The Whys and Wherefores of Navigation/Chapter II

by Gershom Bradford II

The sun is the center of the solar system, with all the planets, including the earth, revolving around it, some with orbits greater and others less than that of the earth. The planets in some instances have satellites revolving in turn around them, as the moon in the case of the earth.

The movements of the earth will be, perhaps, more readily understood by assuming a position at the North Pole; here beneath the observer the earth is rotating lefthanded—against the hands of a watch, once each day; while at the same time it is speeding onward through space in a left-handed curve, which in the course of a year resolves itself into a complete revolution around the sun.

The sun and stars are considered to be stationary for all navigational purposes, the apparent movements of these bodies being entirely due to the motions of the earth. The result of our daily rotation from west to east is, that the heavenly bodies march past our meridian m a majestic procession for 24 hours, after which the performance is repeated. The uninitiated are here forewarned against becoming confused by the assumption used for convenience by navigators that the heavens revolve around the earth.

The planets and moon join the sun and stars in the daily parade past our meridian, but their apparent movements are not entirely an illusion, for they have motions of their own that somewhat affect the precision of their daily revolution. This is readily observed in the moon’s hour of rising, which is very perceptibly later each evening. Her actual revolution around the earth, being from west to east, is contrary to the apparent diurnal motion and thereby each evening finds her farther to the eastward and consequently rising later. As a result of this change in the time of rising, the moon must of necessity rise in every hour of both day and night in the course of a month—the time she requires to revolve around the earth.

The onward movement of the earth in its orbit as we face the sun in latitudes north of the tropics, is toward our right, and this causes the sun to apparently move slowly eastward or to the left among the stars, corresponding exactly to our movement westward. This movement is opposite to his daily course across the heavens. As a year is required for the earth to accomplish our revolution around the sun, it follows that this same length of time is consumed by the sun in making its apparent eastward revolution of the heavens.

The movements of the planets are more complex. They all revolve around the sun in the same direction as the earth, but as their orbits are of vastly different sizes, they will be found in various positions relative to the sun; they overhaul and pass each other, but owing to their uniform direction of revolution they never meet. The planetary system is like the horse race at a county fair; the pole horse has the advantage, but the varying speeds of the contestants soon place them at various parts of the track.

From the earth the movements of the planets, aside from the diurnal movement, are composed of their own actual movement around the sun, combined with an apparent motion, due to the earth’s onward movement precisely as described above in the case of the sun. The combined movement of a planet may be noted by reference to the fixed stars beyond it.

The positions of heavenly bodies are determined by two measurements—coordinates—-the distance north or south of the celestial equator, called the declination, and the distance east from the prime celestial meridian taken as a reference, called the right ascension, each of which will be subsequently treated at length under its individual heading. The movement of the planets eastward or westward as described, constantly changes their right ascension; and as their orbits are inclined at different angles to the celestial equator, they are always changing their declination.

The planets whose orbits are smaller than that of the earth are called inferior, while those whose orbits are of greater dimensions are known as superior planets. Mercury and Venus are inferior planets and consequently are always nearer the sun; their comparatively close proximity making them appear to us as morning and evening stars. In fact, Mercury is so close that it is unavailable, owing to the brilliancy of the sun, for observation with a sextant; while Venus, on the other hand, a little more remote, is an excellent body to observe, and is always found in the east or west, conveniently near the prime vertical, the most favorable place for a time sight for longitude. The twilight or dawn which usually prevails at the time of a Venus sight gives the navigator a good horizon to observe upon. Mars, Jupiter and Saturn are superior planets and their travels are so extended that they may be found almost anywhere in the heavens within the limits of their declinations.

The earth’s orbit is slightly elliptical, with the sun located a little out of center—a little nearer one end. Should a line or axis be drawn through the long diameter, its intersection with that part of the orbit nearest the sun is called the Perihelion while the opposite point is known as the Aphelion. The former is used as a point of reference from which the earth’s position can be located in terms of angular measurement from time to time. This angle, known as the anomaly, is formed by the line from the sun to the Perihelion and that drawn from the sun to the earth. The latter distance is called the radius vector of the earth. We (the earth), are at the Perihelion about January x, and consequently this angle at that date is 0°, but from this time on, the angle increases approximately one degree a day throughout the year.

he plane of the earth’s equator makes at all times an angle of about 23° 28’ with the plane of its orbit. This is a highly important angle to mankind, for upon it depends the climate of the world. The axis of the earth, if we can conceive it as represented by a slender imaginary staff, extends through the unlimited distance to a point in the heavens—the celestial pole; this point is in the zenith for a person at our north pole. Since the distance between these points is mathematically infinite, any number of lines parallel to this “staff” will appear to penetrate the sky at the single point of the celestial pole. Thus the parallel positions of the axis corresponding to the earth’s various positions, even those at opposite sides of the orbit, converge into this common point. To be clearer, the parallel lines representing the different positions of the axis during the year according to our geometry form a group of separate points on the heavens, but the distance being beyond all reckoning, our limited conceptions fail to identify the group of points and it resolves into one point.

By the same line of reasoning the plane of the earth’s equator remains parallel in all its positions throughout the yearly cruise around the sun, and its projection marks but one celestial equator upon the sky.

While the direction of the axis and corresponding position of the equator are constant for all practical purposes, there is, nevertheless, an extremely slow circular movement of the axis, called the precession of the equinoxes, a subject which is reserved for subsequent discussion.

In nautical astronomy the earth is assumed to be the center of space with the heavens forming a globular shell around it, known as the celestial sphere. All fixed stars are assumed to lie on its concave surface from the earth regardless of their actual distances. The tracks of all other bodies moving, or appearing to move, across the sky are considered to be on the surface of this sphere.

It is necessary, in order to conveniently define the position of heavenly bodies to mark this celestial sphere with imaginary circles to serve as coordinates, as we mark the earth with meridians of longitude and parallels of latitude. Before going into the explanation of these coordinates, it may be well to consider a few facts concerning circles. A great circle is of course understood to be one whose plane passes through the center of a sphere, dividing it into two equal parts. There can be an infinite number of these circles whose planes cut the sphere at every possible inclination as long as they pass through its center. A circle may be a great circle of either the celestial sphere, the earth, or even of a baseball. The poles of a great circle are the points on the surface of its sphere, penetrated by the diameter perpendicular to the plane of the great circle. As for example, the poles of the earth are connected by the diameter that is perpendicular to the plane of the equator. An angle at any pole is measured on the great circle which subtends it. For instance, angles at the poles of the earth are measured on the equator; angles at the zenith on the horizon. With these facts well in mind we will proceed, showing the scheme of circles employed in laying off the surface of the heavens.

There are three systems of circles, each designed to fulfil a different requirement.

The first system depends upon the position of the observer and changes its whole imaginary structure to correspond with his movements. The plumb-line, if extended to the heavens overhead, will determine the zenith, the point of origin of this system on the celestial sphere.

The corresponding point directly beneath us is known as the nadir.

The great circle of the celestial sphere everywhere equally distant from both the zenith and the nadir is the horizon. It is plain that a new zenith and new horizon are created with every movement of the observer. The facts that man is on the surface and not at the center of the earth, and that his eye is elevated above its surface, each creates another horizon.

The rational horizon is marked by a plane, perpendicular to a plumb-line and passing through the earth’s center; while the sensible horizon is determined by a plane, also perpendicular to the plumb-line, but passing through the eye of the observer. It will therefore be seen that these two parallel horizons are some 4000 miles apart, the semi-diameter of the earth; but this distance when projected on the celestial sphere becomes insignificant when compared with the infinite distance of this sphere from the earth, and the rational and sensible horizons shrink into a single line so far as we can perceive.

While this statement is true when dealing with the stars, it needs modifying when dealing with the sun and moon, and in very accurate observations of planets, as their distances are insufficient to eliminate the angle formed between the line from the body to the center of the earth, and that from the body to the observer. This is allowed for when observing these bodies by applying the correction of parallax to the observed altitude.

The visible horizon is the boundary seen between the sea and sky. If the observer’s eye were at the level of the sea, his visible horizon would coincide with the sensible horizon, defined above; but the elevation above the surface from which sights are taken causes the line of vision, tangent to the sea, to be depressed below the plane of the sensible horizon making an angle with it called the dip of the horizon. In practice all altitudes of heavenly bodies taken from a vessel are measured to the visible horizon and corrected for the dip to reduce them to the sensible horizon, then again corrected for parallax to obtain the true altitude of the body above the rational horizon; or what is the same thing, the altitude as observed at the center of the earth.

From the zenith, an infinite number of great circles, known as vertical circles, sweep around the celestial sphere, cutting the horizon at right angles and passing through the nadir. The one which cuts the north and south points is called the celestial meridian, and is evidently a projection of the terrestrial meridian. The vertical circle passing through the east and west points is called the prime vertical, and has a distinction above other vertical circles by virtue of its being the most favorable position for a body in observations for longitude. The heavens are further swept by an infinite number of parallels of altitude which are, as their name implies, parallel to the horizon.

The azimuth of a body is its angular distance from the north or south points of the horizon, determined by the angle formed at the zenith, or by the arc of the horizon between the meridian of the observer and the vertical circle passing through the body. Amplitude is the angle at the zenith formed by the prime vertical and the poles.

The angle formed at the pole by the hour circle passing through a body and a local meridian is the hour angle of that body, and is measured westward through 24 hours, although A.M. hour angles of the sun are reckoned eastward through 12 hours.

At the north pole where the zenith is identical with the celestial pole, the vertical circles, parallels of altitude and rational horizon are coincident with the hour circles, parallels of declination and the equator, respectively; but departing from this point they form angles with each other corresponding to the degrees of latitude from the pole; at the equator the angle reaches 90°.

The system of circles described above is by far the most extensively used, and positions determined by its coordinates are comparatively constant, but there is still a third system of circles which was used and handed down to us by the ancients. In the place of the celestial equator, a similar great circle is used, known as the ecliptic. This circle is determined by the extension of the plane of the earth’s orbit to the celestial sphere. The poles of the ecliptic everywhere 90° from this circle are the points from which meridians depart as upon the earth. The prime meridian of this system passes through the intersection of the celestial equator, and the ecliptic— the vernal equinox or First Point of Aries. Celestial latitude and longitude are the coordinates used with this system, but navigators universally prefer to use the well-known declination and right ascension. Hence the path of usefulness of the former seldom leads beyond the observatories.