target="_blank" rel="nofollow" href="#litres_trial_promo">Figure 1.1 shows the two nodes of the Moon’s orbit, the points where the orbit intersects the ecliptic, the path of the Sun on the celestial sphere. The ascending node is defined as the point where the Moon crosses the ecliptic from below to above, south to north. The other node, the descending node, is the point where the Moon crosses the ecliptic from above to below, north to south.
In Figure 1.9 the celestial sphere model has been redrawn with some added features. So that we can further examine the all-important conditions for eclipses, an imaginary line has been drawn connecting the two nodes on the ecliptic. Not surprisingly, this line is known to astronomers as the ‘line of nodes’. The line of nodes rotates slowly in a direction opposite to the Sun’s motion, taking 18.61 years to complete one rotation. When this line is aligned with the Sun, an eclipse can take place. As there are two nodes, the alignment occurs about twice a year. The period of time either side of an alignment is called an eclipse season because if a new Moon occurs at this time, it will be near a node and a solar eclipse is likely.
Figure 1.9. The celestial sphere, showing the ecliptic limits and the line of nodes of the Moon’s orbit.
The motion of the line of nodes affects the time between eclipse seasons. The Sun appears to move about 1° per day eastward around the ecliptic, making a circuit of the sky in 365.25 days – a solar year. The line of nodes slowly regresses westward, in the opposite direction, at a rate of 18.61 years for one cycle. So an eclipse year, the time it takes the Sun to travel from alignment with one node to alignment with the same node again, will be less than a solar year. The length of the eclipse year can easily be calculated.
The line of nodes makes one rotation in 18.61 years, so it moves at a rate of 360/(365.25 x 18.61) = 0.053° per day westward around the ecliptic (365.25 being the length of the solar year). The Sun moves at a rate of 360/365.25 = 0.986° per day in the opposite direction. To get the combined rate of motion of the line of nodes and the Sun towards each other, add the two together to get 1.039° per day. The length of the eclipse year is therefore 346.62 days (the angular values are given here correct to three decimal places – the precise values do yield 346.62 days). From one node to the next, say the ascending node to the descending node, will take half that time, 173.3 days. These values have important consequences for calculating the frequency at which eclipses take place.
THE ECLIPTIC LIMIT
If the Sun at new Moon is aligned exactly with a node, it is obvious that there will be a solar eclipse. But the new Moon does not have to be exactly at a node for a solar eclipse to occur. Whenever the new Moon is close enough to the node for the two disks just to touch, there will be at least a partial solar eclipse. Look at the position of the Sun and new Moon on the celestial sphere with their disks just touching, in Figure 1.9. This position gives the maximum angular distance the new Moon can be from the node for there to be an eclipse. This distance is known as the ecliptic limit. If the total spread of the ecliptic limit, east plus west, is known as well as the speed of the Sun and Moon, it is possible to calculate the frequency at which eclipses can happen. Figure 1.10 shows a close-up of a part of the ecliptic centred on a node. (The node selected is a descending node, for consistency with examples of eclipses discussed in later chapters.)
The angular spread of the ecliptic limit depends on the tilt of the Moon’s orbit, which is a little over 5°. It also depends on the apparent size of the Sun and the Moon which, as we have seen, can vary because the orbital distances of the Earth and the Moon are always changing. Using the appropriate orbital parameters, the maximum angle of the ecliptic limit is found to occur when the Moon is at perigee and the Earth is at perihelion. In this configuration the two disks have their maximum sizes and will ‘touch’ at a point farthest from the node. This is calculated to be 37.02°, corresponding to a window of about 37.5 days. The minimum is 30.70° for the Moon at apogee and Earth at aphelion. In this case, the two disks have their minimum sizes as seen from the Earth, and will just touch at a point closer to the node. The ecliptic limit can be regarded as a ‘danger zone – if a new Moon occurs while the Sun is within this region, there will be a solar eclipse.
Figure 1.10. The ecliptic limits at a descending node (new Moon in each case).
Now, in a synodic month of 29.53 days, from one new Moon to the next, the Sun will move a certain fraction of an eclipse year along the ecliptic. This can be calculated as the ratio of the number of days in a synodic month to the number of days in an eclipse year: or 29.53/346.62, which makes 0.085. This fraction of one total revolution of 360° is 30.67°, very close to but less than the minimum angle of the ecliptic limit.
From this result a very interesting conclusion can be drawn. The minimum value of the ecliptic limit, 30.70°, is greater than the angle the Sun moves through during a complete synodic period of the Moon, 30.67°. The Sun will therefore never have enough time to pass through the ecliptic limit without being over-taken by a new Moon, so at least one solar eclipse is inevitable at each node, or two per calendar year. If the orbital conditions are optimum, with the Moon at perigee and the Earth at perihelion, the maximum ecliptic limit of 37.02° applies. The same calculation indicates that two eclipses are possible at each node, for a total of four solar eclipses in a year. The concept of the ecliptic limit thus provides a convenient way of predicting the properties of solar eclipses. It will also prove useful when it comes to explaining the characteristics of the saros series of eclipses.
THE FUTURE OF THE TOTAL SOLAR ECLIPSE
The Renaissance astronomer Johannes Kepler was as impressed as anyone by the coincidence of the 400 : I distance/size ratio for the Sun and Moon. In fact, he called the solar eclipse ‘a gift to us from the Creator’. He also pointed out in one of his many studies of the heavens that this fortunate situation will not always be true. Millions of years from now, the size of the Moon’s orbit will have increased to the point where total solar eclipses won’t be possible. All central eclipses will be annular. Here is Kepler’s reasoning.
He calculated that when an object’s orbital speed increases, the size of its orbit expands. In the case of the Moon, its orbital speed increases as a result of tidal forces. The gravitational force of the Moon pulls on the Earth’s oceans and crust, creating the twice-daily tides. Likewise, the Earth’s gravity pulls on the Moon, and these tidal effects produce bulges on the Moon’s surface. However, the tidal bulges produced on the Earth by the Moon are not centred at the point on the Earth’s surface directly below the Moon, but are shifted ahead of the Moon because of Earth’s faster rate of spin. This is shown in Figure 1.11. So the tidal bulge on the Earth is dragged ahead of the Moon’s location in the sky. This bulge exerts an additional gravitational force on the Moon which has a component tending to increase the orbital speed. Over many years, this slight increase in the Moon’s orbital speed will cause the Moon to slowly recede from the Earth.
Figure 1.11. The effect of the Earth’s tidal bulge on the Moon’s orbital speed.
The effect of the Moon inching away from the Earth is barely noticeable on human timescales. But millions of years from now the effects will be real. Our distant descendants will never be able to view a total solar eclipse. At some point in the future the Moon’s diameter as seen from the Earth will always appear smaller than the Sun’s, and only annular eclipses will be possible. No doubt this has inspired many of the world’s obsessive eclipse-chasers who travel all over the world to view the extraordinary phenomenon of a total solar eclipse. They know how lucky we Homo sapiens are in the current epoch.