Lunar and planet motions eclipses

The bodies in the solar system are quite close to us compared to the stars; thus they have substantial angular velocities relative to the earth. Their apparent positions on the celestial sphere change radically on an annual scale and by significant amounts day by day. The celestial motions of these bright objects have long fascinated humankind.

Eclipses of the sun and moon

The motions of the sun and moon on the celestial sphere lead to solar and lunar eclipses. The former occurs when the moon comes between the earth and sun and blocks, at least partially, the observer's view of the sun. The latter occurs when the moon is on the far side of the earth from the sun and enters the shadow of the earth.

"Orbits" of the moon and sun

The moon orbits the earth every 27.32 d (the tropical month, the time it takes between south-to-north crossings of the celestial equator, i.e., equinox to equinox. The sidereal period (relative to the stars) is nearly identical to this, within 10-4 d. Thus the moon changes its position on the celestial sphere a great amount each day, 360°/27.3d = 13.2°/d. This large night-to-night motion is ~25 moon diameters and hence is quite dramatic. Watch it move through the stars hour by hour or night to night. When the moon is at its closest to the sun (on the celestial sphere), the sun is far beyond it and illuminates only its far side. This is the new moon. We can see it a couple days later in the evening sky as a narrow sliver when it has moved away from the sun. The period from new moon to new moon, the synodic month, is somewhat longer, 29.53 d, because the moon has to move an additional distance each orbit to catch up to the sun which itself moves through the sky by ~30° in a month.

The orbit of the moon is inclined 5° 08' 43'' from the plane of the ecliptic (Fig. 5), and this value oscillates a small amount (±9' with a period of 173 d) due to torques arising from the earth's equatorial bulge and the sun. These torques (primarily that due to the sun) also cause the moon's orbital angular-momentum vector to precess about the normal to the ecliptic with a period of 18.6 yr. (The underlying physics is similar to that we used for the precession of the earth.)

The moon's orbit is also quite eccentric; the angular diameter of the moon decreases by about 12% from perigee (closest point to earth) to apogee (farthest point). Recall that the solar size varies 3% due to the earth-orbit eccentricity. The above mentioned torques cause the perigee to advance eastward with a period of 8.85 yr relative to stars. Referring to the line connecting the apogee and perigee, this motion is called the advance of the line of apsides.

For these reasons, the path of the moon through the stars, as viewed from the earth, gradually changes from month to month and year to year. Lunar eclipses of radio and x-ray sources that happen to be in the path of the moon have been used to obtain their precise celestial positions by observing the exact times of the occultation and reappearance of the star.

As it orbits the earth, the moon's track on the celestial sphere comes into the vicinity of the sun every 29.53 d. But since the moon's orbit (an approximate great circle) is tilted 5° from the sun's orbit (the ecliptic, also a great circle), the moon is usually well above or below the sun. However, the (apparent) annual solar motion around the ecliptic ensures that the sun will cross the moon's orbit twice a year, and an eclipse could take place.

This is illustrated in Fig. 5 where the great-circle tracks of the sun and moon on the celestial sphere are shown. As the sun moves along the ecliptic, it will pass through the intersections of the two orbital planes, called the nodes at ~6-month intervals.

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Figure 4.5. Tracks of moon and sun around the celestial sphere showing the eclipse limits. The orbits are inclined to one another by only 5° so an eclipse occurs on the earth each time the sun is within the ~35° region surrounding the node (intersection of the orbits). The sun (open circles) passes through the region slowly and the moon (dark circles) passes through rapidly. The nature of the eclipse depends on how far from the node the overlap occurs. The moon's orbit precesses with a period of 18.6 yr so that the node regresses 19.4° per year, leading to solar crossings at 173-d intervals. [Adapted from Littman and Wilcox, Totality, University of Hawaii Press 1991, p. 12]

Figure 4.5. Tracks of moon and sun around the celestial sphere showing the eclipse limits. The orbits are inclined to one another by only 5° so an eclipse occurs on the earth each time the sun is within the ~35° region surrounding the node (intersection of the orbits). The sun (open circles) passes through the region slowly and the moon (dark circles) passes through rapidly. The nature of the eclipse depends on how far from the node the overlap occurs. The moon's orbit precesses with a period of 18.6 yr so that the node regresses 19.4° per year, leading to solar crossings at 173-d intervals. [Adapted from Littman and Wilcox, Totality, University of Hawaii Press 1991, p. 12]

The nodes will slowly drift westward along the ecliptic because of the 18.6-yr precession of the moon's orbit, called the regression of the nodes. This causes the interval between node crossings to be a bit shorter than 6 months, namely 173.3 d. At the nodes, the moon, in principle, could come into the line of sight to the sun. This would result in an eclipse of the sun.

Total and partial solar eclipses

A solar eclipse requires that both the sun and the moon arrive at the node at nearly the same time. This coincidence need not be precise because (i) the two orbits are tilted only 5° from one another, and (ii) the observer can be positioned anywhere north or south on the earth to bring the sun and moon into sufficient alignment (with the help of parallax) to yield at least a partial eclipse wherein the moon covers only a part of the sun. It thus turns out that the sun is susceptible to an eclipse (total or partial) somewhere on the earth for 30 to 37 d while it is in the vicinity of a node. Since the moon takes only 29.5 d to complete its cycle, it will always come through this region while the sun is sufficiently close to the node to be eclipsed. Thus an eclipse somewhere on the earth must occur every —173 d, the time between node crossings by the sun.

A total solar eclipse is the most dramatic because the sun's disk is completely blocked by the moon. This requires the earth-moon-sun alignment to be relatively precise. Also, the moon's angular diameter must be greater than that of the sun which is sometimes not the case because of the elliptical orbits of the moon and earth. Partial solar eclipses are more common. About 30% of solar eclipses are called "total" because the sun is totally eclipsed somewhere on the earth. In fact, a small zone of totality sweeps across the earth creating a "track" of totality. The sun will be partially eclipsed in adjacent regions. The other eclipses are either "partial" or "annular". In the former, no place on earth experiences totality. In the latter, the moon is directly in front of the sun but smaller in angular diameter, for some earth observers. Thus a thin ring of sun surface is seen surrounding the blackness of the moon.

The 18-year saros cycle Since ancient times, it has been noted that a lunar eclipse will always be followed by a similar eclipse 6585.3 d later (18 yr 11.3 d if there are 4 leap years), and later it became known that solar eclipses also recurred with this same interval. Any solar eclipse will surely be followed 6585.3 d later by another of similar degree of occultation and duration. This is due to the approximate coincidences of multiples of four periodicities, each of which yield an interval near 6585 d as follows:

(i) 6585.32 d from 223 cycles of the moon's 29.530 59 d motion from new moon to new moon (synodic month). This establishes the exact interval between the two adjacent eclipses of the series (called the saros).

(ii) 6585.78 d from 19 annual cycles of the sun's 346.620 06-d motion from node to node (2 x 173.3 d = eclipse year). This places the sun at the time of an eclipse, see (i), very close (about 1/2 day or 1/2 degree) from where it was relative to the node at the time of previous eclipse. The eclipse will thus be of similar kind (partial or full) as the previous one; see Fig. 5.

(iii) 6585.54 d from 239 cycles of the moon's orbital period in its elliptical track from perigee to perigee (anomalistic month = 27.554 55 d); the perigee is the time the moon is closest to the earth in its eccentric orbit. This condition ensures that, at the time of the eclipse, see (i), the moon will be almost at the same place in its elliptical orbit and hence of about the same angular size viewed from the earth. This is another component that contributes to the similarity of two adjacent eclipses in the series. If one eclipse is a total eclipse, the next is likely be total also.

(iv) 6574.67 d from 18 cycles of the earth from perihelion to perihelion in its orbit about the sun (anomalistic year = 365.259 64 d). This is similar to (iii); it ensures the sun will be about the same angular size viewed from the earth as for the adjacent eclipse in the series. This too contributes to the similarity and duration of the two eclipses.

These numerical coincidences are not perfect so the eclipses will slowly evolve in character. A series of such eclipses (saros) will begin with a brief partial eclipse with the moon barely touching the sun, and observable only from the Arctic (or Antarctica). Gradually, the eclipses of the series move southward (or northward), become more and more complete until they become total or annular (or both, sequentially). Thereafter (possibly after ~20 total or annular eclipses), the eclipses become partial again with gradually decreasing degree of occultation. Eventually, after about 75 cycles or ~1350 yr from the start, the last minimal partial eclipse of the saros appears near the other pole of the earth. With this the saros dies; there are no more eclipses in the series. At any time, there are ~38 saros in progress, one for every node crossing at 173.3 d intervals during the 18-yr period.

The total eclipse of 1919 was used to demonstrate that Einstein's prediction of the bending of starlight was quantitatively correct. Although there is no physical connection between the eclipses in a given saros cycle, much was made of the fact that the 1991 July 11 eclipse belonged to the same saros as the 1919 eclipse. Both were total eclipses of rather long duration (almost 7 min at the center of the shadow track) because the moon's angular diameter was large and sun's was small on those dates. That is, the moon was near perigee (closest point to the earth) and the earth near aphelion (most distant from the sun). At the times of other contemporary eclipses in this saros, e.g., 1919, 1937, 1955, 1973, 1991, and 2009, the earth and moon are at similar positions in their orbits, so these eclipses are all total and of similarly long duration.

Finally, one should note that the earth spin period (1.0 d) is not commensurate with the 6585.3 d saros period; the earth will have rotated 0.3 revolution beyond its original orientation at the time of the previous eclipse of the series. The eclipse track on the earth's surface is displaced westward ~1/3 of the way around the earth, and a bit south or north, compared to the track of the previous eclipse in the saros. This and the limited width of the eclipse track made it more difficult to discover the 18-yr recurrence cycle of solar eclipses than was the case for lunar eclipses (see below).

Wonder and science

Total eclipses are a wondrous experience. During the eclipse, one can see with naked eye the solar corona extending several solar diameters beyond the (covered) solar disk. Beautiful red prominences can sometimes be seen with the naked eye. These are giant loops of hot gas emitting the red Balmer line of hydrogen. The shadow of totality on the earth at one instant is quite small, less than 300 km in diameter. It can be viewed only by those who are along its track across the earth's surface. Astronomers have a history of transporting equipment to remote places with great difficulty in order to carry out studies of the sun during a total solar eclipse.

Eclipses provided astronomers the opportunity to study the chromosphere and corona of the sun. For example, the discovery of highly ionized iron indicated that the corona is exceedingly hot, >1 x 106 K. This was surprising since the photosphere is only 5800 K. The light we see from the corona is light from the photosphere of the sun, its visible surface, scattered toward us by either the energetic electrons of the ionized hot gas (the inner K corona), or by "dust" grains in the solar system (the outer F corona). Dust grains consist mostly of ice, silicates, and graphite and are also found in interstellar space.

The shadow of the total solar eclipse of 1991 July 11 passed directly over the mountain-top observatory of Mauna Kea, Hawaii. This allowed astronomers to use major telescopes for observations to study various aspects of solar energetics, e.g.,the mechanism by which the corona becomes so hot, most likely, the release of magnetic energy by annihilation or twisting of magnetic fields. The relation between the various components of the solar surface and atmosphere that appear at radio, optical and x-ray wavelengths could also be studied.

A note about safety: one must use a safe sun filter when observing the disk of the sun before and after totality, by naked eye and especially with binoculars or telescope. But during totality, when the disk is completely covered, it is perfectly safe to use the naked eye with or without optical aids. The corona is roughly 106 times fainter than the solar disk. With a sun filter you would see nothing at all during totality!

Corona in x rays and visible light An x-ray photo of the sun was obtained during the 1991 July 11 eclipse from a rocket launched at White Sands, New Mexico (USA) exactly when the sun was totally eclipsed in Hawaii (Fig. 6a). At this time in New Mexico, the (partial) eclipse had not quite started. The looming and approaching shadow of the moon is seen to the right at about 4 o'clock; the x rays can not penetrate the moon. The bright features in the figure are typically located at active regions where sunspots (in optical light)

Figure 4.6. (a) X-ray image of sun taken from a rocket flown from White Sands, New Mexico at the time of the 1991 July 11 total eclipse in Hawaii, showing hot plasmas above active regions. The approaching moon shadow is seen at about 4 o'clock. (b) Composite of the x-ray image and optical images taken at the same time during totality with the CFH Telescope on Mauna Kea in Hawaii. It shows the x-ray features within the solar disk and optical features outside it. The extended corona is seen in photospheric light scattered toward the telescope by coronal electrons and dust. The outer ring is an artifact of the graded filter used to bring out the features of the outer (and fainter) portions of the corona. [L. Golub and S. Koutchmy/NASA/CFHT]

Figure 4.6. (a) X-ray image of sun taken from a rocket flown from White Sands, New Mexico at the time of the 1991 July 11 total eclipse in Hawaii, showing hot plasmas above active regions. The approaching moon shadow is seen at about 4 o'clock. (b) Composite of the x-ray image and optical images taken at the same time during totality with the CFH Telescope on Mauna Kea in Hawaii. It shows the x-ray features within the solar disk and optical features outside it. The extended corona is seen in photospheric light scattered toward the telescope by coronal electrons and dust. The outer ring is an artifact of the graded filter used to bring out the features of the outer (and fainter) portions of the corona. [L. Golub and S. Koutchmy/NASA/CFHT]

are present. The x-ray emission indicates the presence of extremely hot coronal plasmas (~106 K), which are confined and guided by magnetic fields to form loops and other structures.

An optical photograph obtained during totality with the Canada-France-Hawaii Telescope at Mauna Kea is combined with the x-ray image in Fig. 6b. The optical contribution to Fig. 6 shows the extended corona reaching out to several solar radii and a small "prominence" at about 3 o'clock. This prominence was easily visible with the naked eye as red light from the hydrogen Balmer Ha line. The coronal configuration varies in appearance from eclipse to eclipse; in this case, it extends approximately north-south (up-down).

The optical exposure contributes nothing to the disk of the sun in Fig. 6b because the moon covers it; it is totally black. Note the black circle of the moon shadow surrounding the disk of the x-ray image in Fig. 6b. The x-ray contribution in contrast shows bright features across the face of the sun and around the edges just above the surface. Thus in this picture, one sees the corona, mostly transversely in the optical and mostly face-on in x rays, in a three-dimensional perspective reaching nearly down to the photosphere.

Lunar eclipses

Fifteen days before or after a solar eclipse, the moon has moved to the opposite side of the earth, almost directly opposite the sun. Since the earth-moon-sun alignment was quite precise just 15 d earlier, it is not unlikely that the moon will now enter into the shadow cast by the earth. This is known as a lunar eclipse. Lunar eclipses can be partial or total. At this time the moon is full, so the phenomenon can be quite dramatic. During a lunar eclipse, the moon does not completely disappear because sunlight scattered by the earth's atmosphere illuminates the lunar disk weakly. It may appear to have a dark reddish-brown color.

Observation of such an eclipse is not restricted to observers along a narrow path on the earth, as is the case for solar eclipses. Anyone on the moon side of the earth at the time of the eclipse can see the darkened moon. Thus for a given location on the earth, lunar eclipses are more probable than solar eclipses. It was these more frequent sightings of lunar eclipses that allowed the ancients (Chaldeans, over 2000 yr ago) to discover the 18 yr 11.3 d saros cycle, which could then be used to predict future lunar eclipses.

Planets

The motions of the sun and moon on the celestial sphere are both steadily eastward. In contrast the motions of the planets on the celestial sphere are quite complex. The planets all orbit the sun in the same direction as the earth (eastward viewed from the sun), but, viewed from the earth, they will move both eastward and at times westward (retrograde motion) due to the earth's motion. It is amazing that Kepler was able to deduce from such motions that a planet is actually moving in a simple elliptical orbit. The planetary motions are quite dramatic and easily noticeable by eye, more or less on a week-to-week time scale.

Sometimes two or three of the brighter planets will come close together in the sky. The novelty of this and its beauty, especially if the moon happens to be nearby, is anoteworthy occurrence. The orbits of the planets all lie within 3.5° of the ecliptic with the exception of Mercury (7°) and Pluto (17°). (One can argue that Pluto is not really a planet.) Thus the planets move along the sky very closely to the track of the sun, the ecliptic. Sometimes after sunset, one can see several planets (e.g., Venus, Mars, and Jupiter) spread along a great circle in the sky. They and the just-set sun nicely map out the ecliptic.

80 4 Gravity, celestial motions, and time 4.5 Measures of time

The study of celestial motions requires a definition of "time". It turns out that time keeping is not a simple process. There are many factors that contribute to this complexity. It was difficult in early times when clocks at different geographic locations could not be easily synchronized, and it is difficult in modern times as one pushes to obtain greater and greater accuracy, even with the aid of atomic clocks.

We humans instinctively conceptualize time as a smoothly running entity that can be agreed upon by all observers. This is the Newtonian model of time. That is, they can synchronize their watches (time and rate) and agree on the time of events at different locations. However, when high velocities or strong gravity are encountered, the comparisons are no longer so simple. General relativity (GR) gives a different model of time, taking into account the effects of speed and gravity. Special relativity takes into account the effect of speed.

In contrast to these models, there are nature's clocks which can be used to keep track of time. We have already seen that Caesar's model of time (his calendar; Section 3) did not track well nature's clock, the passing of the seasons. Manufactured clocks make use of nature's clocks. Traditionally they have been based on the natural frequency of the pendulum or that of the spring/mass oscillator. More recently the quartz crystal is commonly used to produce much more stable electrical oscillations. Each oscillation can be thought of as a tick of time, and time in "seconds" is defined in terms of a number of ticks. The SI standard of time is now based on oscillations of the cesium atom in atomic clocks.

Nature's astronomical clocks are the daily rotation period of the earth, the earth's annual rotation about the sun, and a rapidly rotating neutron star, seen as a regularly (up to a point) pulsing radio pulsar. A few of these pulsars exhibit an extremely high degree of stability that begins to rival that of atomic clocks, but they have not yet been used as a time standard.

One must keep in mind the distinction between a model of time and a clock of nature. One can tell time no better than the most stable clock one has. It may drift in rate, but there is no way to know it until a more stable clock is invented or discovered. In contrast, the model of time specifies ideally how time behaves, as a function of location, speed, and gravity. For example, the Newtonian model describes a time that is infinitely stable and everywhere the same. The story of time is one of the development and discovery of increasingly more stable clocks together with the development of models that successfully explain the subtleties of time keeping these improved clocks reveal.

Time according to the stars and sun

The most fundamental time keeping is based on the motions of the sun and stars as they pass overhead with seasonal variations.

Sidereal time

As the earth rotates under the sky, the zenith of a given observatory (e.g., Palomar Mountain) moves along the celestial sphere from west to east (Fig. 3.1). Thus, as we have described, a typical celestial object will rise in the east and move westward relative to Palomar Mountain. At any given time, the meridian of right ascension directly over the observatory is by definition the sidereal time, e.g., 15 h 25 m 35 s. A star at right ascension, a = 15 h 25 m 35 s, crosses over the longitude meridian of Mt. Palomar at sidereal time 15 h 25 m 35 s (by definition). At this sidereal time, stars at a = 16 h will not yet have transited the observer's meridian, and stars at a = 15 h 00 m will have already transited it.

Sidereal time is strictly a local time. To set his sidereal clock, the observer need only look overhead to see what meridian of the celestial sphere is passing overhead; its right ascension is the time to which he would set the clock. If, at some instant, the observer compares his sidereal time to that of another observer at a different longitude, by radio signal for example, they would report different sidereal times.

Mean solar time

Solar time is another local time; it indicates the location of the observer's meridian with respect to the sun. By definition, the sun is overhead (or due south or north of the zenith) at noon, solar time. Solar time varies in rate relative to a fixed-rate universal time because of the eccentricity of the earth's orbit and the tilt of the earth's axis relative to the normal to the earth's orbital plane. Mean solar time averages out these variations. The difference between solar time and mean solar time kept by clocks varies by as much as 16 min during the year; the difference is known as the equation of time. Clock makers used to put a table of the time difference for each day of the year on their clocks.

Mean solar time still lacks something in convenience; it will differ in two nearby towns if one is west or east of the other because the sun will pass over one meridian before it passes over that of the other. To solve this problem, everyone in a given region (time zone) agrees to keep the time of an observer at the central meridian of the region. This is called zone time, e.g., Eastern Standard Time. There are 24 such time zones, each 15° wide, more or less. The historic observatory at Greenwich England lies at the zero of geographic longitude. The zone time of this region has been called Greenwich Mean Time (GMT). This system of time zones dates back to 1884.

Astronomers find it convenient to use the modern equivalent of GMT (Universal Coordinated Time UTC, see below) during their observations. Frequently, observations at different observatories must be compared, and comparison of data from different time zones can lead to confusion. Astronomers tend to use the date of the beginning of an observing night to label that night's data even if the data were actually taken after midnight. This can add more confusion. In contrast, the UTC time and date (at Greenwich) of each specific observation during the night is unambiguous.

Sidereal and solar (or zone) times are not synchronized to one another because the former is referenced to the stars and the latter to the sun which moves through the stars as viewed from the earth. Since the sun moves along the ecliptic, the lengths of the two types of days must differ slightly, by about 1 part in 365, or by about 4 min. Which is the longer? Since the sun moves along the celestial sphere in the same direction that the earth spins, a given point on the earth must move more than 360° (relative to the stars) in order to catch the sun. The solar day is thus longer than a sidereal day.

The mean solar day is divided into 24 x 60 x 60 = 86 400 UT seconds which are slightly elastic(!) due to small irregular variations in the earth spin rate (see below). In civil life, we keep track of time with fixed atomic or SI seconds. Thus,

The shorter sidereal day is

1 Sidereal day = 23 h 56 m 04.09 s (mean solar time)

which tells us that the sidereal day is shorter than the solar day by 235.91 s ~ 4 min.

The sidereal day is referenced to the vernal equinox which moves slowly through the sky in the westward direction (Fig. 4). The day relative to the fixed stars is thus about 0.01 s longer than that given in (13). The relation (13) is valid even as the earth rotation rate varies irregularly because the mean solar day lengthens proportionally with the sidereal day.

The mean solar day is now defined such that, in about 1820, it would have been exactly 86 400 SI seconds in duration. It is now somewhat greater, by about 2 ms, due to the decreasing (on average) rotation rate of the earth. See discussion of UTC time and leap seconds below.

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