The whys and wherefores of navigation

The observed altitude of a heavenly body, as measured with the sextant, requires treatment for numerous errors to reduce it to the true altitude and make it ready for use in working any of the navigational observations. The amount of error varies in different bodies, the moon requiring the maximum and the fixed star the minimum correction. All the errors are not common to all bodies, that is, with some, certain errors become so insignificant that they are cast aside.

These errors comprise the index error of the sextant, refraction, dip, semi-diameter, and parallax. In Table 46, Bowditch, will be found the combined corrections (index error excepted) to be applied to an observed altitude of a star or planet and to that of the sun’s lower limb. A supplementary table furnishes an additional correction to be applied to the semi-diameter of the sun when accuracy is desired. These corrections will now be discussed in the order named:

Index Correction

The index and horizon glasses of a sextant are supposed to be parallel when the zero of the vernier and the zero of the limb are in one, and with this the case, the true and reflected images seen in the horizon glass should exactly coincide. Any difference between them is the index error.

It is seldom that a sextant is so well adjusted that no index error exists, but it is not desirable to keep tampering with the instrument with an attempt to eliminate this error, for it will in time injure its accuracy.

By testing the sextant at each sight, the error can be closely watched and allowance made for it in correcting altitude. The easiest and most accurate method of ascertaining this error is by using a star in the following manner: Set the zero of the vernier a little to either side of the zero of the limb, and observe a 2d or 3d magnitude star—move the reflected image past the real and note if they pass directly over one another. If not, the horizon glass is not perpendicular and needs adjustment. Bring the reflected star in exact conjunction with the real star, and read off the index correction—if the zero of the vernier is to the left of the zero of the limb—on the arc—the difference is minus (-) and subtracted from the observed altitude; and if to the right—off the arc—it is plus (+) and added. A well-known rule of thumb expresses it thus: if it’s on it’s off, and if it’s off it’s on. The sea horizon is also available for determining this correction and serves the purpose with fairly accurate results.

Semi-diameter

In measuring the altitude of certain bodies for navigational purposes, it is necessary to determine the distance of the center of the body above the horizon. To accomplish this in an accurate manner the lower edge or limb is brought down to the horizon and the semi-diameter applied to this measured altitude. When the lower limb is used, as is the usual practice, the correction for semi-diameter is obviously plus (+). The upper limb can be resorted to, however, should the lower side of the body become veiled by cloud, and in this case the correction is minus (-).

Semi-diameter of the sun is obtained readily from the Nautical Almanac for each ten-day period, for it must be remembered that the sun is continually changing its distance from the earth, and consequently the diameter of the former is increased and lessened slightly at different times of the year. For instance: On January 2d, when the earth is near perihelion and we are at our nearest point to the sun, the semi-diameter is 16´ 17´´.90, while on July 2d, when we are in the remote parts of the orbit, the semi-diameter is only 15´ 45´´.69, making a difference of over 32´´.

The moon being such a near neighbor of ours gives more trouble in determining her diameter. Besides being greatly affected by her rapidly changing distance from the earth, a further correction is occasioned by the fact that our position on the surface is nearer the moon at times than is the center of the earth. That is, when the moon is in the zenith we are 4000 miles, the earth’s radius, nearer that body than when she is in our horizon. It is evident that the direction of the moon in our sensible horizon is at right angles to a perpendicular erected at our place of observation and passing through the earth’s center, and this again makes it evident that the moon is about equally distant from the earth’s center and our position on the surface; but as she ascends the heavens she comes nearer our position until in the zenith the distance has been reduced by 4000 miles and the diameter appears correspondingly larger. Draw a diagram and see for yourself. This Augmentation of the Semi-diameter, due to the altitude, is found tabulated in Bowditch, Table 18.

The semi-diameter becomes too small to consider in ordinary navigation when observing any of the planets, and of course fixed stars are beyond its scope.

Refraction

Everyone knows that the blade of an oar when dipped in the water appears to be bent in a remarkable manner at the surface. This is a clear case of refraction. Should the oar, however, be held under everywhere at an equal depth, a square look downward at it would fail to show any refraction. So it becomes evident that refraction is caused by the rays of light passing obliquely from a rarer to a denser medium or vice versa. A ray of light coming from a heavenly body to the earth passes through a medium of gradually increasing density, from the thin outer air to the denser atmosphere at the surface of the earth. The ray of light consequently becomes curved downward and reaches the earth at a point nearer the heavenly body than would be the case if the light ray traveled in a straight line. The effect of this to the observer is that the body appears higher than it really is. The difference between the actual direction of the ray of light unaffected by the air, and our line of vision as we see the body, is the refraction.

The amount of refraction ordinarily affecting an observed altitude depends upon the distance of the body above the horizon. At the zenith, the rays of light, entering our atmosphere perpendicularly, are not deflected and refraction is nil. But, on the other hand, when the body is near the horizon, the rays of light pass through the atmosphere at a sharp angle and are consequently subject to the greatest bending, thus giving us our maximum refraction. In fact, this element becomes so unreliable in low altitudes that it is not advisable to observe a body when less than 10° or 13° above the horizon. This in no way concerns bearings taken of bodies in the horizon for amplitude, as refraction affects the altitude and not the azimuth of a body.

Dip

It is a well-known fact to every seaman that by going aloft, he can pick up a light sooner than on deck; that is, the higher his elevation the wider his horizon becomes. The horizon of a man in a small boat is only about 3 miles away, but, if he climbs to the bridge of a steamer some 60 feet above the water, he finds that the horizon has receded until he has a range of about 9 miles.

The fact that the horizon can be altered by changing the altitude should appeal to every navigator as a possible means of getting a horizon in foggy weather, by going aloft or getting as low as possible, provided the fog bank is lying above or close to the water.

The altitude of a body is measured to the visible horizon, yet the measurement must be adjusted to the sensible horizon before the true altitude can be obtained. This correction is accomplished by applying to the observed altitude the amount of the angle formed at the observers eye by the planes of the sensible and visible horizons. The angle is known as the dip of the horizon. It is readily seen that this angle always makes the observed altitude too large, for the eye if located at the exact surface of the sea, theoretically sees the sensible and visible horizons in one, while at every elevation above the surface it depresses the visible horizon correspondingly. It is, therefore, always necessary to apply the dip as found in Table 14, Bowditch, with a minus (-) sign.

An inspection of the table of dip will show that the rate of increase of this error becomes more rapid as the height of the eye is diminished. To illustrate: The reader will note that between an elevation of 4 feet and one of 9 feet there is a difference of 1 minute in the dip, while higher up, say between 26 feet and 38 feet, a difference of 1 minute is likewise found, yet in the first instance there was a range of 5 feet and in the second a range of 12 feet. This fact in itself is an argument in favor of observing altitudes at a good height above the water.

In calculating a meridian altitude, an error in the dip directly affects the result by a corresponding amount, so extra care should be exercised in this respect. In this instance, we endeavor to locate the body relative to our zenith and anything that affects its altitude directly affects the latitude. In a time sight, a different principle is involved. Here the position of the body as defined by the latitude locates the apex of one angle of the astronomical triangle and hence a small error in the altitude will very likely cause a greater effect on the longitude.

An allusion was made under the caption of Refraction to the displacement of the visible horizon by terrestrial refraction to detect which requires watchfulness on the navigator’s part. The familiar “loom” seen along the coast is an example of the workings of variable refraction. Now imagine this distortion less aggravated with no land to show its existence and you have a good illustration of this error.

Refraction of this nature is usually found during light airs and calms when the different layers of air arrange themselves according to their temperatures. The heated air over land below the horizon in hot weather will displace the intervening horizon; moreover, when the air is warmer than the sea, the horizon is elevated above the normal and, when the conditions are reversed, the horizon is unduly depressed. Thus lights become visible a little sooner after a hot day ashore. The Red Sea, Gulf Stream, mouth of the Amazon, and other large rivers are places where the horizon should be especially distrusted. Capt. Lecky, in his famous Wrinkles in Practical Navigation, refers to an experience he once had with this error. The latitude had been found “by an excellent meridian altitude of the sun to be as much as 14´ in error. The time was mid-winter—the day a clear cloudless one—the sea smooth, and the horizon clean-cut. Five observers at noon agreed within the usual minute or half minute of arc, nevertheless, on making Long Island (U. S. A.) in less than two hours afterwards, the latitude was found wrong to the amount stated. Many such cases have come under the writer’s notice, but this one alone is cited on account of the magnitude of the phenomenon.”

What Captain Lecky said in his work on navigation is reliable and this should serve to make an impression as to the dangers of such occurrences.

In clear weather this displacement of the horizon may be lessened somewhat by observing from aloft. By extending the horizon, such disturbing influences as the motion of the vessel and an irregular horizon caused by rough sea are minimized. In hazy weather, however, it is recommended to observe low, bringing the horizon as close as possible.

Parallax

In calculating the true altitude of a body the distance of its center above the horizon is supposed to be measured from the center of the earth, or what is the same thing, the altitude above the rational horizon.

The application of semi-diameter adjusts the measured angle with the center of the body, while parallax corrects the error due to our observing from the surface of the earth to the sensible horizon, instead of from the center to the rational horizon.

Parallax, in other words, is the angle formed at the body by the lines drawn from the observer’s position, and from the center of the earth, respectively. This angle is subtended by the radius of the earth, and it is obvious that the farther away a body is, the smaller the angle, and consequently the less the parallax. So when dealing with planets or fixed stars, it becomes insignificant and no parallax is considered.

The moon, on the contrary, is so close aboard that the angle of parallax reaches a value of nearly 1´; as a minute of altitude means a minute of latitude and in turn a mile, so with this body the error due to parallax must be carefully determined.

In the case of the sun, however, it is somewhat of a waste of time to bother with parallax, for it never exceeds 8´´ or 9´´ and such fine calculation is uncalled for in ordinary navigation where so many greater errors must be kept in sight. However, we desire to eliminate every known element of error without undue figures, so it is recommended that Table 20B, Bowditch, be used, where without extra trouble the parallax may be found conveniently combined with the refraction.

When a body is in the sensible horizon, the parallax is greatest. The angle of parallax subtended by the radius of the earth is then an acute angle of a right-angled triangle and is as large as it can possibly be with the body at the same distance. As the body obtains altitude above the horizon, the right angle of the triangle (at the observer) becomes obtuse and our acute angle of parallax becomes smaller and smaller until the body reaches an altitude of 90°—in our zenith, when the obtuse-angled triangle has resolved itself into the perpendicular line that passes through our position and the earth’s center. The angle of parallax here disappears.

When a body is in the horizon, its parallax is known as Horizontal Parallax in contra-distinction to Parallax of Altitude. The latter has become generally known among navigators merely as parallax.

Our position on the surface causes a body to appear lower than if viewed from the center of the earth, so the error of parallax is added to the observed altitude; when, however, it is combined with refraction it is subtracted in an observation of the sun, but added when the moon is used.

The parallax of the moon is excessive because the radius of the earth becomes a considerable amount when compared with the close proximity of the body, and causes a considerable angle at the body between the lines drawn from the observer and that drawn from the center of the earth. The change in parallax is so great that it becomes necessary in order to preserve accuracy to correct the observed altitude for index correction, dip and semi-diameter, to secure an approximate corrected altitude before attempting to correct for parallax. The horizontal parallax, which is the angle subtended by the earth’s radius when the moon is in the horizon, is taken from the Nautical Almanac, and with this and the approximate altitude as arguments, enter Table 18, Bowditch, and pick out, having regard for correction tables at the right, the parallax and refraction combined.

The usual corrections to the observed altitude of the sun or stars can be picked out at once from Table 46, Bowditch, where they are all combined for a quick correction.