4. ALTITUDES AND LATITUDE BY MERIDIAN ALTITUDE.
13 MEASURING AND CORRECTING THE ALTITUDE OF A CELESTIAL BODY
52. TAKING A SEXTANT ALTITUDE – A “SIGHT”
“Taking Sights” is the nautical term for measuring the altitude of one or more celestial bodies with a sextant in order to ascertain either a position line or a vessel’s position at sea. The actual taking of an observation is very simple and fig. 13.1 illustrates what the sextant actually does. The index and horizon glasses are shown together with the sextant arc, and the Sun is shown in the sky. The observer places his eye at the end of the sextant telescope (not shown) and looks directly through the horizon glass at the horizon. When he has moved the index bar 30º from zero he has really made the angle at his eye 60º (owing to the sextant being an instrument of double reflection – see the beginning of this chapter), and as this is the altitude of the sun, he should just see the reflected image of the Sun (through the silver portion of the horizon glass) resting on the horizon.
The altitude of a celestial body is the angle of the centre of that body above the sea horizon, and in the case of the stars and planets this is the angle measured with the sextant, but as the Sun and the Moon each have a perceptible disc, their centres cannot be observed accurately. The part of the circumference of the Sun or Moon which we observe is one of the edges called the limb; thus in fig. 13-1.
An observation is being taken of the Sun’s lower limb. In this case, we would say we had taken a “sextant altitude of the Sun’s lower limb”, sometimes abbreviated to “sext. Alt. Sun’s L.L.,” but more usually simply by the symbol. ☉ On occasions when the Sun’s upper limb is observed, the symbol ☉. would be used. For the Moon, the symbols used are Ͼ whith ¯. for the upper limb and Ͼ For the lower limb.
In taking an observation, the navigator must choose a position in the vessel where the greatest steadiness will be obtained and in which he can stand comfortably with his feet and legs firmly braced and the upper part of his body free to bend and sway – generally it is better to be near the centre line of the vessel since there will be less motion there.
In a small yacht the best positions are the forward end of the cockpit close to the cabin, or on deck supported by the mast or shrouds. In a seaway a high position is preferable, where the effect of nearby waves obscuring the horizon will be less.
53 Taking a Sextant Altitude of the Sun
The beginner should practice continually bringing the Sun down to the horizon, and at first, the telescope may be dispensed with – just look through the telescope collar, thus getting a much bigger field of view.
Before taking any observation of the Sun, the Shades must be used to eliminate glare, otherwise, it will be impossible to see the Sun at all and considerable damage may be done to the eyes. The Shades may be used singularly or in combination, and when the Sun is very brilliant, two Shades are often necessary. Generally, it is best to turn in the lightest coloured Index and Horizon Shades to commence with, altering this to “deeper” Shades if the Sun appears too strong.
With the Index bar clamped approximately at zero (0º) on the arc, hold the sextant vertically and point the telescope collar and horizon glass directly at the Sun. It will be obvious at once whether the correct Shades have been used, since the Sun’s face should be clearly defined as a bright coloured circle, neither too strong or too weak. If the glare makes it impossible to see the Sun properly it will be necessary to turn in deeper Shades before again directing the sextant at the Sun. At this moment, the true image of the Sun (as seen through the un-silvered portion of the horizon glass) is coincident with the reflected image (as seen through the silvered portion of the horizon glass) because the index bar is set at zero.
Now the index bar clamp is pressed, and kept pressed, to allow the index bar to slide easily along the arc, and, with the eye fixed on the reflected image of the Sun (in the silvered portion of the horizon glass) the index bar is pushed slowly and smoothly away from the body. The true and reflected images will be seen to separate, and the reflected image will descend towards the horizon at the same rate as the observer pushes the index bar along the arc, the sextant being swung downwards to follow the reflected image.
When the reflected image is at or near the sea horizon, which can be seen through the plain part of the horizon glass, the angle on the sextant will be the approximate altitude. The index bar clamp should now be released, and exact contact of the Sun’s lower limb on the horizon made with the tangent or micrometer screw. When the bottom edge of the Sun just touches the horizon it is termed “kissing” the horizon.
Since a celestial body will be either rising or falling it can only “kiss” the horizon momentarily unless the observer turns his tangent or micrometer screw at the same speed as the movement of the body, so as to make certain the body is just touching the horizon when the observation is made and recorded. It is also important that the reflected image of the Sun be brought down exactly underneath the actual Sun and the angle measured with the sextant held vertically.
Fig. 13-2 shows what the observer sees when he has brought the reflected image of the Sun down to the sea horizon. The large circle shows the line of sight through the telescope collar to the horizon glass, in which the reflected image of the Sun is seen kissing the horizon, one half visible on either side of the line joining the plain and silvered glass portions of the horizon glass.
Now the observer will see that the Sun’s lower limb could be made to “kiss” the horizon at a number of places so the sextant must be rotated slightly to the right and left (pivoted on the handle by the wrist round the index glass as in centre-pendulum fashion), so that the reflected Sun is made to swing off the horizon on each side describing the lower arc of a circle exactly as shown in fig. 13-2. The altitude should be taken at the lowest part of the swing when the Sun just skims the horizon, and when this happens the plane of the sextant must be vertical and the point be directly under the Sun. Fig. 13-3 shows when the reflected image of the Sun is correctly kissing the horizon, as seen in the horizon glass.
Beginners should practice the whole of the operation described above repeatedly and frequently until it can be done quickly and accurately. Once experience has been gained in this method, however, the more usual way of observing the Sun’s altitude can be adopted. In this, the sextant is simply pointed at the horizon directly underneath the Sun and the index bar moved backwards and forwards until the Sun is located. To observe the Sun properly, the telescope should, of course, be shipped and focussed to suit the eye of the observer, index and horizon Shades as necessary turned in, and the sextant directed at the horizon beneath the Sun. The index bar is then moved as described above until the Sun is near the horizon, clamped and exact contact made with the micrometer or tangent screw. At the instant of “good contact” the observer must call out “stop” to an assistant stationed by the chronometer or deck watch, who takes the exact time the altitude was observed.
The observer must not move the micrometer screw after ‘Stop’ has been called, but must read the altitude on the sextant at this instant, to be recorded together with the time in the sight notebook.
54. TAKING A SEXTANT ALTITUDE OF A STAR OR PLANET
Stars and Planets should always be observed during morning and evening twilight when the horizon is clearly defined, although Venus ♀and Jupiter♃can occasionally be observed during daylight if the approximate altitude is calculated in advance and set on the sextant.
No sextant Shades are required for star work, so they should be folded back out of the way. A bright star can be brought down to the horizon in the same way as has been described for the Sun, by pointing the sextant telescope at the star with the index bar clamped at zero, then slowly and smoothly pushing the index bar forward until the star is at or near the sea horizon and making final contact with the tangent or micrometer screw. In this case, the centre of the star should cut the horizon.
It is however, much simpler and generally quicker to “take the horizon up to the star (or planet)” This is especially valuable with a high altitude star, or a faint star (2nd or 3rd magnitude). In this method, the sextant is held in the LEFT hand with the index bar approximately at zero and the arc uppermost (as shown in fig. 13-4) with the end of the arc resting on the forehead and the telescope pointed straight at the star so that the star is seen in the plain portion of the horizon glass. Next, keeping the star fixed in sight in the plain glass, unclamp the index bar with the right hand and move the index bar away until the sea horizon appears near the star. Then clamp the index bar, reverse the sextant to the normal position and bring the star into exact contact with the horizon using the micrometer screw. A star or planet, of course, has no appreciable limb, so the centre of the star is brought to the horizon and made to cut it as accurately as possible.
55. THE THREE HORIZONS AND FOUR ALTITUDES.
The measured altitude is referred to as the sextant altitude. If the sextant possesses index error the sextant altitude must be corrected by applying the index error, to give an angle which is known as the observed altitude. The observed altitude of a celestial body may be defined as the angle at the eye of the observer contained between the apparent direction of the object and the sea-line, measured in the plane of the vertical circle on which the body lies (see fig. 13-5).
The sea-line or visible horizon is that circle which bounds the observers view at sea. It may be defined as a circle, at every point on which the sea “meets” the sky. In order to determine his position a navigator is concerned not only with the altitude of a celestial body above the horizon which he actually sees, but also with its altitude above the celestial horizon.
The celestial horizon is a great circle on the celestial sphere, every point of which is 90º from the observer’s zenith and whose plane passes through the Earth’s centre (see § 1). The celestial horizon is sometimes referred to as the rational horizon (RCH2 in fig. 13-6).
The true altitude of a celestial body is the arc of a vertical circle between the body and the celestial horizon. Having measured the sextant altitude of the body above the visible horizon, a navigator applies certain corrections to this measured altitude in order to obtain the true altitude above the celestial horizon. In order to obtain the true altitude, five corrections must be applied to the sextant altitude:-
1. Index Error (I.E.) – 2. Dip. – 3. Refraction.
4. Semi-Diameter. – 5. Parallax
For reasons which will become apparent, they must always be applied in the above order.
Index Error is an instrumental error inherent in the particular sextant used to measure the altitude and the methods of finding and applying it have been described earlier in this chapter. The Index Error is added to the sextant altitude when plus, and subtracted when minus.
Dip is the angle between the horizontal plane through the observer’s eye (called the sensible horizon) and the apparent direction of the visible horizon. In fig. 13-6, the visible horizon to an observer at 0 is the small circle on the Earth’s surface, VV₁ . It can be seen that the visible horizon is below the horizontal plane through the observer’s eye (SOH₁) – the sensible horizon. The angle of dip (SOH) is the depression of the visible horizon below then sensible horizon. Obviously the angle of dip depends upon the height of eye of the observer above sea level, being greater the higher he is. Any altitude measured by an observer will therefore be increased by the angle of dip, which must therefore be subtracted from the observed altitude to give what is called the apparent altitude. The apparent altitude is defined as the sextant altitude corrected for Index Error, and Dip. A table giving the ‘Dip of the Sea Horizon’ is included in all ‘Nautical Tables’.
At this point we have referred to three different altitudes, which the student may find a little confusing, so let us stop for a moment and sum these factors up in one single diagram, (fig. 13-7).
In the figure, the small circle represents the Earth and the outer circle the vertical circle through an observer at A and a celestial body X. The point C is the centre of the Earth and Z the observer’s zenith. The three horizons for the observer whose height of eye is at A are:-
The Visible Horizon – AVH₁ – the circle bounding the observer’s view at sea.
The Sensible Horizon – SAH₂ – the horizontal plane through the observer’s eye.
The Celestial (or Rational) Horizon – RCH₃ – The horizontal plane through the centre of the Earth parallel to the sensible horizon.
The four altitudes for the observer whose height of eye is at A and who observes the celestial body X are:-
- The Sextant Altitude – angle XAV uncorrected for index error of the sextant.
- The Observed Altitude – angle XAV after correction for index error of the sextant.
- The Apparent Altitude – angle XAS – the altitude of a body above the sensible horizon.
- The True Altitude – angle XCR – the altitude of a body above the celestial horizon.
Fig. 13-7. it can be seen that the observer measures ∠XAV with his sextant. He should really measure ∠XAS, the angular height of the body above his sensible horizon SAH₂ ; but ∠SAV is the angle of dip which must be subtracted from ∠XAV to give XAS indirectly.
The sensible and celestial horizons are purely imaginary and are astronomical conveniences introduced to facilitate the steps in correcting from observed to true altitude, the centre of the Earth being selected as the common point of reference to which observed altitudes must be reduced before a “sight” can be utilised for position finding.
We can now move on to discuss the three remaining corrections in converting a sextant altitude to a true altitude.
Refraction is the bending of light from its path, and it occurs when a ray of light passes from one medium to another of different density. Since the density of the air surrounding the Earth grows less as the distance from the Earth’s surface increases, a ray of light from a celestial body passes continually from one medium to another of greater density, from the moment it enters the surrounding envelope of air until it reaches the observer. Its path is therefore curved, as shown by the line XTSRQO in fig. 13-8. For this reason an observer sees the celestial body X as if it were at X’, with an apparent altitude HOX’ which is greater than its actual altitude above the horizontal plane through O. The difference between these two altitudes is the angle of refraction, and it must be subtracted from the apparent altitude.
The value of refraction depends upon the altitude of the celestial body. It is greatest when the altitude is zero i.e. when the body is on the horizon, with a maximum value of about 33’. When the altitude of the body is 90º, the refraction is nil, because the ray of light which enters the observer’s eye from the object passes through the atmosphere without bending. (ZO in fig 13-8).
The positions of the Sun and Moon given in the “Nautical Almanac” are the positions of their centres. Because of their size and nearness to the Earth, an observer sees the Sun and Moon not as points of light but as bodies which have an appreciable diameter, and when finding their altitudes, he measures the angles between the horizon and their lower or upper limbs, then applies a correction known as semi-diameter to determine the altitudes of their centres.
Semi-Diameter (S.D.) is the angle at the Earth subtended by the radius of the body observed, as shown in fig. 13-9. When the lower limb is observed, the S.D. correction is added to the altitude of the limb, and when the upper limb is observed, it is subtracted from the altitude of the limb.
The Sun’s S.D. varies between 16.3’ and 15.8’ during the course of the year, and the Moon’s S.D. varies between 16.7’ and 14.7’. The precise values of S.D. for both the Sun and the Moon are given in the daily pages of the ‘Nautical Almanac’ at the foot of the Sun and Moon columns.
By applying the corrections so far considered to the sextant altitude of a celestial body’s upper or lower limb, an observer is able to find the altitude of the celestial body’s centre above the sensible horizon. But the true altitude is the altitude of the celestial body’s centre above the celestial or rational horizon through the centre of the Earth. It remains for the observer to allow for the Earth’s radius.
The true altitude of any body of the solar system (Sun, Moon or Planet) is always slightly greater than the altitude of the body above the sensible horizon by an angle which depends on –
The distance of the body from the Earth and
The apparent altitude of the body.
The angle by which the true altitude exceeds the altitude of any celestial body above the sensible horizon is known as parallax in altitude, and this may be defined as the angle at the centre of the observed body between the directions of the observer and the Earth.
In fig. 13-10, C is the centre of the Earth, O the observer, OH the sensible horizon and X the centre of a celestial body. The altitude of X above the sensible horizon is angle HOX, but the true altitude of X is the angle RCX. The correction to apply to HOX is the parallax in altitude of X, which is the angle OXC.
Parallax for any body is greatest when the body is on the sensible horizon (known as the horizontal parallax) and decreases with altitude until it is zero when the body is at the observer’s zenith. Only in the case of the Moon is the horizontal parallax large (about 60’), For the Planets the maximum horizontal parallax is about 30”, for the Sun about 8”, and for the stars it is negligible. Precise values will be found in Nautical Tables.
59. CORRECTING THE SEXTANT ALTITUDE TO TRUE ALTITUDE
The individual corrections to the sextant altitude (except of course Index Error) may be obtained from any of the standard volumes of nautical tables (Burton’s, Nories’ and Inman’s). Rather than apply each of the corrections described in this chapter individually, it is more usual to apply them in the form of as total correction and again each of the standard volumes of Nautical tables provides a Total Correction table for the Sun, one for the Stars and Planets, and another for the Moon.
It is probably most convenient, however, to use the total correction tables which are now included in the Nautical Almanac and which will be found on the backs of its covers and on the fly-leaves. These include a table for Dip, which is applied in all cases, and additional separate tables for the Sun, Stars and Planets (at the front of the Almanac), and for the Moon (at the back of the Almanac). THESE ALTITUDE CORRECTION TABLES WILL BE FOUND ON PAGES 11-13 OF THE A.N. PAMPHLET SUPPLIED WITH THIS STUDY.
The Dip correction is first to be applied to the observed altitude (i.e. the sextant altitude after the application of any Index Error) in order to obtain the apparent altitude. The apparent altitude is used as an argument when entering the main Correction Table.
The Sun Altitude Correction table provides for lower and upper limb observations. The corrections for lower limb observations are printed in heavy type and those for the les frequently used upper limb observations are printed in light type. Toa allow for the change in the Sun’s semi-diameter the Sun Altitude Correction table is in two parts, one for use during October-March, and the other for use during April-September, and includes corrections for refraction, semi-diameter and parallax.
The Altitude Correction table for the stars and planets is entered with the body’s apparent altitude and the main correction is extracted. An additional correction required for Venus ♀and Mars ♂ allowing for parallax and phase; this additional correction varies with the time of year and with the altitude of the planet. The Correction Tables for the Moon will be described in a later chapter of the study.
All the tables described above for the Nautical Almanac are so-called “critical” tables, so that no interpolation is required, the figure given in the right-hand column being nearly enough correct for all values between the two tabulated figures (one slightly above it and the other slightly below it) in the left-hand column.
The figure with which to enter the tables, if different from a tabulated figure in the left-hand column, will always be in the interval between two tabulated figures, and the correction to extract from the right-hand column is that opposite the interval in question. If the entered figure is exactly the same as a tabulated figure, regard it as the larger limit of the interval for which the correction is to be taken. For instance, in the Sun Altitude Correction Table, if the apparent altitude is 35o 30’ this lies in the interval between 35o 17’ and 37o 26’ (April – September) and the correction to extract would be +14.7’ but if the apparent altitude is 52o 44’ this corresponds exactly to a tabulated altitude (April – September) and the correction would be +15.2’.
Some examples are now given of the correction from sextant altitudes to true altitudes for each of the Sun, a star and a planet. These should be studied closely in conjunction with the ALTITUDE CORRECTION TABLES ON PAGE 11 OF THE A.N. COURSE PAMPHLET.
The Altitude Correction Tables in the Nautical Almanac are based on a mean refraction corresponding to that in which the air has a se-level pressure of 1016 mb (30” of Mercury) and a temperature of 10o C (50o F.)
An additional correction table is provided in the Nautical Almanac for use when the atmospheric conditions of pressure and temperature are non-standard, but inspection of this table will show that the additional correction to apply in such cases only becomes significant at altitudes below 10o. Since it is normal to avoid taking altitudes of less that 10o, most navigators avoid this extra complication.
Practice correcting sextant altitudes to true altitudes with the following self-test exercise, comparing results with the answers given below. Use the Altitude Correction tables on page 11 of the AN pamphlet supplied with this ‘Ocean Navigation’.
The table below shows the sextant altitudes, the celestial body observed, the index error of the sextant, the height of eye of the observer and the date of the observation. In each case find the observed altitude, the apparent altitude and the true altitude of the body.