The whys and wherefores of navigation

Owing to the important place that declination holds in nautical astronomy, a detailed explanation will appropriately follow closely in the wake of the preceding remarks. It must be made clear, before getting under way, that declination is the distance, in degrees, minutes and seconds, of a body north (+) or south (-) of the celestial equator measured on the hour circle passing through the body. This distance is identical with the latitude of the place in the zenith of which the body happens to be. What declination is to a body in the heavens, latitude is to the place on the earth directly beneath it.

The declination of fixed stars changes very slowly from month to month, but the planets meander about on the celestial sphere in a way that is liable to puzzle anyone other than an astronomer. This element, however, is worked out in the observatory and given in the nautical almanac in a way that relieves the navigator of worry concerning the complex movements of these latter bodies. The same may be said of the moon, but the subject will be treated, somewhat superficially though sufficiently for the needs and desires of the practical mariner, in a special talk on the moon. This eliminates all the celestial bodies except the sun, the most important; and for this reason the facts relative to its declination will be considered at some length.

As has already been stated, the sun is stationary, but our movements around it to the right causes it to appear to move to the left; precisely as you see, when under way, an anchored vessel’s masts move to the left along the land behind her, while you move on to the right. We have no landmarks behind the sun by which to observe his apparent movements, so in lieu of such ranges, we resort to the fixed stars, which serve as excellent marks to get a bearing on Old Sol and keep tab on him as he moves eastward among them. This movement must in no way be confounded with his apparent daily motion westward. As an illustration, we may see Orion—a familiar friend—swinging high in the western sky in the early evening; some weeks later he is riding low, and yet a little later still, he is swallowed up in the brilliancy of the setting sun. In other words, the sun and Orion have approached and passed each other. We know Orion does not move, for he is composed of fixed stars, and this seeming westward movement of his is in reality the apparent eastward marching of the sun, which is due to the earth’s movement of revolution. The sun in this apparent movement eastward follows a course at a rate equal to that of the earth, along a great circle of the celestial sphere called the ecliptic, a circle that plays an important part in the explanation of declination, particularly that of the sun. The ecliptic is marked by the extension of the earth’s orbit to the celestial sphere.

A few more words concerning great circles will be introduced here, and the following statements, while they apply to great circles in general, especially fit the relationship of the equinoctial or celestial equator to the ecliptic. These two great circles cut each other at an angle of 23° 28´. Great circles always bisect each other, and hence any two great circles of the celestial sphere, regardless of the angle they may take with the celestial equator, must intersect each other at exactly opposite points, 180° apart. What is true in this regard of the celestial sphere is equally true of the great circles of the earth. A vertex of a great circle is the point which departs the greatest distance from the equator—the highest point of the circle reached in declination. There are two vertices 180° apart with the two points of intersection 90° in either direction. The declination or latitude of either vertex is equal to the angle at which the circles intersect each other. The intersections are called the equinoxes, and it may be well to say here that the word equinox has several meanings in navigation, often rendering it necessary to judge by the text which is intended. The vernal equinox, for instance, refers to a certain time of year—March 21st. The sun is that day directly overhead at the intersection of the equator and the terrestrial ecliptic and this point is sometimes called the vernal equinox. Again, the sun at the same time occupies a point on the heavens also known as the vernal equinox, which is at the intersection of the celestial equator and the ecliptic. The point in the orbit occupied by the earth at this time is also spoken of as the vernal equinox.

The reader is now asked to arouse his imagination and if possible to conceive himself a passenger in an aeroplane equipped with some remarkable power capable of carrying him to a position in space, above, yet a little outside, the earth’s orbit, near the Perihelion, and there to heave to and view awhile an astronomical picture. Spread out before his unrestricted vision will be the earth, its orbit, and the sun. It is to be hoped that the imagination of the reader is still sufficiently supple to suppose the plane of the orbit to be the surface of an infinite ocean stretching away beyond human conception of distance and “breaking” against the celestial sphere; the “surfline” there marks the ecliptic; the “ocean’s” surface representing the great plane of the ecliptic. The sun will be seen as if at anchor in his proper place within the orbit. The earth is “underway,” half submerged, and listed 23° 28´ toward our point of vantage. This inclination, or direction of the axis, is in a general way toward the perihelion, and within a few degrees of being parallel with the long diameter of the orbit. The earth maintains this nearly parallel position of its axis with the long diameter throughout the period of its revolution; a fact of importance to remember.

It will be readily seen that during the encircling of the sun there must be one position where the northern axis is inclined directly toward that body, another opposite where it is headed away from him, and two positions midway where the bearing of the axis (projected on the plane of the orbit) is at right angles to the bearing of the sun from the earth; another feature to be “salted down” in the memory.

If the earth revolved on an even keel, the equator and the “waterline” would be coincident, but fortunately this is not the case, and owing to the inclination of the axis another great circle is defined by the “waterline,” called the terrestrial ecliptic, being directly beneath its celestial namesake. The inclination of the northern pole being in a general way toward the perihelion, correspondingly depresses or “submerges” that half of the equator below the plane of the ecliptic, represented by the “water surface,” and at the same time the opposite side rolls the equator above it. At two points (the equinoxes) on opposite sides of the earth, and at right angles to the direction of its inclination, the equator and terrestrial ecliptic cross each other at the “water’s edge.”

The sun is always exactly overhead for that point of the earth which is nearest to it. This is an essential fact to remember in navigation. Bearing in mind that the sun is stationary and ignoring for a time the rotation of the earth, each advance in its orbit brings about a change of bearing of the sun and a new position becomes the nearest point, and thereby directly beneath the sun. The constant changing of the sun’s bearing continues throughout the year, or one revolution, and a circle of these overhead positions is marked upon the earth, which is coincident with the terrestrial ecliptic—the visionary “waterline.” It is obvious that the vertical rays of the sun must apparently follow this line, for it can only be overhead for places that are in the same plane, and this again is the level of the “ocean.”

This circle of overhead positions projected on the celestial sphere marks the ecliptic—the “margin” of the infinite ocean, and the path that the sun seems to follow eastward among the stars.

The above paragraphs show us that the sun in following this line around the earth crosses the equator twice, and twice he attains a distance of 23° 28´ from it, and so must be on the equator twice and reach a declination of 23° 28´ north and 23° 28´ south in the course of one year.

Returning to our imaginary illustration, we will now follow the peregrinations of the earth for a year and note the effect of its inclination in the different parts of the orbit upon the declination of the sun.

It will be assumed that it is the 21st of March and from our airy position we see the earth away on our right nearly 90° from the Perihelion. As this is the vernal equinox, there are a number of interesting points to be considered: The direction of the earth’s axis, projected on the plane of the orbit, is at right angles to the bearing of the sun from the earth; the sun is directly over the intersection of the equator and terrestrial ecliptic, and being overhead for this point on the equator, the declination must be 0°. Moreover, a line drawn from this intersection, or terrestrial vernal equinox, through the center of the sun and extended to the celestial sphere would strike the corresponding intersection of the ecliptic and the equinoctial or celestial equator—the celestial vernal equinox. The arrival of the earth at this position is the signal of spring for the northern hemisphere, likewise it announces the advent of autumn to our southern neighbors below the “Line.” The sun this day rises in the east (approximately) and passing through the zenith, sets in the west for those living on the equator. The explorer at the north pole is cheered by the first light as the sun appears in the horizon, while the south pole becomes enshrouded in the long Antarctic night. Without lingering for ceremonies over the change of seasons, the earth continues steadily on its way toward the aphelion; the sun’s vertical rays leave the intersection of the equator and the terrestrial ecliptic, and follow along the latter, thus widening its distance from the equator as the earth proceeds. As the ecliptic in this half of the orbit is above, or north, of the equator the former is in north latitude and the sun, following along it, is thereby also in north declination. A line from any place having the vertical rays, through the sun to the celestial sphere, always terminates on the celestial ecliptic, all being in the same plane, and shows the corresponding celestial position of the sun on it. Its declination distance from the celestial equator, in degrees, minutes and seconds, is identical with that of the place on the earth directly beneath it relative to our equator. So by showing the course of the sun’s overhead positions on the earth its celestial positions are, at the same time, indicated. The overhead position of the sun on the terrestrial ecliptic gradually departs from the equator culminating about June 21st, the summer solstice, in a declination of 23° 28´ at a point near the aphelion in the orbit, 90° (approximately) from the equinox.

The positions in the orbit of the summer and winter solstices are reached by the earth several days before the points of the aphelion and perihelion. These respective positions would be in conjunction were it not for a slow and remarkable motion of the earth’s axis before spoken of, and later to be described, called the precession of the equinoxes.

The summer solstice is the great half-way point of the earth’s annual circumnavigation of the sun; it is a matter of moment all over the world, and another great change of seasons is at hand. The sun is overhead for places along the parallel of 23° 28´ N. and bears north 23° 28´ from the zenith at noon from places on the equator.

At the north pole, since its appearance on the horizon on March 21st, the sun has mounted to an altitude of 23° 28´ and to nearly 67° at places on the Arctic circle. The earth’s northern axis is, in this position, inclined 23° 28´ directly toward the sun, which pours its rays continuously upon the northern regions, uninterrupted even by the earth’s daily rotation. It is on this day that the whole Arctic zone enjoys the full glory of the midnight sun. The earth’s continuous movement of revolution does not allow a delay of this favorable season in northern latitudes, but continues to make the sun’s vertical rays follow the terrestrial ecliptic as before on its way toward the intersection with the equator 90° away. On this leg of the journey, the sun is traveling on the upper one of two converging lines and thereby gradually lessening its distance from the other—the equator—or, in other words, reducing its declination. This continues until September 21st when the autumnal equinox is reached and the sun’s declination becomes 0°. The sun now being overhead at the intersection of the equator and the terrestrial ecliptic, is on the opposite side of the earth from the intersection of March 21st. In fact the conditions are similar, but now the earth is on the opposite side of the sun, and the change of seasons is the entrance of spring for the dwellers in southern latitudes.

The sun has dropped lower and lower in the sky at the north pole since June, until on this day it is in the horizon and it is time for the Esquimos to seek their igloos and prepare to hibernate during the long Arctic night now ushered in.

The sunshine at the time of the equinoxes is equally distributed over the northern and southern zones, and the zenith distance of the sun at noon at any place is, theoretically, equal to the latitude of the place (except a small error due to change of declination accumulated subsequent, or previous, to the instant of the equinox).

The conditions during the next six months are reversed as the earth proceeds into that half of the orbit containing the perihelion. Now the sun following the terrestrial ecliptic enters southern latitudes or south declination, for in this part of the orbit the equator is above (or north) the plane of the ecliptic. The sun’s diverging course from the equator leads it farther and farther southward until on or about December 21st it arrives at the winter solstice with a culmination of 23° 28´ south declination. At this point the earth is but a few degrees from the perihelion as it was from the aphelion at the summer solstice.

The earth’s north pole is now inclined directly away from the sun and its rays have entirely forsaken the Arctic for the Antarctic zone; notwithstanding the earth’s daily rotation, which brings alternating light and darkness to the greater part of the world, the northern polar regions are in a continuous shadow, and no sunlight reaches these remote parts. At this time of the year the northern hemisphere above the tropic of Cancer, is in an unfavorable position relative to the sun, and as a result places situated on parallels less remote than the Arctic are having long nights and short days in proportion to their latitude north. On the other hand, in the southern hemisphere the days are longer and the nights shorter, as the southern latitude increases until at the Antarctic circle night disappears and the sunshine is uninterrupted. It is seen that this is an exact reversal of the conditions at the summer solstice.

The earth enters the last quadrant of the great ellipse of its orbit, the sun now approaches the equator as the earth nears the vernal equinox. The south declination diminishes until on March 21st it becomes 0° and the earth has completed its revolution. We will now go on another tack and instead of considering only the effects of declination due to the earth’s revolution, will assume that the earth has been halted in its onward course of revolution and is making its daily rotation in the same position. The earth turning from west to east causes the sun to appear to proceed from east to west in its diurnal motion. Each rotation, requiring 24 hours, marks upon the earth a circle of overhead positions parallel to the equator and hence without change of declination. The result of such a remarkable condition would be, no change of seasons and no change in the length of the days and nights. In reality, however, we are saved from such monotony, for both the motion of rotation and revolution of the earth are acting together and giving a compound effect on the apparent movements of the sun. This alters the daily circles just mentioned to a fine spiral of overhead positions, ever changing in declination. The daily difference of the sun’s declination shown in the Nautical Almanac is equivalent to the distance between two threads of this spiral.

The change of declination is most directly seen and felt in the polar regions, where the activities of the denizens are mostly limited to the favorable phases of this change. At the north pole, after the sun has appeared above the horizon, this spiral of declination can be continuously followed. The sextant will disclose a constant increase in altitude as the sun circles round and round the sky, winding itself up and finally culminating at 23° 28´. The process is then immediately reversed. The stars here make daily circles of equal altitudes as their change of declination is insignificant; but the circles of the planets and the moon are converted into spirals, the fineness of which is in proportion to the rate of their change of declination.

The fact that the sun reaches an altitude of 23° 28´ at the pole at the summer solstice with its declination of a like amount and that on March 21st, when the sun is in the horizon with the altitude 0°, it is directly over the equator with 0° declination, shows that at this place (the pole) the altitude is equal to the declination. Should an explorer travel southward 1°, his sextant would show an altitude 1° greater than at the pole, yet moving about does not affect the declination at a given time. It follows by taking his altitude at noon the explorer in the polar regions may readily learn his distance from the pole by subtracting the declination in the Nautical Almanac from his sextant reading.

It may not generally be known that the southern summer is shorter than the summer of the northern hemisphere, but such is the case by approximately eight days. The reason of this inequality lies in the fact that the sun is nearer one end of the orbital ellipse, and the short diameter passing through this body divides the orbit into unequal parts. The smaller part being that traveled by the earth during the southern summer. Furthermore the nearer proximity of the sun causes an accelerated motion which further tends to lessen the time spent by the earth in this part of the orbit.

Right Ascension

Declination and right ascension being used together as coordinates, we will not separate them. It will be remembered that the equator and the terrestrial ecliptic cross each other on opposite sides of the earth; that on or about March 21, the sun is overhead at the intersection that is the vernal equinox. Now if at this intersection on this day a plumb-line were carried upward, it would at length reach the sun, and continued to infinity and projected on the celestial sphere would locate a point called the celestial vernal equinox, known by many as the First Point of Aries. This point is one of the most important celestial “landmarks” used in astronomy and navigation, but, unfortunately, no heavenly body marks its place. However, as its relative position among the neighboring stars is well known, its exact location is easily ascertained.

The hour circle which passes through this point is known as the equinoctial colure, and may be considered the prime meridian of the heavens, for from it is measured the right ascension of all bodies. Right ascension of a body is the angle at the celestial pole between this meridian of reference and the hour circle passing through the body. It is always measured eastward through 24 hours of sidereal time (360°). The angle is measured by the arc intercepted on the celestial equator. For example, a star 15° east of the equinoctial colure has a right ascension of 1 hour or 15°, but, if the star is 15° west, its right ascension is 23 hours or 345°.

The positions of heavenly bodies are defined by right ascension and declination exactly as positions upon the earth are expressed by longitude and latitude, right ascension corresponding to longitude and declination to latitude.

In the discussion of Time, to follow, more facts concerning right ascension will be found.

Precession of the Equinoxes

A comparison of the present positions of the fixed stars with their places as recorded in ancient times shows a great discrepancy. The celestial latitudes, which were reckoned from the ecliptic, show no appreciable change; but in the declinations and right ascensions there is a great departure from the old positions. The error of right ascension was found by the old Greek astronomer, Hipparchus, to appear as a uniform eastward movement of all the stars, which led him to reason that, instead of the stars themselves changing, their point of reference was moving westward, thus lengthening all right ascensions.

The famous astronomer after further reasoning decided that the position of the celestial pole was changing, in fact that the line of the earth’s axis was describing a circle on the heavens, which was left-handed or against the hands of a watch as viewed from the north pole of the earth. This movement was found to be extremely slow, requiring 25,800 years to complete the circle which has as its radius the amount of the inclination of the earth’s axis—23° 28´.

If a match is put through a piece of cardboard about the size of a half dollar to the distance of ¼ inch, and spun, the motion of the cardboard just as it staggers through loss of speed, gives some idea, although exaggerated, of the precession movement of the plane of the equator, which is of course infinitely slower. The movement of the top of the match is a semblance of the corresponding motion of the vanishing point of the axis on the celestial sphere.

The earth, as already explained, points its axis at practically the same spot in the heavens throughout the year, and if it were not for this annual precession of 50´´ it would for all intents and purposes hold a permanent direction. About December 21, the winter solstice, while the earth is still some degrees from the perihelion, its northern axis, is inclined directly away from the sun. Each year this distance from the perihelion is becoming greater, widening this angle between the direction of the axis, projected on the plane of the orbit, and the major diameter of the orbit, until in time the north pole will be headed directly away from the sun in that part of the orbit which the earth now occupies in September, and so on.

In the year A.D. 1250 the winter solstice occurred at perihelion and in the year 6400 A.D. the vernal equinox will occur at this point of the orbit. That is, the axis of the earth was inclined directly away from the sun at perihelion in the former year but in the latter year the inclination will have changed about 90° backward against the earth’s course about the sun, and it will be the beginning of spring (the vernal equinox) when the earth is at perihelion instead of the first of winter as in 1250 A.D. Since 1250 A.D. the inclination has changed an equivalent of about 11 days for now the earth is at perihelion about January 1st, and the solstice occurring about December 21st, shows the present relative situation.

In other words, the vernal equinox is slipping back in the orbit towards the perihelion, and as the solstices maintain their positions at 90° from the equinoxes they must likewise be “slipping a cog” each year.

The vernal equinox was situated many centuries ago in the first part of the constellation of Aries, and was known as the First Point of Aries, but owing to the movement of precession it has dropped back or westward (as we face our southern horizon) 50´´ a year until it has left that constellation entirely and is now about leaving the constellation of Pisces, some 30° from the position used by Hipparchus in his calculations. The majority of navigators still call this point of the celestial vernal equinox the First Point of Aries.

Holding these facts in mind, it may be clear that as the earth approaches that part of the orbit where the vernal equinox occurs it has turned its pole, and correspondingly points on the equator, 50´´ to the right or west during the year; thus causing the point of the terrestrial equinox to meet (or come under) the sun that much sooner. In other words, referring to the effect as seen on the heavens, the celestial equinox was advanced to the westward that much to meet the sun in its eastward movement among the stars and will become the nearest point to the sun, 50´´ before the position of the equinox of last year. As the points in the orbit where the vernal equinox occurs year by year works back toward the perihelion, the range line through the sun to the heavens beyond must each year correspondingly edge its way westward along the celestial ecliptic through different constellations. This is what is known as the precession of the equinoxes.

The course of the celestial pole in the heavens is shown by a circle drawn about the pole of the ecliptic using 23° 28´ as a radius. This path will pass 1¼° from our pole star and this position marks the present termination of our extended axis; half way around the circle it passes the first magnitude star Vega close aboard, thus making this the future pole star some 12,000 years hence. If there be such creatures as navigators in those far-away days, latitude by Vega will no doubt be a popular sight among them.

The cause of this remarkable movement of the earth is due to the fact that the earth is not a true sphere, and the influence of the sun is not exerted equally upon its mass. Its flattening at the poles is attended by a corresponding bulging along the equatorial belt. When the earth is in the vicinity of the perihelion, leaning away from the sun, the half of this ring of extra matter on the side towards the sun is above the plane of the ecliptic or orbit. The tendency of the added attraction exerted upon it, is to draw the earth to an upright position, or in other words, at this time the sun is pulling stronger on the northern or upper side than on the lower. Again, when near the aphelion and summer solstice, leaning towards the sun, that part of the ring of extra matter on the side towards the sun is below the level of the orbit, and the attraction is again tending as before to pull the earth upright. At the equinoxes there is an equal amount of this extra matter above and below the plane of the orbit evenly distributing the attraction.

The effect of this influence would in time bring the earth’s equator and the plane of the ecliptic into coincidence and the earth’s pole would be directly beneath the pole of the ecliptic, were it not for its rotation. The two forces acting upon the earth result in the slow revolution of the axis. The exact effect of these forces is rather complex but it is a demonstration of the principle of the gyroscope. The movement of the axis is affected very slightly by other influences than that of the sun, the most notable of which is the moon, whose monthly revolutions around the earth produce a similar influence in the bulging mass within the tropics, but as its revolutions are so rapid, it has but a slight effect on the precession movement of the earth. It is sufficient, however, to cause the extended axis to nod slightly and make a waved circle of precession on the heavens. This is called “Nutation,” from the Latin word nuto, meaning to nod.