^{s} 
Beginning of autumn → beginning of autumn:  365 ^{d} 5 ^{h} 48 ^{m} 30 ^{s} 
Beginning of winter → beginning of winter:  365 ^{d} 5 ^{h} 49 ^{m} 33 ^{s} 
The length of the tropical year according to this definition depends on the arbitrary choice of the vernal equinox as the starting point.
Precession fluctuations
Finally, as a third complication, the speed at which the vernal equinox precesses along the ecliptic is not strictly constant. The precession is caused by the gravitational influence of the moon , sun and planets. The trajectories of the latter, however, are subject to mutual interference and change slightly (a typical multibody problem ), which in turn means that the precession movement is currently slightly accelerated. This effect is discussed in more detail in the next section.
Modern definition: traversing 360 °
Because of the described inadequacies of the earlier definition of the tropical year as the (mean) time interval between two passages of the sun through the vernal equinox, the International Astronomical Union adopted the modern definition in 1955:
The tropical year is the period in which the mean length of the sun increases by 360 ° . 
The length is related to the “mean equinox of the date ”, which slowly moves opposite to the sun due to the precession with respect to the fixed star background . The change in length of exactly 360 ° must therefore be determined in this slowly  but evenly  rotating reference system . Compared to a reference system defined by the fixed stars, the sun then covered only 359 ° 59 ′ 10 ″ in the ecliptic.
Medium length
The ecliptical length of a real celestial body on an elliptical orbit traversed at variable speed can be calculated for any point in time by first determining the length of a fictional celestial body increasing at constant speed on a circular orbit with the same orbital duration and then by adding a relatively easy to calculating correction, the socalled midpoint equation, which receives the length on the elliptical orbit. In the case of higher accuracy requirements, the orbital disturbances caused by other celestial bodies must also be added. The midpoint equation and most perturbations are periodic quantities. Additional terms in , etc. take into account nonperiodic (socalled secular ) drifts, which are also caused by disturbances:
The mean length is defined as the above expression without the ( averaged away ) periodic terms:
Mean length of the earth
The mean longitude of the earth in relation to the mean equinox of the date is given by
The time calculated from epoch J2000.0 in Julian millennia is 365,250 days each, measured in terrestrial time . If the Julian date counted in terrestrial time is the observed point in time, the following applies:
 .
Note that the coefficients in the length formula are partly given in degrees and partly in arc seconds . The formula is applicable for the years −4000 to +8000. After adding 180 °, it also applies to the apparent movement of the sun. The nonlinear terms are mainly generated by the acceleration of the precession already mentioned.
The instant tropical year
The speed at which the mean length changes is obtained by deriving it from time:
On January 1st of the year 2000 at 12 o'clock terrestrial time, i.e. for , the mean length of the Sun changed at a speed of
In order to cover a distance of 360 ° = 1,296,000 ″ at this speed, she needed (while converting Julian millennia into days)
(A more recent value is 365 ^{d} 5 ^{h} 48 ^{m} 45.261 ^{s} for the beginning of 2000.)
Note that this is the time it took for the Sun to travel 360 degrees while maintaining its speed on January 1, 2000. It is not the period after which she has actually traveled 360 ° (this would only be 365 ^{d} 5 ^{h} 48 ^{m} 45.248.085 ^{s} ), because during this year her speed increased slightly because of the secular terms. The length of the year calculated above is just another way of expressing the speed with which the mean length of the sun changes at a given point in time ; it is the socalled instantaneous year length. This is comparable to the statement that a vehicle is moving instantaneously (instantaneously) at a speed of 100 kilometers per hour. In order to be able to provide this information, one does not have to wait until the vehicle has actually covered 100 km. Rather, it means: it would take an hour to travel 100 kilometers if it kept the current speed unchanged. For a real travel distance of 100 km, it can also take less than an hour if the speed increases while driving. Accordingly, the length of the year for the "beginning of the year 2000" was given at the beginning, not as the length of the year 2000 itself.
Since the periodic influence of the orbital ellipticity was "averaged away" at the transition to the mean length and only the current speed is considered anyway, it is no longer important for the definition at which point on the orbit the starting point is located. The modern definition is therefore independent of the vernal equinox.
Variability of the tropical year
In the previous section, the length of the tropical year was derived specifically for January 1, 2000 by calculating the speed with which the mean length changes for the special case . If we keep the general expression for instead , the instantaneous tropical year length for an arbitrary point in time is given by
. 
Is a power series
given, its reciprocal can also be developed into a power series, and it is
 .
For the above expression, the length of the year follows (with simultaneous conversion of Julian millennia into days):
The instantaneous length of the tropical year on January 1, 2000 was 365 ^{d} 5 ^{h} 48 ^{m} 45,250,742 ^{s} , on July 1, 2000 365 ^{d} 5 ^{h} 48 ^{m} 45,248,093 ^{s} and on December 31, 2000 365 ^{d} 5 ^{h} 48 ^{m} 45,245,415 ^{s} . The time it took for the sun to cover a total of 360 °  starting on January 1st at 0 °  was 365 ^{d} 5 ^{h} 48 ^{m} 45,248,085 ^{s} ; this is the mean of the instantaneous years that occurred during this period. This can be compared to the fact that the driving time that a vehicle needs for a certain route results from the mean value of its current speeds over the course of the route. In both cases, the total time required is the reciprocal of the mean value over the current speeds.
In all of these formulas, the day is to be understood as the idealized ephemeris day of the same length , each of 86,400 seconds . For the question of how many real rotations of the earth or how many mean sunny days there are in a tropical year, the fluctuations and the longterm slowdown of the earth's rotation should also be taken into account (see below).
The instantaneous length of the tropical year on January 0, 1900 (= December 31, 1899) 12h UT served as the basis for the definition of the ephemeris second , the forerunner of today's atomically defined second (the International System of Units ).
The length of the tropical year changes because the precession movement of the vernal equinox, which serves as the reference direction, is currently accelerating slightly. The length of the sidereal year related to the fixed star background, however, is only subject to shortterm periodic fluctuations, but no longterm change.
Comparison of definitions
For comparison, the illustration on the right shows the lengths of the tropical years of different definitions over a period of 16 millennia.
The colored curves represent the time it takes for the sun to return to the same reference point on the ecliptic after a full orbit, namely for the reference points spring equinox, summer solstice, autumn equinox and winter solstice. As can be clearly seen, this period depends on the choice of the reference point, but runs through comparable oscillations with an amplitude of just under a minute and a period length of about 21,000 years (after which the precessing reference points again assume the same position with respect to the perihelion).
The gray curve shows the length of the tropical year according to the 360 ° definition. It is independent of reference points and shows only a small fluctuation with a fairly long period, which is related to irregularities in the precession.
Historical development of the measurement
year  use  Year length 

Babylon  365 ^{d} 4 ^{h} 365 ^{d} 6 ^{h} 36 ^{m} 

432 BC Chr.  Metonic calendar  365 ^{d} 6 ^{h} 19 ^{m} 
Callippian calendar  365 ^{d} 6 ^{h}  
200 BC Chr.  Ptolemy  365 ^{d} 5 ^{h} 55 ^{m} 
882  alBattani  365 ^{d} 5 ^{h} 46 ^{m} 24 ^{s} 
1252  Alfonso X (Castile)  365 ^{d} 5 ^{h} 49 ^{m} 16 ^{s} 
1551  Prutenic tablets  365 ^{d} 5 ^{h} 55 ^{m} 58 ^{s} 
1627  Rudolfine tablets  365 ^{d} 5 ^{h} 48 ^{m} 45 ^{s} 
Jérôme Lalande  365 ^{d} 5 ^{h} 48 ^{m} 45.5 ^{s} 
The knowledge that the positions of the sun and thus the seasons repeat in the rhythm of about 365 days comes from prehistoric times. Unfortunately, very vague information about their knowledge of the length of the year has been handed down from the older cultures.
In Babylonian astronomy there was no generally binding numerical value for the length of the year expressed in days. The parameters used in various astronomical calculation systems correspond to lengths of the year between 365 ^{d} 4 ^{h} and 365 ^{d} 6.6 ^{h} .
The Greek astronomer Meton led in 432 BC. BC in Athens a calendar based on the Metonic cycle , which corresponded to a year length of 365 1/4 + 1/76 days. A hundred years later, Callippus modified this cycle by omitting one day in each of four Metonic cycles, thus obtaining the Callippic cycle , which corresponded to a year length of 365 1/4 days.
The earliest recorded description of a determination of the length of the year comes from Ptolemy , who in the Almagest wrote that of Hipparchus in the 2nd century BC. Used methods and observations. The distinction between sidereal and tropical year goes back to Hipparch's discovery of precession . By the latter, Hipparchus understood the period between two corresponding equinoxes or solstices. Hipparchus determined the times of some equinoxes and solstices and compared them with observations made by Meton and Euctemon (5th century BC) and Aristarchus (3rd century BC). He received 365 1/4  1/300 days for the tropical year, which corresponds to about 365 ^{d} 5 ^{h} 55 ^{m} , while the actual value at that time was 365 ^{d} 5 ^{h} 49 ^{m} 9 ^{s} . ^{}^{}^{}^{}^{}^{}^{}
Hipparchus had expressed doubts whether the tropical year really had a constant length. Ptolemy (2nd century AD) determined the length of the year again using the same method, got exactly the same result and saw no reason to doubt the constancy of the year length.
In the year 882 alBattani observed the autumn equinox and obtained from a comparison with an observation handed down by Ptolemy a year length of 365 ^{d} 14 '26 "(in sexagesimal notation), which corresponds to about 365 ^{d} 5 ^{h} 46 ^{m} 24 ^{s} . For other examples of the numerous longitude determinations during the Arab period of astronomy see Al Sufi , Ulug Beg .)
Towards the end of the Middle Ages, inaccuracies in the planet tables of the Almagest and its Arab successors had grown into significant errors, so that a revision of the tables was necessary. The result was the Alfonsine Tables published in 1252 . These tables used a year length of 365 ^{d} 5 ^{h} 49 ^{m} 16 ^{s} .
In 1551, the published by Erasmus Reinhold worked Prutenischen panels that on the heliocentric planetary theory of Nicolaus Copernicus were based. To this end, Reinhold improved the numerical values originally given by Copernicus and used a year length of 365 ^{d} 5 ^{h} 55 ^{m} 58 ^{s} .
Finally, in 1627 Johannes Kepler published his Rudolphine tables . He had compared his own observations with those of the astronomer Bernhard Walther and obtained an annual length of 365 ^{d} 5 ^{h} 48 ^{m} 45 ^{s} .
During the next few centuries almost every astronomer dealt with determining the length of the year. For example, Jérôme Lalande (1732–1807) found 365 ^{d} 5 ^{h} 48 ^{m} 45.5 ^{s} . Lalande also began to pay attention to the celestial mechanical complications in determining the length of the year, namely the movement of the perihelion, the secular acceleration of precession and the orbital disturbances mainly caused by the moon, Venus and Jupiter. In the meantime it had become clear that the times of individual equinoxes or solstices are subject to fluctuations of several minutes because of these influences and that the mere measurement of their time intervals therefore had to lead to different results depending on the observation pairs used.
Only when analytical celestial mechanics had developed sufficiently in the 18th century to derive the subtleties of the sun's mean movement and its temporal variability from the theory of gravity, could the tropical year be defined in a way that was independent of periodic disturbances. Only the secular shortening of the tropical year caused by the acceleration of precession was defined as a property of the same and not factored out; the tropical year was thus seen as changeable over the long term.
In 1840 , J. H. von Mädler gave the (then) current length of the tropical year as 365 ^{d} 5 ^{h} 48 ^{m} 47.5711 ^{s} with a decrease of 0.595 s per century.
UJJ LeVerrier described the current length of the tropical year and its variability
 ,
and S. Newcomb received from his solar theory
In the last two expressions, the time measured from 1900 January 0.5 ephemeris time in Julian centuries is 36525 days each.
According to the planetary theory VSOP 87 , the length of the tropical year is
 .
Measurements are made here in Julian millennia for 365250 days since the epoch J2000.0. A Tag is in the last three formulas each have a Ephemeridentag one whose length mean solar day approximately corresponding to the year 1820th The slow increase in day length should also be taken into account (see next section).
Tropical year and calendar year
Calendars are used for calculating time, but with very different objectives (e.g. for the establishment of religious festivals, for agricultural planning, etc.) and with very different procedures (based on pure observation, based on nonastronomical mathematical cycles, based on astronomically derived mathematical cycles, etc. .). Numerous calendars try to understand the sequence of the seasons by using an arithmetically formulated switching rule to approximate the length of the tropical year by means of a suitable sequence of calendar years of different lengths but each containing whole days. When comparing such switching rules with the astronomical tropical year, the different definitions mentioned above must be observed.
In the Gregorian calendar , the calendar year has an average length of 365 + 1/4  1/100 + 1/400 = 365,242.5 days. In order to determine the error of the Gregorian switching rule , this number is often compared with the length of the tropical year of 365,242.19 ... days taken from tables. The difference is 0.000.31 days per year or one day after about 3200 years. After this period of time, according to the usual argument, the Gregorian calendar will deviate from the tropical course of the year by one day. However, the variability of the tropical year length and the question of the definition of the tropical year to be used are not taken into account.
The argumentation is based on the current numerical value, which corresponds to the 360 ° definition of the tropical year. As explained above, the length of this tropical year is decreasing by about 0.5 seconds per century (gray curve in the adjacent picture). The tropical year, which is already shorter than the Gregorian calendar year, becomes even shorter in the course of the following centuries, so that the error grows faster than expected.
Pope Gregory XIII however, in his own words, had introduced the new switching rule "so that in the future the spring equinox does not deviate again from March 21" (ne in posterum a xii. Cal. April. aequinoctium recedat). According to this, the calendar year should correspond to the period between two passes of the sun through the spring equinox and thus the earlier definition of the tropical year. This period is currently 365,242,375 days (see table above) and is increasing (green curve in the picture opposite). The error of currently only 0.000.125 days per year will therefore continue to decrease in the future, and over several millennia the Gregorian calendar year will be an excellent approximation of the tropical year in the traditional definition.
So far, the slight but continuous slowing down of the earth's rotation has not been taken into account . The calendar counts the real daynight changes, i.e. the mean sunny days that have become longer over the centuries . The formulas given above for the length of the tropical years are based on constant ephemeris days of 86,400 seconds each . If the tropical years are instead measured in mean sunny days, in addition to the previously described variability in the length of the year, there is a continuous apparent shortening of the years (because the time unit now used is constantly expanding). The second picture shows the effect on the length of the year, whereby according to current studies it was assumed that the day length increases in the long term by 1.7 ± 0.05 milliseconds per century. The difference between calendar year and 360 ° year is now increasing even faster. The difference between the calendar year and the vernal equinox year will continue to decrease in the near future, but will already reach a minimum around the year 3000 with approx. 0.000.12 days per year and then increase again.
For the switching rules of other calendars, see the article Leap year .
See also
literature
 KM Borkowski: The Tropical Year and Solar Calendar . In: J. Roy. Astron. Soc. Can. Volume 85, No. 3, 1991, bibcode : 1991JRASC..85..121B
 J. Meeus, D. Savoie: The history of the tropical year . In: J. Br. Astron. Assoc. Volume 102, No. 1, 1992, bibcode : 1992JBAA..102 ... 40M
Individual evidence
Definitions:
 Definitions: (Meeus 2002), p. 359
 Table, distance between two spring beginnings: calculated from the equinoxes in (Meeus 1995)
 Table, duration of the equinox and solstitial years: (Meeus 2002), p. 362
 The IAU 1955 redefined it: (Seidelmann 1992), p. 80
 Formula for the mean length of the earth: (Meeus 2002), p. 360; Original source: (Simon 1994)
 Newer value 365 ^{d} 5 ^{h} 48 ^{m} 45.261 ^{s} : (Bretagnon, Rocher 2001)
 Power series development of the reciprocal of a power series: (Bronstein 1993)
 Graphic, comparison of different definitions: according to orbit elements from (Meeus 2002), chap. 63; Original source (Simon 1994). The 360 ° curve is the reciprocal of the instantaneous speed of the mean length of the sun, expressed in ephemeris days per 360 °. The other curves are the time intervals, expressed in ephemeris days, which the ecliptical length of the sun (calculated from the mean orbital elements mentioned) needs for a full orbit, for the start and end points of the spring equinox, summer solstice, autumn equinox and winter solstice.
Historical development of the measurement:
 History: mainly (Meeus 1992)
 Babylonian year lengths: (Neugebauer 1975), p. 528
 Meton, Kallippos, Hipparch, Ptolemäus: (Ptolemäus 0150), p. 12, 131 ff.
 alBattani: (alBattani 900), p. 42
 Mädler: (Mädler 1852), p. 147
Tropical year and Gregorian calendar:
 Tropical year and Gregorian calendar: (Meeus 2002), chap. 63
 Quote Gregory XIII: (GregorXIII 1581)
 Increase in day length 1.7 ± 0.05 milliseconds per century: (Stephenson 1997), p. 514
Sources used:
 (alBattani 900): alBattani, M .: Zij . ArRaqqah, about 900; Latin translation: CA Nallino: AlBattani sive Albatenii Opus Astronomicum . Milan 18991907; Reprinted by Olms, Hildesheim 1977
 (Bretagnon, Rocher 2001): Bretagnon, P., Rocher, P .: Du Temps universel au Temps coordonnée barycentrique . Découverte, No. 285, pp. 3947 (2001)
 (Bronstein 1993): Bronstein, IN, Semendjajew, KA, Musiol, G., Mühlig, H .: Taschenbuch der Mathematik . Harri Deutsch, Frankfurt / M. 1993, ISBN 381712001X
 (GregorXIII 1581): Gregor XIII: Bulle Inter Gravissimas , Tusculum 1581 ( online source , accessed July 10, 2006)
 (Mädler 1852): Mädler, JH: Popular Astronomy . Carl Heymann, Berlin 1852
 (Meeus 1992): Meeus, J., Savoie, D .: The history of the tropical year , see literature
 (Meeus 1995): Meeus, J .: Astronomical Tables of the Sun, Moon and Planets . WillmannBell, Richmond 1995, ISBN 094339645X
 (Meeus 2002): Meeus, J .: More Mathematical Astronomy Morsels . WillmannBell, Richmond 2002, ISBN 0943396743
 (Neugebauer 1975): Neugebauer, O .: A History of Ancient Mathematical Astronomy . Springer, Berlin 1975, ISBN 354006995X
 (Ptolemäus 0150): Ptolemäus, C .: Almagest . Alexandria, about 150; engl. Translation: GJ Toomer (transl.): Ptolemy's Almagest , Princeton University Press, Princeton 1998, ISBN 0691002606
 (Seidelmann 1992): Seidelmann PK (Ed.): Explanatory Supplement to the Astronomical Almanac . University Science Books, Mill Valley 1992, ISBN 0935702687
 (Simon 1994): Simon, JL et al .: Numerical expressions for precession formulas and mean elements for the Moon and the planets , Astronomy and Astrophysics, vol. 282, pp. 663683 (1994) ( bibcode : 1994A & A ... 282..663S )
 (Stephenson 1997): Stephenson, FR: Historical Eclipses and Earth's Rotation , Cambridge University Press, Cambridge 1997, ISBN 0521461944