## S

Fig. 5. a Extreme values, north (N) and south (S), of the geocentric declination of the Moon, 1900 to 2100. Actually, as we have seen in Chapter 1, the latter inclination is itself variable, being affected by an inequality whose period is 173.31 days and whose amplitude is 8'. (There are other, smaller periodic terms). The inclination reaches its greatest value (5 17') when the line of the nodes is directed to the Sun, that is at each conjunction of the Sun with one of the nodes. When 0 is...

## At At T T T

-1.52 -1.00 -0.95 -0.78 -0.77 -0.73 -0.73 -0.00 +0.96 + 1.03 + 1.49 The three Saros series described above all started in the southern hemisphere and ended in the northern one. This is always so when the eclipses take place near the descending node of the Moon's orbit. (All the eclipses of the same Saros series occur near the same node). When, however, the eclipses occur near the ascending node of the lunar orbit, they gradually shift southward in the series. This is the case for the series...

## Info

Now we can start with the calculation. We wish to obtain the opposition place for any arbitrarily given ecliptical (celestial) longitude X. In other words suppose that the Earth and the planet have the same heliocentric longitude X then the planet is in opposition (in longitude) with the Sun, and we wish to find its right ascension a and declination 5 at that instant. This can be performed as follows. The formulae will be given without proof. Radius-vector of the Earth (this is the distance of...

## N

Fig. 38. a The paths of Mercury over the Sun's disk at the November transits from 1953 to 2078. North (N) is at the top. In each case, Mercury enters from the east (left) and leaves the Sun at the western limb (right). Note the periodicities of 6, 7, 13, 33 and 46 years, which are mentioned in the text. The dashed line is the chord described by Mercury at the transit of 2190 November 12, which occurs 217 years after that of 1973 November 10. Fig. 38. a The paths of Mercury over the Sun's disk...

## The equation of time

Former versions of the following text were published in the Dutch journal ttemel en Dampkring, Vol. 68, No. 2, pages 21-27 (February 1970) and in L'Astronomie (Soci t Astronomique de France), Vol. 109, pages 188-193 (June 1995). In ancient astronomy, the word 'equation' designated a correction to be added algebraically to a mean value in order to obtain a true value. For example, Ptolemy as well as Copernicus used an 'equation of the center', a quantity to be added to the mean longitude of a...

## J 8 17 25 42 151 2 15 32 47 79 284

Besides the very rough periodicity of 2 years, corresponding to one synodic revolution of Mars, we have the ever better periods of 15, 32, 47, 79, and 284 years. Note that 47 15 + 32, while the period of 79 years is the sum of those of 32 and 47 years. The following opposition dates of Mars illustrate these periodicities. In the last column, note the jump of 10 extra days between the events of 1412 and 1696, due to the Gregorian calendar reform in A.D. 1582, when it was decided to suppress ten...

## The equation of time in the course of the year

The equation of time slowly varies from one day to the next, and this variation repeats almost identically from one year to the next. It is the sum of two components, corresponding to the two effects mentioned before the first component, due to the fact that the Sun moves in the ecliptic and not along the celestial equator, has a period of six months. It is zero four times a year at the equinoxes and at the solstices. Its extreme values are -9.87 and +9.87 minutes (for the value of the...

## Back to Neptune

Something similar occurs with Neptune. We can say that, in a first approximation, Neptune does not revolve around the Sun, but around the barycenter of the system Sun-Jupiter. Indeed, Jupiter is the planet with the largest mass in the solar system, and the distance Sun-Jupiter (5 astronomical units) is small in comparison to the distance Sun-Neptune (30 AU). Of course, contrarily to the fictive example of Figure 28.a, where A and B had almost the same mass, the mass of the Sun is much greater...

## The duration of the true solar day

As we have seen, the variation of the equation of time causes a variation of the length of the true solar day. With the present values of the obliquity of the ecliptic, the eccentricity of the orbit of the Earth and the longitude of the perigee of the Sun, the true solar day can exceed the mean solar day by up to 30 seconds. This happens near the time of the December solstice. On the other hand, near the equinoxes the true solar day is smaller than the mean one by about 20 seconds. Four times a...

## Using the osculating orbital elements

The peculiar motion of Neptune can also be described by means of the so-called osculating elements of its orbit. Osculating elements at a particular epoch are defined as the elements of an unperturbed elliptical orbit, referred to as the osculating orbit, in which the position and velocity of the planet at the epoch are identical with the actual position and velocity of the planet in its perturbed orbit at the same instant. The osculating elements therefore contain the effects of the...

## Exact commensurability

Besides the above-mentioned commensurabilities, which are only approximate, there exists an exact commensurability between the motions of the satellites I, II and III. In what follows, we will suppose to be on the Sun, so all the phenomena we will describe will concern an heliocentric observer. At the very end of this chapter, we will return to geocentrism. At a certain instant, one of the satellites is exactly in front of Jupiter, in inferior conjunction. We will call this the position u 0 ....

## Occultations of bright stars by planets

Occultations of bright stars by planets, though relatively rare phenomena, have been observed on several occasions. Perhaps the most famous cases were the Venus-Regulus event of 1959 July 7 which was observed in Europe in full daylight (the author saw it easily with a 6-inch refractor), the occultation of Sco by Jupiter and its satellite Io in 1971, and the occultation of e Gem by Mars on 1976 April 8. During the period 1900-2100, there are only fourteen occultations of stars brighter than...

## The declination of Polaris

Polaris, the Pole Star, or a Ursae Minoris, is not situated exactly at the northern celestial pole. As a result of the precession, the star is now drawing closer to the pole or, more precisely, the pole is approaching the star. Table 50. A gives the right ascension and declination of Polaris for some epochs, referred to the mean equator and equinox of that epoch. In the last column the star's north polar distance, 90 minus the declination, is given in arcseconds. In the first column, the...

## Ecliptic and galactic equator

The solstitial points are the points of the ecliptic at celestial longitude 90 and 270 , or at right ascension 6*00 and 18h00m. The center of the solar disk reaches these points at the times of summer and winter solstices. The galactic equator is the great circle on the celestial sphere that passes through the plane of the Milky Way. (The Milky Way system, or Galaxy, is the huge star system of which the Sun is a member). In some modern star atlases, such as Tirion's Sky Atlas 2000.0 (Sky...

## Regions of visibility of solar eclipses

Many astronomical almanacs give for every solar eclipse of the current year a map showing the region of visibility of the event. In this chapter we will describe the various forms such a region can take. Let us first recall some classical concepts. In Figure 11.a the Sun, the Moon and the Earth are represented (not to scale ). ABC is the Moon's umbral cone all points inside this cone see a total solar eclipse (or an annular one in the extension of this cone). JABK is the Moon's penumbral cone,...

## Alpha is not always the brightest

Castor and Pollux, the two brightest stars of Gemini, form a prominent pair, 4Vi degrees apart. Castor, the northern star, is the slightly fainter of the two, although it is designed by the Greek letter a. Gemini is not the only 'exception'. There are no less than 34 constellations, out of 88, where the brightest star does not bear the Greek-letter designation a. Of course, here we are speaking of the apparent visual magnitudes of the stars. For the magnitudes I made use of those given in the...

## The distribution of Easter Sundays

In the Julian calendar, the Christian Easter date has a periodicity of 532 years. This means that after a period of 532 years the Easter Sundays repeat in the same order on the same dates of the year. So for example Table 59. A gives the frequency of the different dates of Easter in the 532-year period. We note that in the Julian calendar the two extreme possible dates, March 22 and April 25, occur four times each in a period of 532 years (second column), so this gives a mean frequency of one...

## Jupiter and triple shadow phenomena

As we have seen in the preceding chapter, simultaneous transits of the shadows of two satellites of Jupiter are reasonably frequent. Simultaneous shadow transits of three satellites, however, are much rarer events. Since satellites I, II and III can never be in conjunction simultaneously, satellite IV is always involved in such triple shadow phenomena. In 1979, the Belgian amateur astronomer and calculator Christian Steyaert calculated all triple shadow phenomena taking place from 1900 to 2100....

## Type II

It often happens that only a part of the lunar penumbra passes over the Earth, although there exists a line of central eclipse. Just as for eclipses of type I, the event is either total, annular, or annular-total. In Figures 11. and 11. the event takes place in the northern part of the Earth. In this case, there exists a southern limit of partial eclipse (M 'Rthe projection of MR on the Earth's surface), but there is no northern limit because the northern edge of the penumbra moves S along the...

## Pollux Delta Sagittarii and the period of 183 years

As was mentioned at the end of the preceding chapter, stars with a celestial latitude larger than 6 36' cannot be occulted by the Moon. The latitude of Pollux (0 Geminorum) is +6 4l'. So this star is just outside this limit and hence cannot be occulted for observers on the Earth's surface. However, due to the star's proper motion and the rotation of the ecliptic, the latitude of Pollux is presently increasing at the rate of +26 per century. Near the year -400, the latitude of the star was +6...

## Y

Fig. 11. y The region of visibility of the partial solar eclipse of2000 December 25, which is visible in the United States. It is clearly seen that the curve 'maximum on the horizon' the curve passing over Vancouver Island doesn't pass over the node of the distorted curve '8', forming a small triangular region (called in the text) north of the Hudson Bay. Fig. 11. y The region of visibility of the partial solar eclipse of2000 December 25, which is visible in the United States. It is clearly...

## The equinoctial and solstitial points and the constellations

It is a well-known fact that the so-called 'signs' of the zodiac those empty, rectangular areas of the astrologers no longer coincide with the constellations of the same name. The reason is the precession, which I already mentioned in the previous chapter. Due to the precession, the shift is approximately one sign every 2000 years. Nowadays, the difference is almost one sign. On September 23, for instance, when the Sun enters the 'sign' of Scorpius (which the astrologers call Scorpio), it is...

## How often are the planets aligned

The following text was originally published in Heelal (Belgium), Vol. 40, No. 10, pages 270-272 (October 1995). Sometimes the following question is posed to an astronomer suppose that at some instant all nine planets (Mercury to Pluto) are exactly aligned with the Sun after what time interval will they again be situated on a straight line More precisely stated if all planets have the same heliocentric longitude, when will they again have the same heliocentric longitude (not necessarily the same...

## Calculation of the equation of time

Because the elements of the Earth's orbit slowly vary, the curve representing the yearly variation of the equation of time will take different shapes in the course of the centuries. How can such a curve be calculated for a given year Here we shall neglect the small irregularities due to the nutation and the planetary perturbations. From what has been said, the equation of time E is obtained by subtracting the right ascension a of the true Sun from the right ascension of the mean Sun, that is,...

## The frequency of total and annular solar eclipses for a given place

The following text was first published in the Journal of the British Astronomical Association, Vol. 92, No. 3, pages 124-126 (April 1982). A classical question concerning solar eclipses is the following one How often can a total or an annular eclipse of the Sun be expected at a given point on the Earth's surface In their classical textbook Astronomy (Boston, 1926), H.N. Russell, R. S. Dugan and J. Q. Stewart write (vol. I, page 227) Solar eclipses that are total somewhere or other on the...

## Occultations of stars by Uranus and Neptune

Occultations of relatively bright stars by Uranus and Neptune are very rare events, because these two planets have a small angular diameter and because they are moving slowly on the starry background. Moreover, few relatively bright stars can be occulted by these planets as seen from the Earth, Uranus and Neptune always remain in a small band situated near the ecliptic. For Uranus, this belt is only 0.08 degree wide for Neptune, the width is 0.12 degree. Of all stars brighter than visual...

## A

In a and in c the eclipsed Sun is up though nothing is seen of the eclipse. In b the eclipse is seen, but the center of the solar disk is under the horizon. The original version of this chapter was published under the title 'Eclipsgebieden' in the April 1971 issue of the Dutch journal Hemel en Dampkring. 22. When is the northern limit the southern one On 1985 November 12 a total solar eclipse took place in the southern hemisphere. The path of totality passed...

## Cm

A mentions all passages of Neptune through the perihelion and aphelion between A.D. 1500 and 2750. Beside the years, the succession of the extreme values of the distance of Neptune to the Sun is indicated. P means the main perihelion, A the main aphelion p and a are the secondary perihelion and aphelion, respectively (p) is a periheloid, and (a) an apheloid. For instance, in 1876 first the principal perihelion was reached then came an apheloid in 1881, and finally in 1886 there was a...

## Ecclesiastical and astronomical Easter

Most amateurs know the rule Easter falls on the Sunday which follows the first Full Moon occurring on or after the day of spring equinox. In fact, according to the ecclesiastical rules, the vernal equinox is fixed at March 21, not by the actual motion of the Sun. Moreover, the date of the Full Moon that occurs on or next after the vernal equinox the Paschal Full Moon is taken from ecclesiastical tables, not from astronomical ephemerides. Hence, the 'ecclesiastical Moon' is not strictly...

## Table 43e

Middle of the series of simultaneous phenomena Time interval between the phenomena of the series The series (f), (g) and (h) occur halfway between two successive instants t0. For instance, in 1994 there were simultaneous shadow transits of I and II from August 11 to September 19. Moreover, we see that the following series occur only once in the course of the period of 437.64 days simultaneous transits of I and II (h), simultaneous eclipses of I and II (c), simultaneous transits of II and III...

## Total and annular solar eclipses in close succession at a given place

For a given place on the surface of the Earth, a total or an annular solar eclipse is a rare event. In the preceding Chapter their mean frequencies were discussed. However, it is not rare at all that the path of one event crosses that of another one after a short time interval. When this happens, people living in the common region of the two paths can experience two total or annular eclipses in an unusually short interval. The table which follows lists all such cases, between the years 1900 and...

## The positions of Uranus Neptune Pluto and Ceres at their discovery dates

The maps on pages 298 and 299 show the positions of the planets Uranus, Neptune, and Pluto, and of the minor planet Ceres among the stars at the time of their discovery. Each map is 8 wide and 4 30' degrees high. North is up, and the right ascensions and declinations mentioned along the borders are referred to the standard equinox of 2000.0. Stars are plotted to visual magnitude 7.7. Uranus was in opposition with the Sun on 1780 December 17, and stationary on 1781 March 2. The planet had just...

## 1

Fig. 29. a The path (dashed line) of the spacecraft Voyager 2 through the solar system. The central star represents the Sun. The smallest circle is the orbit of the Earth. J Jupiter, S Saturn, U Uranus, N Neptune. For each of these planets, the open circle is the position on 1977August 20, when Voyager 2 left the Earth, while the black dots are the positions at the time of the ftyby of Voyager 2. Consequently, the open circles are simultaneous positions, while the black dots are not. Fig. 29.b...

## A periodicity of 179 years

The American spacecraft Voyager 2, launched on 1977 August 20, arrived close to the planet Jupiter on 1979 July 9. Its path was chosen in such a manner that, after being deflected by the gravity of the giant planet, Voyager 2 was directed to Saturn. The craft arrived at this planet on 1981 August 25. Later, it had close encounters with Uranus (1986 January 24) and Neptune (1989 August 25) see the drawing below. This 'Grand Tour' to Neptune, with relatively little fuel on board, and using the...

## The mean frequency yes but

Suppose that the hourly rate of a meteor stream is 60. This means that the mean number of meteors seen by a single observer watching a clear sky is 60 per hour, or one per minute. But we may not expect to observe exactly one meteor every minute Sometimes none will be seen during several minutes, and another time two meteors will appear almost simultaneously. In practice, the conception of 'mean frequency' is meaningfull only when a sufficiently long period is considered. In some cases, the...

## Heliacal risings and settings

We will consider stars which are situated not too far from the ecliptic. Some time after its conjunction with the Sun, such a star reappears in the morning sky the beginning of its new period of visibility. First, we can ask on what date of the year a star and the Sun are rising simultaneously. This is called the cosmic rising of the star. In the case of Aldebaran (a Tauri) and at latitude 45 N, the date is June 8, as we shall later see. Actually, the date is different for other geographical...

## Rounding numbers

Results of measures or of calculations should be rounded correctly and meaningfully, where it is needed. Rounding should be made to the nearest value. For instance, 15.88 is to be rounded to 15.9, or to 16, not to 15. However, calendar dates and years are exceptions. For example, March 15.88 denotes an instant belonging to March 15 it means 0.88 day after March 15, 0 Hence, if we read that an event occurs on March 15.88, it takes place on March 15, not on March 16. Similarly, 1984.69 denotes an...

## March 10 1982 Planetary Alignment

The results of the calculations are given in Table 30. A. A similar list, for the period 0-3000, had been published earlier (Sky & Telescope, Vol. 63, No. 1, page 6 January 1982). But now I have calculated the data again from scratch, the list is extended to A.D. 4000, and my new calculations are based on the VSOP87 planetary theory of Bretagnon and Francou (Bureau des Longitudes, Paris). During the period 0-4000, there are 39 cases when the eight major planets come inside a 90 heliocentric...

## The Moon

In 1991 Michelle Chapront-Touz6 and Jean Chapront, two astronomers at the Bureau des Longitudes, Paris, France, published their Lunar Tables and Programs from 4000 B.C. to A.D. 8000 (Willmann-Bell, ed.). These tables enable the calculation, for any instant over the years -4000 to +8000, of the geocentric position of the Moon (longitude, latitude, distance) the osculating elements of the lunar orbit. The osculating elements are those of the 'instantaneous' orbit of the Moon, that is, the...

## Enzo Mora

Enzo Mora is short for Gian Vincenzo Mora. He was born in Sequals, a small village near Udine, in 1870 into a noble family. Soon interested in art and mathematics, he studied drawing and architecture at the Fine Arts School of Venice. At the same time he studied astronomy and languages. (He had full command of Latin, Greek, English, French, German and Spanish). From 1910 to 1915 he worked as a draughtsman in Padua and then, until 1943 (aged 73 years ), in Sesto S. Giovanni, near Milan, living a...

## Earth

Moon's horizontal parallax x and apparent semidiameter s. In other words, we should have B - 0 < x + s. This is shown in Figure 19.b, where there is just occultation for point P of the Earth. In this case, the difference B j3 is equal to the angle SEM, from which we deduce SEM EMX EMP + PMX EMP + MPL x + J. For the Moon, the mean values are x 57'02.6 and s - 15'32.6, and hence the mean value of the sum x + s is 72.6 arcminutes, or 1.21 degree. Consequently, there is occultation of the star by...

## Some particular dates

Accurate times of the equinoxes and solstices for the years 1 to 3000 are given on pages 101-175 of my Astronomical Tables of the Sun, Moon and Planets (2nd ed., Willmann-Bell 1995). These instants are expressed in Dynamical Time. If we use dates measured in Universal Time, the following facts are found. During the second half of the 21st century and the end of the 22nd, spring equinox sometimes falls on March 19. For the first time since 1796, this will occur in A.D. 2044. In 2007, the equinox...

## Uranus and Neptune

Table 41.H lists all TCs involving the distant planets Uranus and Neptune in the period A.D. 1900-2100. If we don't take Pluto into consideration, the rarest planet-planet conjunctions are those of Uranus-Neptune. The mean interval between successive heliocentric conjunctions of these planets is 171.4 years. Before the TC of 1993 there were TCs in 1478-1479, 1650, and 1821. In the future, there will be a single conjunction in January 2165, and TCs in 2336-2337 and 2508-2509. (In 2164-2165,...

## Triple conjunctions

The outer (exterior) planets usually move eastward against the stars. But for a few months around their opposition with the Sun they appear to change direction and move backward, or 'retrograde', because our moving Earth is overtaking them. This is illustrated in Figures 38.c and 39.c. In the first case, we see that Mars is three times in conjunction with the star Regulus, while in the second case it is three times in conjunction with Spica. These are examples of so-called triple conjunctions...

## Planetary groupings

It may happen that three or more planets are near each other on the sky. Such a grouping is sometimes erroneously called a 'conjunction'. Two celestial bodies are said to be in conjunction if their celestial longitudes or their right ascensions are equal. But three or more planets never can be simultaneously in conjunction. Although this is theoretically possible, in practice the probability of the event is zero. Let us first consider the groupings of three planets, A, B, and C. For any...

## Foreword

Every time two full moons occur in the same month, pundits in the media take note. They explain that, according to folklore, the rare second one is called a blue moon whence the saying, once in a blue moon. However, they have got it backwards, says Philip Hiscock of the Folklore and Language Archive, Memorial University of Newfoundland. The fabled blue moon, meaning any rare or unusual occurrence, dates back at least 150 years in the English language. The link to lunar phases in a calendar...

## Opposition loops

During the weeks around its opposition with the Sun, an exterior planet has an apparent retrograde (westward) motion among the stars. This is caused by the fact that the Earth overtakes and passes the planet, which thus seems to move 'backward*. But since the planetary orbits are not situated exactly in the plane of that of the Earth the ecliptic the motions are not just a simple back-and-forth shift on one and the same line. Instead, they occur along characteristic curves. These elegant loops...

## The halfsaros

The original version of the following text was published in the journal Hemel en Dampkring Netherlands , Vol. 63, pages 141-143 1965 . On pages 181-182 of his Calender ftir Sternfreunde 1965, Dr. Paul Ahnert 1897-1989 draws the attention on a remarkable fact related to the Saros period, which he had never seen mentioned in the literature. As is well known, the Saros is equal to 223 synodic months this is 6585.32 days, or 18 years approximately 10 days. This period is almost equal to 242...

## Quintuplets

Uptain, of Aberdeen, Mississippi, calculated all groupings of two, three, four and five planets taking place from 4000 B.C. to A.D. 2750. He adopted the following criteria as limits for the four types of groupings a smallest distance of 3 for the planetary pairs, a smallest circle with a diameter of 9 for the trios, of 16 for the quadruplets, and of 25 for the quintuplets. For the period 2000 B.C. to A.D. 2750, Mr. Uptain found 86 quintuplets diameter of circle lt...