Wednesday, 6 June 2012

The Celestial Clockwork

Ask most people what they associate with astronomy and they will mention planets, stars, galaxies, etc. Ask them what they associate with astronomers, however, and the answer will probably be telescopes – but it is less than four hundred years since the invention of the telescope. By contrast, human awareness of the heavens goes back millennia; almost certainly deep into prehistoric times. Until comparatively recently, astronomers have studied the heavens with nothing more sophisticated than the naked eye. From earliest times, they would have tried to make sense of the great complexity of what can be seen in the skies.

Even the most casual observer will be aware the skies present a differing appearance from hour to hour, as the Sun, Moon and stars march steadily across the heavens. They will realise, however, that the stars are fixed in relation to one another, often making distinctive patterns in the sky. They will also realise that the Moon changes its appearance from night to night, sometimes waxing sometimes waning. But why does the Sun rise and set in different places at different times of the year? Why is the Moon sometimes visible in broad daylight? Why do some stars remain close to the same point in the sky and never set? And what is to be made of the bright star-like objects that do not remain in a fixed position in relation to their neighbours, but move at differing speeds across the starry background, always keeping to the roughly same plane as the Moon in its wanderings.

Today, we know that the Moon goes round the Earth and the Earth and other planets go round the Sun. However, this does not even begin to tell the full picture and the achievements of people such as the Maya and the ancient Babylonians – who were not even armed with these basic facts – cannot be overstated. The workings of the celestial clockwork make the achievements of the finest Swiss watchmaker pale into insignificance, yet from a modern perspective they are not difficult to understand; the object of this short (9,000 word) work is to give the reader just such an understanding.  

The Celestial Sphere
Astronomers often use a model known as a celestial sphere to illustrate the movements of the Sun, Moon, planets and stars as seen from Earth. This can be thought of as being similar to a geographer’s globe, except it surrounds the Earth and we look at it from the inside. Another way of thinking of it is as a grid system projected onto the heavens to help us find our way around. To demonstrate such a grid would at one time have required the facilities of a planetarium, but there are now many ‘augmented reality’ smartphone apps available that can achieve almost the same effect.

Let’s first consider the finer points of the celestial sphere itself. Like the geographer’s globe, it will have two poles, an equator and lines of latitude and longitude. The celestial North and South Poles and the celestial equator (usually referred to simply as the “equator”) are projections of their terrestrial counterparts onto the grid system. The coordinate system used to define a point on the grid system differs slightly to that used by geographers. The position of an object to the north or south of the equator is given by declination (Dec.). Like latitude, it is measured in degrees, but instead of suffixing the declination with an N or an S, northern declinations are given positive values and southern declinations negative values. The equivalent of longitude is right ascension (R.A.), which has a zero line similar to the Greenwich meridian that passes through a point known as the vernal point (to be discussed shortly) or first point of Ares. Right ascension is measured eastwards from the first point of Ares. It can be measured in degrees but is usually measured in hours, minutes and seconds. An hour corresponds to 15 degrees, so 24 hours is equivalent to 360 degrees.

From anywhere in the world, our field of view will be bounded by the horizon, which divides the celestial sphere exactly into two. The point directly above us is known as the zenith. Running through both poles and the zenith is a great circle known as the meridian. The angle of elevation of any object above the horizon is known as the altitude; the angular distance around the horizon, measured clockwise from due North is known as the azimuth.

Let us take an imaginary or smartphone augmented reality trip to the (geographical) North Pole. The celestial North Pole is directly overhead, at the zenith, and the equator lies exactly on the horizon. Between the equator and the Pole are a series of concentric circles, growing ever smaller the nearer they are to the Pole. These represent differing declinations. A series of lines extend upwards from the equator, converging at the Pole. These are the lines of right ascension. Note that because the declination circles are parallel to the horizon, we can see them in their entirety. However, we cannot see anything south of the equator at all.

If we observe the sky for a few hours, the grid and stars will appears to revolve in a clockwise around the celestial North Pole. This is happening because the Earth is rotating on its axis, in a west to east direction, or anticlockwise as viewed from “above” the North Pole. This motion is known as diurnal motion. During the course of its diurnal motion, a star will reach its highest point in the sky when it is on the meridian. When a star reaches the meridian, it is said to culminate.

If we look close to the Pole, we will see a bright star. This is Polaris, the Pole Star. It appears to be almost fixed while everything else wheels around it. The further a star is from the Pole, the larger the circle in which it moves, with those near the equator moving in the largest circle of all. Note that no star ever rises or sets; we can see half of the stars the whole of the time.

Before we move on, how long do you think it takes the stars to make a complete circuit of the skies? The answer is of course the same length of time it takes the Earth to make a complete turn on its axis – but this is not 24 hours. The Earth makes a complete turn on its axis once every 23 hours 56 minutes and four seconds. This period of time is known as the sidereal day. However, we reckon time by the Sun rather than the stars, and as we shall see, the solar day is slightly longer.

Now we will travel to London, which is at latitude 51.5 degrees N. The first thing we notice is that the grid now appears tipped over towards one side. The celestial North Pole is no longer at the zenith; in fact its altitude will be equal to the geographical latitude. The declination circles are now no longer parallel to the horizon, so only those close to the Pole will be visible in their entirety and the others will be visible only as ever-decreasing arcs. However, we can now partially at least see declination circles that are south of the equator. Exactly half of the equator is visible, and it cuts the horizon at points due east and due west. The celestial North Pole lies due north.

There is a declination circle whose southernmost point just touches the northern horizon, and only the stars within this circle now remain permanently above the horizon or, as Homer put it in The Odyssey, “never bathe in Ocean’s stream”. Such stars are said to be circumpolar from that latitude (at either Pole, as we have seen, all the stars are circumpolar). Stars further south do spend increasing amounts of time below the horizon, though those North of the equator are still up for more than half of a sidereal day. Stars lying directly on the equator are visible for exactly half of a sidereal day. Note that these stars rise due east and set due west. Stars located still further south are visible for less than half of a sidereal day. Finally, on the southern horizon, is the northernmost point of a declination circle that lies entirely below the horizon. Stars lying within this circle are permanently out of view and include those making up the Southern Cross and our nearest stellar neighbour, Alpha Centauri.

Next let us go down to the equator. The grid will now appear to be completely tipped onto its side, and the declination circles will now lie at right-angles to the horizon. Exactly half of each circle will be visible, but our observer can now see all of them. The celestial North Pole lies exactly on the northern horizon and, 180 degrees away, the celestial South Pole has come into view. Every single star will be above the horizon for half of the sidereal day.

It will now be clear that were we to continue south, the celestial North Pole would dip below the horizon and the process we have just witnessed would occur in reverse until upon reaching the geographical South Pole, we would see the celestial South Pole at the zenith.

The Sun and Seasons
Now let us observe the Sun and stars as a sidereal day passes. At the end of one sidereal day, all the stars will be back where they started – but the Sun will be lagging behind. In fact, it will take approximately four minutes for the Sun to catch up. This is because while the Earth has been spinning on its axis, it has also been moving in its orbit around the Sun, completing a 1/365.24th of a circuit. Like the axial rotation, the orbital motion is west to east, or anticlockwise. (Astronomers refer to such motion as direct; clockwise motion (which is rare in the Solar System) is said to be retrograde.) The Sun, as viewed from Earth, appears to have changed its position slightly with respect to the stars. A solar day is defined as the time between successive crossings of the meridian by the Sun, but because the Earth’s orbital speed varies slightly over the course of a year, the solar day is not constant in length. It is only over the course of a year that it averages out to the familiar 24 hours.

The result of the solar day being about four minutes longer than the sidereal day is that any given star will rise four minutes earlier each (solar) day. This is why the constellations visible at a given time vary over the course of the year. After a year has passed, the solar and sidereal days come back into step. Our star will rise at the same time that it did on the corresponding day a year ago.

If we follow the Sun’s apparent movement over the course of a year, we will see that it will make a complete circuit of the celestial sphere. The path it traces out is known as the ecliptic. The ecliptic represents the plane of the Earth’s orbit around the Sun and it is inclined to the equator at an angle of 23.5 degrees. The reason for this is that the Earth’s axis of rotation is not perpendicular to the plane of its orbit, but inclined at an angle of 23.5 degrees. This inclination is known as the obliquity of the ecliptic.

Constellations straddling the ecliptic are said to be zodiacal. These include the familiar twelve signs of the Zodiac, but to make matters confusing there is actually a thirteenth zodiacal constellation, Ophiuchus the Serpent Bearer, that has been ignored by astrologers and is not considered to be part of the Zodiac. The two points at which the ecliptic and the equator intersect are known as the vernal point (which we have already encountered) and the autumnal point.

What effect will the Sun’s movement along the ecliptic have over the course of a year? The ecliptic is inclined to the equator, so the Sun will spend half of the year north of the equator and the other half of the year south of it. Recall that for the Northern Hemisphere, if a star is north of the equator it will be up for more than half of the sidereal day, but if it is south of the equator it will be up for less than half of the sidereal day. The same applies to anything on the celestial sphere, and this includes the Sun.
Thus for half of the year, day will be longer than night and for the other half it will be shorter. On the two days of the year known as equinoxes, day and night will be of equal length; this will occur when the Sun is at the vernal or autumnal point. The vernal point is known as the Sun’s ascending node because this is where it crosses into the northern hemisphere from the south. Similarly, the autumnal point is known as the descending node. Mid-way between the equinox points, the Sun will reach its most northerly and most southerly positions; these points are known as the solstices. In the northern hemisphere, the summer solstice occurs when the Sun is at the most northerly point on the celestial sphere and the winter solstice when it is at the most southerly point. This is the explanation for the seasons.

In lower latitudes, the Sun attains greater elevations above the horizon. When it is at either equinox, the Sun will be directly overhead at midday along the equator. At the summer solstice, it will be directly overhead at midday in the latitude defined by the Tropic of Cancer and at the winter solstice it will be directly overhead at midday in the latitude defined by Tropic of Capricorn. This is why it gets rather hot in these parts of the world. Conversely, the Sun is circumpolar during the summer months within the Arctic Circle, but never rises at all during the winter months. This is the explanation for the famous ‘Midnight Sun’. The situation is reversed for the Antarctic Circle.

Most people think of the Sun rising in the East and setting in the West. But the rising point of the Sun is actually only due East on two days of the year, those when it is at one or other of the equinox points. In the summer months, the azimuth of rising point moves north, reaching its maximum extent at the summer solstice, before returning south. In the winter months, the azimuth of the rising point moves south reaching its maximum extent at the winter solstice, before returning north. Around the solstices, the rising appears to stand still for a few days; the word ‘solstice’ is derived from this phenomenon. The setting points move in the same manner, with the winter sunset limit lying opposite the summer sunrise limit, and the summer sunset limit lying opposite the winter sunrise limit. These seasonal variations in the rising and setting points vary with latitude, being more pronounced in higher latitudes.

The Earth’s Orbit 
Let us now consider the Earth’s orbit around the Sun in a little more detail. The orbit is not circular but elliptical, meaning that distance between the Earth and the Sun isn’t constant but varies over the course of the year, ranging from 147 million km (when Earth is said to be at perihelion) to 152 million km (aphelion). The Earth’s orbital speed is at its greatest at perihelion and at its least at aphelion. This is because its movements are governed by Kepler’s Laws of Planetary Motion, which we will examine in more detail presently. The mean distance is 149.6 million km (93 million miles) and this distance is referred to as the astronomical unit (AU). The departure of the orbit from a perfect circle is known as the orbital eccentricity. Perihelion does not occur in the same place each year, but advances by 11.64 seconds of an arc on each orbit. This is due to gravitational effects of other planets in the Solar System.

Precession, Nutation and other cycles
There are in addition a number of more gradual motions which are only significant over a long term. The most important of these is precession. In addition to rotating on its axis, the Earth also oscillates like a spinning top, each oscillation taking about 25,800 years and causing the Earth’s spatial orientation to gradually change. This motion is due chiefly to the pull of the Sun and the Moon, though the planets make a small contribution.

The observable effect is to make the nodes of the ecliptic gradually move westwards at about 50 seconds of an arc per year. The stars will remain fixed in relation to the ecliptic, but as our celestial sphere grid system uses the Earth as its frame of reference, they will appear to move very slightly against it. This means that star maps have to be calibrated for a particular epoch which since 1984 has been Epoch 2000.0, the start of the year 2000.

The effect, though small, is cumulative and just about visible to the naked eye over a lifetime (a good amateur telescope on a suitable mount could show it in a matter of weeks if not days). More significant effects are experienced over longer periods – in antiquity, for example, the Southern Cross could be seen from Greece and the ancient Greeks included it in their star charts. 14,000 years from now it will be visible all over Britain. The precessional motion is not smooth but slightly wavy. This irregularity is known as nutation and it is the result of a slight nodding of the Earth due to variations in the distances and positions of the Sun and the Moon.

In addition to these effects, the obliquity of the ecliptic varies with time, chiefly due to nutation, though the gravitational effects of other planets also play a part. Finally the eccentricity also varies, albeit very gradually. Again, this is due to the gravitational effects of other planets. The precessional cycle and the cyclical changes in the obliquity and orbital eccentricity are now known as the Milanković cycles. They are named for the Serbian mathematician Milutin Milanković, who proposed a link between them and cyclical changes in Earth’s climate while interned in Budapest during World War I. Climatologists now accept that Milanković was right, but his views attracted little interest in his lifetime.  

Four types of year
Up until now, we have used the term ‘year’ rather loosely. Most people think of a ‘year’ as being the time it takes the Earth to go once round the Sun. That is certainly a type of year, but not the only type. The Earth in fact has four different types of year. The first is the sidereal year (365.256 days), and this is indeed the time it takes the Earth to go once round the Sun – but this is not the ‘year’ we base our calendar on.

The Gregorian calendar, which is now used throughout the Christian world, is actually based on the tropical year (365.243 days), which is defined as the time between the Sun making two passages through the vernal point. The vernal point is moving slowly in the opposite direction to the Sun along the ecliptic as a result of precession, so the Sun ‘arrives’ there about twenty minutes before it completes its circuit of the celestial sphere. Hence the tropical year is slightly shorter than the sidereal year. However, the progression of seasons is dictated by the former, so the calendar is based upon it.

The third type of year is the anomalistic year (365.260 days), which is defined as the time between the Earth making two returns to perihelion. As we have seen, the perihelion advances with each circuit, the Earth requires a bit longer to ‘catch up’; hence the anomalistic year is fractionally longer than the sidereal year.

Finally there is the eclipse year (346.620 days), which we shall encounter presently.

The Moon and its Orbit
Most people will think nothing if they happen to see the Moon in the night sky, but are often surprised to see it in broad daylight. In fact, the Moon spends on average as much of its time above the horizon in day time as it does in night time. The most singular feature of the Moon is, in fact, something which most of us take completely for granted. This is that it appears almost exactly the same size as the Sun. The explanation is simple – the Sun is about four hundred times larger in diameter than the Moon, but it is also about four hundred times further away. The odds against this happening – if not quite astronomical – are pretty low.

The Moon’s orbit around the Earth is inclined at an angle of 5 degrees to the ecliptic. The Moon’s apparent path around the celestial sphere intersects the ecliptic at two points again known as nodes; as with the intersections between the ecliptic and equator there is an ascending node and a descending node. These nodes do not remain fixed, but move westwards on the celestial sphere at 19 degrees per year due to perturbation by the Sun, taking 18.61 years to complete a nodal cycle. This phenomenon is known as the regression of nodes.

The orbit itself is rather more elliptical than that of the Earth around the Sun. Distance from Earth (centre to centre) varies from between 356,410 kilometres (minimum distance, or perigee) to 406,697 kilometres (maximum distance, or apogee). The Moon’s orbital speed increases at perigee, and it decreases at apogee. Like the Earth, this is due to Kepler’s Laws of Planetary Motion, which apply to all orbiting bodies. The eccentricity of the orbit is quite pronounced, so the effect is quite noticeable in terms of nightly movement on the celestial sphere, and this has been known since ancient times. In a manner similar to the Earth’s perihelion, the Moon’s perigee advances with each orbit, taking 8.85 years to complete a cycle.

Phases of the Moon
The most noticeable feature of the Moon is that its appearance changes from night to night. These phases are due to differing portions of its day-lit side being presented to us as it moves around the Earth. At the start of the cycle it cannot be seen because it lies in the same direction as the Sun and its illuminated side faces away from us. A few days later it will have moved eastwards away from the Sun and will be seen as a slim crescent in the evening sky. As the days pass, the Moon is seen ever higher in the evening sky as it continues to grow or wax. After seven to eight days the Moon will be 90 degrees east of the Sun in the sky; this point is known as the first quarter and the right half of the Earth-facing side is illuminated. At around fifteen days, the Moon’s distance from the Sun reaches 180 degrees. At this point the Moon rises at sunset. The entire Earth-facing side is now illuminated and we see a full Moon. Thereafter, the Moon begins to wane, going through its phases in reverse as its angular distance from the Sun begins to decrease once more. After about 22 days the Moon is 90 degrees west of the Sun; this point is known as the third quarter and the left half of the Earth-facing side is illuminated. Subsequently the Moon becomes an increasingly slim crescent, moving ever closer to the Sun and appearing only just before sunrise. Finally, after 29.531 days on average, it disappears from view and the cycle begins again.

Five types of month
Most people think of this cycle of 29.531 days as being a month, but they also think of the Moon going round the Earth once a month. In fact, the Moon takes only 27.321 days to go round the Earth. So which ‘month’ is right? Well actually both are. It all depends on what is meant by a month. As we have seen, the Earth has four types of year; the Moon, not to be outdone, has five types of month.
The most obvious, perhaps, is the time the moon takes to go once round the Earth. This is known as the sidereal month and as we have seen, it is 27.321 days. But because the Earth is moving round the Sun at the same time the Moon is moving round the Earth, it takes the Moon a bit longer than a sidereal month to return to the same position with respect to the Sun and the Earth. As it is this which governs the phases, it takes more than a sidereal month to go through a complete cycle or lunation. The time for a lunation is known as the synodic month. The Earth’s orbital speed varies slightly over the course of a year, so the synodic month is not fixed. 29.531 days is only the average figure.

The tropical month is slightly shorter than the sidereal month. It is defined as the time from one lunar equinox to another. The lunar equinox occurs when the Moon crosses the equator; this takes slightly less than a sidereal month due to the effects of precession (c.f. tropical year).

Next is the anomalistic month, the time taken for the Moon to go from perigee to perigee. The perigee advances, so this is longer than the sidereal month and is 27.554 days.

Finally we have the draconic month of 27.212 days. This is the time between successive passages by the Moon through the same node. The nodes are moving westwards and the Moon is moving eastwards along the celestial sphere, so it takes less than a complete orbit for the Moon to return to the node, and thus the draconic month is shorter than the sidereal month. The word ‘draconic’ refers to a mythical dragon thought to devour the Sun and the Moon during solar and lunar eclipses; the eclipse year is also sometimes referred to as the draconic year for this reason.
The interrelationship of various types of month and year are of great importance when it comes to predicting eclipses, and these cycles may have been understood as far back as prehistoric times.

Lunar and Solar Calendars
In the Western world, we have long been accustomed to a year of 365 days, with a leap day inserted into February every fourth year. The Gregorian calendar, now the most widely used civil calendar in the world, does have exceptions to this leap year every fourth year rule, but the last such ‘non-leap year’ was in 1900 and the next will not occur until 2100. Most of us will live our lives without ever having been troubled by such details, but they are important.

The Gregorian calendar is an example of a solar calendar, or one based on the tropical year. Since this is not an exact number of days, a leap day must be intercalated (inserted) at intervals, and the Gregorian calendar provides for an extra day in February if the year is divisible by four. An exception to the rule is made if the year is divisible by 100 but not by 400, as is the case for 1900 and 2100, but not 2000. The Gregorian calendar was introduced in 1582 in the time of Pope Gregory XIII. It was a refinement to the earlier Julian calendar, which inserted the leap day every fourth year without exception. This gave a year of 365.25 days, which is slightly longer than the tropical year of 365.243 days. The Julian calendar was introduced by Julius Caesar in 46 BC and the error, though small, had amounted to several days by the sixteenth century. The Gregorian was not immediately adopted everywhere, due to resistance in Protestant countries to a Catholic innovation. In Britain, the changeover did not occur until 1752, by which time the correction amounted to eleven days and so Wednesday, 2 September was followed by Thursday, 14 September. The story that this led to riots by people demanding the return of their eleven days is probably apocryphal. In Russia the new system was not adopted until early in the communist era, by which time thirteen days had to be dropped from the calendar. An ironic consequence was that the date of the Great October Socialist Revolution was shifted into November.

Solar calendars follow the seasons, but the months do not follow the phases of the Moon because there are not an exact number of synodic months in a tropical year. A lunar calendar is one based on the phases of the Moon and examples include the Islamic calendar, which comprises twelve synodic months and therefore lags the solar calendar by 11 to 12 days each (tropical) year. The Islamic calendar is the official calendar of Saudi Arabia, but elsewhere in the Islamic world it is used mainly for religious purposes.

To get round the problem of a lunar calendar fairly rapidly drifting out of synch with the tropical year, some calendrical systems insert an intercalary month every so often, though various calendars use different systems for determining how and when these occur. Such systems are known as lunisolar; examples include the Hebrew and Chinese calendars.

Lunar and solar calendars generally come into line every 19 years. This is because 19 tropical years are almost exactly 235 synodic months; thus every 19 years the Moon will have the same phase on the same day of the year. This 19-year cycle is known as the Metonic cycle after the Greek philosopher Meton of Athens (ca 440 BC) who noticed it, though it was undoubtedly known earlier. The Metonic cycle formed the basis of the Greek calendar until 46 BC, when the Julian calendar was adopted.

Moonrise, Moonset and lunar movements
Like the Sun, the Moon does not rise and set in exactly the same place every day. The azimuth of the rising and setting points varies cyclically over the course of a sidereal month between northern and southern limits and, as with the Sun, these variations are more pronounced in higher latitudes. Note that the ‘month’ in question here is the sidereal rather than synodic month hence the Moon will not be at the same phase at two successive risings or settings at a particular point. Another way of looking at this is to consider only the azimuth of rising and setting of the full Moon, which will vary between the same limits over the course of a year.

However, these limits themselves open out and close up over the course of the 18.61 year nodal cycle. In the Northern hemisphere, the variation reaches a maximum when the ascending node is co-incident with the summer solstice; these are the major standstill points. When the descending node reaches this point, the variation is at a minimum; these are the minor standstill points. Between these limits, the standstill points gradually close up and then re-open. The situation is reversed in the Southern Hemisphere.

In simpler terms, at the major standstill the Moon’s 5 degree orbital inclination is added to the effect of the Earth’s axial tilt; at the minor standstill it is subtracted. Thus the variation exceeds that of the Sun at major standstill, but is less than it at the minor standstill.

As we have seen, the cycle is driven by the sidereal and not the synodic month, so different phases of the Moon will be best observed at different times of the year. The full Moon, for example, rides majestically high in the winter skies, but in summer its performance is decidedly lacklustre. It struggles into the sky, staggers wearily along the southern horizon for a few hours before giving up and disappearing again. The explanation is straightforward enough: when full the Moon is in the opposite part of the sky to the Sun, so in winter it behaves as the Sun in summer, and vice-versa. In spring, the waxing first quarter Moon is most favourably presented for observation, and in autumn it is the turn of the waning last quarter Moon. The waxing crescent is best seen in mid-spring; the waning crescent in mid-summer. These rules hold in both hemispheres, because the seasons are reversed in the Southern Hemisphere. As with the standstill points, these effects are accentuated and diminished over the course of the 18.61 year nodal cycle.

The Dark side of the Moon
When people refer to ‘the dark side of the Moon’ they really mean the side that cannot be seen from here on Earth. As is correctly pointed out in the eponymous Pink Floyd album, there is no dark side of the Moon and both sides experience equal portions of day and night. It is, however, true that the Moon’s sidereal day is exactly one sidereal month, so in the main one side permanently faces the Earth. However, it is not strictly speaking true to say that we can only see one side from Earth.
The orbital speed is not constant, so the orbit and rotation get slightly out of step at times, which causes a slightly different face to be presented. This effect is known as libration in longitude. In addition, because the Moon’s axis is inclined by 6.5 degrees to its orbit, it appears to ‘nod’ back and forth over the course of a month – this is libration in latitude. Finally, parallax effects result in slightly different faces being presented to the observer at different times of the day; in total 59 percent of the Moon’s surface may be seen from Earth (though of course no more than 50 percent at any one time).

The Wanderers
The word ‘planet’ comes from the Greek word planetes, meaning ‘wanderer’. Long before the time of the Classical Greek civilisation, man would have been aware of bright star-like objects that did not did not remain fixed in relation to the stars but moved in roughly the same narrow band to which the Sun and Moon are constrained. Five planets (excluding the Earth) have been known since prehistoric times – Mercury, Venus, Mars, Jupiter and Saturn. They fall into two groups, the inferior planets, whose orbits lie close to the sun that of the Earth (Mercury and Venus) and the superior planets whose orbits whose orbits lie further away from the Sun (all the other planets, excluding Earth). The distance of each planet from the Sun is often given in astronomical units. Incidentally, the terms ‘superior’ and ‘inferior’ do not mean that the superior planets are ‘better’ planets.

The motion of each planet around the Sun is governed by Kepler’s Laws of Planetary Motion, which were formulated by the German mathematician Johannes Kepler between 1609 and 1618 and they apply not just to planets but all orbiting bodies, such as the Moon, satellites of other planets and even artificial Earth satellites.

The First Law states that the orbit of any planet around the Sun will be an ellipse, with the Sun at one focus. (If you add the distances of any point on an ellipse from each of the two foci you will always get the same result. By comparison, if you measure the distance of any point on a circle from the centre of that circle, you will always get the same result. In fact these properties define circles and ellipses, which are both examples of what are termed conic sections by mathematicians.

The Second Law states that the movement of any planet in its orbit is such that its radius vector (an imaginary line joining the planet to the Sun) sweeps out equal areas in equal times. This explains why the Earth and other planets move faster when they are close to perihelion and why the Moon moves faster when it is close to perigee. The sector swept out in, say, 24 hours, is shorter at these times, but because the Earth (or Moon) is moving faster, it is also ‘fatter’ and these two effects exactly cancel out.

The Third Law states that the square of a planet’s orbital period in years is equal to the cube of its mean distance from the Sun in astronomical units. More generally, the square of the orbital period of any orbiting body is proportional to its mean distance from the body it orbits.

These laws arise naturally from Newton’s Law of Universal Gravitation, which states that between any two objects, there exists an attractive force that is proportional to their masses multiplied together and divided by the square of their distance apart. Objects under consideration can be stars, planets, satellites or even the apocryphal apple that is said to have given Newton the idea in the first place.

Aspects of the planets
As seen from the Earth, certain positions of the planets relative to the Sun are known as aspects. For superior planets the two principal aspects are opposition and conjunction.

At opposition, a planet is opposite to the Sun in the sky, i.e. they are 180 degrees apart. It will be visible throughout the night and will reach the meridian it midnight. Opposition is the best time to observe a superior planet, because it is at its closest to the Earth. At conjunction, a superior planet is on the opposite side of the Sun to the Earth. It will not be visible from earth at this time, being lost in the Sun’s glare.

When a planet is at either opposition or conjunction (i.e. it, the Earth and the Sun are in a straight line) it is said to be at syzgy. The Moon is at syzgy when it is both new and full. When a planet is at an angle of 90 degrees from the Sun as seen from Earth, it is said to be at quadrature. We see a half-Moon when it is at quadrature.

Inferior planets cannot reach opposition or quadrature, but have two types of conjunction, inferior conjunction, when they lie between the Earth and the Sun and superior conjunction, when they are on the far side of the Sun. When an inferior planet is at its greatest angular separation from the Sun it is at greatest elongation. At its greatest elongation west it will appear in the morning sky; at greatest elongation east it will appear in the evening sky. An inferior planet can never be seen throughout the night.

The inferior planets display phases like the Moon but when best seen (i.e. at elongation) they are crescent. They will be at full phase at superior conjunction and “new” at inferior conjunction, but cannot be seen at these times. The superior planets show very little phase effect; only Mars shows a pronounced gibbous phase when it is at quadrature.

Movements of the planets
As seen from the Earth, the planets normally appear to move from west to east. However, around opposition a superior planet can appear to halt and then move briefly in an east to west direction before resuming its normal progress. This retrograde motion, so beloved of astrologers, occurs because the Earth, which is moving more rapidly, catches up and overtakes the planet in question. The points where the planet halts before changing direction are known as stationary points.

The planets all keep fairly close to the ecliptic, but all have orbits that are slightly inclined to it. Orbits are defined in terms of six elements or quantities. These are the semi-major axis (a) or mean distance from the Sun; the eccentricity (e); the inclination to the ecliptic (i); the longitude of the ascending node (Ω); the argument of perihelion (ω) which is angular displacement from Ω; and the time of perihelion passage (T).

A planet’s ‘year’ is known as its sidereal period, corresponding to the Earth’s sidereal year. The time taken for a planet to return to a particular aspect (such as opposition) as seen from Earth is known as the synodic period (c.f. the Moon’s synodic month).

There is little doubt that a total eclipse of the Sun is one of the most awesome spectacles of Nature available anywhere in the Solar System. On no other planet is there such an exact match between the apparent size of the Sun and the apparent size of a satellite – despite some planets having upwards on fifty of the latter to choose from, while we on Earth have to make do with just the one. Not quite as spectacular, perhaps, but still noteworthy is the sight of the Moon turning a deep blood-red as it enters the Earth’s shadow during a lunar eclipse.

The phenomena are related, but strictly speaking a solar eclipse is an occultation or hiding of a self-luminous body (in this case the Sun) by the Moon. In principle there is no difference between this and the occultation of stars that occur throughout the month as the Moon pursues its course around the Earth. By contrast, a lunar eclipse entails the Moon being cut off from the source of its illumination as it enters the Earth’s shadow.

Unlike point sources (such as a distant searchlight), extended luminous objects such as the Sun do not cast sharp shadows. A shadow will of course be cast when an object is interposed between the observer and the light source, but it will have two regions: the umbra in which the light source is wholly obscured and the penumbra in which it is only partially obscured.
For a disc such as the Moon, the Earth as seen from the Moon’s surface or a hot-air balloon drifting in front of the Sun as seen by an observer on the ground, the umbra will be cone-shaped, converging to a point; the umbra will be fan-shaped and diverging.

Types of Solar eclipse
The Moon’s umbra under favourable conditions will just reach the Earth. It does not remain stationary but races across the Earth’s surface as the Moon moves in its orbit. The path it follows is known as the track. Observers inside the umbra will see a total solar eclipse; those outside it but still within the penumbra will see a partial solar eclipse; those completely outside the Moon’s shadow will see nothing.

The degree of obscuration of the Sun by the Moon or magnitude will increase the closer an observer is to the zone of totality. Magnitude ranges for 0 (no obscuration) to 1 (totality) and it refers to the solar diameter covered, not area. A 0.5 magnitude eclipse is one in which half the Sun’s diameter is covered, but a little geometry will show that only 40 percent of the Sun’s area will actually be hidden by such an eclipse.

The actual duration of totality for any eclipse varies and is dictated by three factors: the distance of the Earth from the Sun when the eclipse occurs; the distance of Moon from the Earth when the eclipse occurs; and the latitude at which the eclipse occurs.

If the Earth is at its maximum distance from the Sun its apparent diameter will be diminished and if the Moon is at its minimum distance from Earth its apparent diameter will be increased; these factors favour long eclipses.

The Earth is rotating in the same direction as the Moon’s shadow is moving, and this has the effect of prolonging the time the latter will linger over a particular region. The speed the Earth’s surface is moving depends on latitude – at 40 degrees north or south of the equator, the west to east motion is 1,270 kilometres per hour but at the equator it is 1,670 kilometres per hour. The relative speed of the Moon’s shadow is thus lower at lower latitudes and thus eclipses that take place in tropical latitudes tend to be of greater duration than those occurring in temperate latitudes.

If the Moon is at or close to its maximum distance from Earth, even if it passes directly in front of the Sun the umbra will not quite reach Earth and a ring of sunlight is left showing. Such eclipses are said to be annular. Total and annular eclipses are referred to as central eclipses, and annular eclipses are the slightly more frequent of the two types.

Occasionally, an eclipse is just total at mid-track, but due to the curvature of the Earth the umbra doesn’t touch the end-points. The result is a hybrid total/annular eclipse with observers at mid-track experiencing a total eclipse but those at either end-point viewing only an annular eclipse.
Finally in about one third of all solar eclipses only the penumbra reaches the Earth with the umbra missing it altogether. Such eclipses are partial only; nowhere on Earth is a total eclipse seen.

Stages of a Solar eclipse
The key events in a solar eclipse as viewed from a particular site are known as contacts. First Contact occurs when the Moon’s western edge begins to slide across the Sun and is the point at which the penumbra first begins to move across the site. It is abbreviated to P1, for first penumbral contact. Second Contact occurs when the Moon’s eastern edge touches the Sun’s eastern edge. The marks the onset of totality or annularity, and for a total eclipse is the point at which the umbra begins to move across the site. It is abbreviated to U1 for first umbral contact (though strictly speaking this term is only appropriate for a total eclipse). Third Contact occurs when the Moon’s western edge leaves the Sun’s western edge. This marks the end of totality or annularity and is the point at which the umbra leaves the site. It is abbreviated to U1. Finally Fourth Contact, abbreviated to P2, marks the departure of the penumbra from the site and the end of the eclipse. In a partial eclipse, only P1 and P2 occur.

Types of Lunar eclipse
Whereas the Moon’s umbra will affect only a small portion of the Earth, the Earth’s umbra is large enough to fully immerse the Moon. During a lunar eclipse, the Moon never entirely disappears from view but appears reddish. This is due to refraction or bending of sunlight by the Earth atmosphere into the umbra; red light is more easily refracted. There are three types of lunar eclipse; total, when the whole of the Moon enters the umbra; partial when only a portion does; and penumbral when the Moon just grazes the penumbra. The latter type generally results in only a slight dimming of a portion of the Moon and is often undetectable to the naked eye. Unlike a solar eclipse, a lunar eclipse may be viewed anywhere on Earth where the Moon is above the horizon.

Stages of a Lunar eclipse
As with solar eclipses, the key stages of a lunar eclipse are referred to as contacts though unlike a solar eclipse these are the same from any point on Earth. P1 occurs when the Moon begins to enter the Earth’s penumbra. U1 is the point at which the Moon begins to enter the umbra; U2 is the point at which it is fully inside the umbra, marking the onset of totality. U3 is the point at which the Moon begins to leave the umbra, marking the end of totality; U4 is the point at which the Moon leaves the umbra altogether. P2 is the point at which the Moon leaves the penumbra and the eclipse ends. U2 and U3 do not occur in a partial eclipse. In a penumbral eclipse, U1 and U4 do not occur either.

When do solar eclipses occur?
As you might have inferred, a solar eclipse can only occur at new Moon – but why don’t they occur at every new Moon, i.e. once every lunation? Recall that the Moon’s orbit is inclined at about 5 degrees to the ecliptic. So the Moon usually ‘misses’ the Sun. Recall that there are two nodes where the Moon’s path crosses the ecliptic. Only when the Sun is close to a node at new Moon can an eclipse occur, although it does not have to be exactly at a node for an eclipse to occur; for the two discs to touch in a ‘grazing’ encounter will at minimum produce a partial eclipse.

The region the Sun has to occupy at new Moon to produce an eclipse is known as the eclipse limit. This varies, depending on the distance of the Moon from Earth at the time the new Moon occurs, and that of the Earth from the Sun. It ranges from between 30.70 degrees to 37.02 degrees in total, or from 15.35 to 18.51 degrees each side of the node. For a central eclipse to occur, the limit is less, ranging from 9.92 to 11.83 degrees each side of the node.

With the Sun moving along the celestial sphere at just under one degree per day, it will be apparent that it will take it more than a synodic month of 29.53 days to traverse even the minimum distance. In other words, the Sun will never be able to get through one of these ‘danger zones’ without the Moon catching up with it at some stage and causing an eclipse. Furthermore, if the Sun has only just entered the eclipse limit when the Moon comes around, the latter will have time to cause a second eclipse before the Sun can get out of the way.

The time period during which the Sun is within the eclipse limit is known as an eclipse season. Eclipses can only occur during an eclipse season and as we have just seen, at least one must occur. How many eclipses will occur in a calendar year, given at least one must occur whenever the Sun approaches a node?

Recall the nodes are moving along the celestial sphere in the opposite direction to the Sun, completing a complete cycle every 18.61 years. It will therefore take the Sun slightly less than a year to make successive passages through the same node. This is the ‘fourth kind of year’, the eclipse year mentioned earlier, of 346.62 days. There will be two eclipse seasons in each eclipse year and a minimum of two solar eclipses and a maximum of four.

The calendar year is longer than an eclipse year and so the eclipse year will end at different times of the calendar year. Normally there will only be two eclipse seasons (and hence a minimum of two eclipses) in a calendar year, but if an eclipse year ends in December, a portion of a third eclipse season can be squeezed into that calendar year, meaning that a maximum of five solar eclipses could occur. Unfortunately, you will have to wait until 2206 before this next happens.

When do lunar eclipses occur?
Just as solar eclipse can only occur at new Moon, so a lunar eclipse can only occur when the Moon is full. The condition for a lunar eclipse is for the Moon to pass through the opposite node to the one through which the Sun is passing during an eclipse season. As with a solar eclipse, this must happen at least once during an eclipse season, and can happen twice.

The maximum number of both types combined in an eclipse season is only three, because it would take 1 ½ lunations to produce two solar and two lunar eclipses, which is longer than the maximum length of an eclipse season. However, there will always be at least one of each. This rule does include penumbral lunar eclipses, which many authorities omit from eclipse statistics.

The Saros
The word ‘saros’ is taken from an ancient Babylonian word meaning ‘repetitive’ and was adopted by Sir Edmund Halley to describe an 18-year cycle of eclipses first recorded by the Babylonians in 400 BC, though it may well have been known much earlier.

The saros results from a series of coincidences of nature: 223 synodic months (6585.32 days) is almost exactly the same length of time as 19 eclipse years (6585.78 days) and also coincides with 239 anomalistic months (6585.54). The net effect is that at the conclusion of 223 synodic months from the time of an eclipse, not only are the Sun and Moon in the same places in the sky (thus giving rise to another eclipse) but the Moon will be at the same distance from the Earth as for the previous eclipse and the eclipse limit will thus be the same. This latter factor is equally important because were the Moon to be at a greater distance from Earth than previously, the eclipse limit would be smaller and an eclipse might not occur at all.

However, there is one important difference. 223 synodic months does not contain a whole number of days. The odd 0.32 of a day means the second eclipse will occur at a longitude of 0.32 times 360 degrees, i.e. approximately 115 degrees west of the first eclipse due to the Earth’s additional rotation.
Eclipses occur more frequently than every 18 years, so there are a number of saros cycles in operation at any one time, and each one is given a number. Saros cycles involving the Moon’s descending node receive even numbers and those involving the ascending node receive odd numbers. Each saros cycle evolves and has a finite life. For a saros cycle involving the Moon’s descending node, the series begins with an eclipse at the South Pole. Each successive eclipse then has a track more northerly than the last, until a final eclipse at the North Pole concludes the cycle. For a saros cycle involving the Moon’s ascending node, the reverse happens, with the series beginning at the North Pole and concluding at the South Pole.

At any one time there will be 43 saros cycles in operation and as soon as one concludes at one Pole another one will begin at the other Pole. The length of a cycle ranges from between 1,206 to 1,442 years.

This all happens because 19 eclipse years are actually 0.46 days longer than 223 synodic months, and the Sun will not be in exactly the same place for successive eclipses. Given the Sun moves approximately one degree per day along the celestial sphere, each eclipse will occur about 0.46 degrees west of its predecessor. The Moon of course will also be 0.46 degrees further west than before. For the descending node, this will additionally put the Moon slightly further north than before; for the ascending node, the Moon is slightly further south than before.