Chapter 6
Now we must go back to some of the formulas we learned when discussing star time and apply them with the information we now have from the Nautical Almanac. If the G.M.T. on April 20th is 4h--16m--30s, what is the G.S.T. for the same moment? That is, when Greenwich is 4h--16m--30s from the sun, how far is Greenwich from the First Point of Aries? You remember the formula was G.S.T. = G.M.T. + (.).R.A. + (+).C.P.
G.M.T. 4h--16m--30s (.).R.A. 1 --50 -- 6 (+).C.P. 0--42 -------------- G.S.T. 6h--07m--18s
Suppose you were in Lo. 74 deg. W. What would be the R.A.M. (L.S.T.)? You remember the formula for L.S.T. from G.S.T. was the same relatively as L.M.T. from G.M.T., i.e.,
L.S.T. = G.S.T. - W. Lo. + E. Lo,
Here it would be
G.S.T. 6h--07m--18s (74 deg. W) - 4 --56 --00 --------------- L.S.T. 1h--11m--18s
Now these are not a collection of abstruse formulas that you are learning just for the sake of practice. They are used every clear night on board ship, or should be, and are just as vital to know as time by the sun.
Suppose you are at sea in Lo. 70 deg. W and your CT is October 20th 6h--4m--30s A.M., CC 2m--30s fast. You wish to get the R.A. of your M, i.e., the L.S.T. How would you go about it? The first thing to do would be to get your G.M.T. It is CT--CC.
20d--06h--04m--30s A.M. --12 ------------------ CT 19d--18h--04m--30s CC --02 --30 ------------------ G.M.T. 19d--18h--02m--00s
Then get your G.S.T.
Oct. 19d--18h--02m--00s (.).R.A. 13 --47 --38.5 (+).C.P 2 --57.7 -------------------- 19d--31h--52m--36.2s --24 -------------------- G.S.T. 19d-- 7h--52m--36.2s
Then get your L.S.T.
G.S.T. 7h--52m--36.2s W.Lo (--) 4 --40 -------------- L.S.T. 3h--12m--36.2s
The last fact to know at this time about the Almanac is found on pages 94-95. Here is given a list of the brighter stars with their positions respectively in the heavens, i.e., their celestial longitude or R.A. on page 94 and their celestial latitude or declination on page 95. These stars have very little apparent motion. They are practically fixed. Hence, their position in the heavens is almost the same from January to December though, of course, their position with relation to you is constantly changing, since you on the earth are constantly moving.
The relationship between these various kinds of time is clearly expressed by the following diagram, which put in your Note Book:
Assign for reading in Bowditch, Articles 294-295-296-297-299-300-301-302-303-304-305-306-307.
If any time is left, have the class work out such examples as these:
1. G.M.T. June 20th, 1919, 5h--14m--39s. In Lo. 68 deg. 49' W. Required L.S.T., G.S.T., L.M.T., L.A.T.
2. L.M.T. Oct. 15th, 1919, 6h--30m--20s A.M. In Lo. 49 deg. 35' 16" E. Required L.S.T.
3. L.M.T. May 14th, 1919, 10h--15m.--20s A.M. Lo. 56 deg. 21' 39" W. Required L.A.T.
4. W.T. April 20th, 1919, 11h--30m--14s C-W 2h--14m--59s CC 4m--30s slow. In Lo. 89 deg. 48' 30" W. Required G.M.T., G.A.T., L.M.T., L.A.T., G.S.T., L.S.T.
5. What is Declination and R.A. on May 15th, 1919, of Polaris, Arcturus, Capella, Regulus, Altair, Deneb, Vega, Aldebaran?
6. What is the sun's declination and R.A., Time at Greenwich, July 30th:
7h--14m--39s A.M. 4h--29m--14s A.M. 3h--04m--06s 11h--49m--59s 2h--14m--30s A.M.?
SATURDAY LECTURE
CORRECTION OF OBSERVED ALTITUDES
The true altitude of a heavenly body is the angular distance of its center as measured from the center of the earth. The observed altitude of a heavenly body as seen at sea by the sextant may be converted to the true altitude by the application of the following four corrections: Dip, Refraction, Parallax and Semi-diameter.
Dip of the horizon means an increase in the altitude caused by the elevation of the eye above the level of the sea. The following diagram illustrates this clearly:
If the eye is on the level of the sea at A, it is in the plane of the horizon CD, and the angles EAC and EAD are right angles or 90 deg. each. If the eye is elevated above A, say to B, it is plain that the angles EBC and EBD are greater than right angles, or in other words, that the observer sees more than a semi-circle of sky. Hence all measurements made by the sextant are too large. In other words, the elevation of the eye makes the angle too great and therefore the correction for dip is always subtracted.
Refraction is a curving of the rays of light caused by their entering the earth's atmosphere, which is a denser medium than the very light ether of the outer sky. The effect of refraction is seen when an oar is thrust into the water and looks as if it were bent. Refraction always causes a celestial object to appear higher than it really is. This refraction is greatest at the horizon and diminishes toward the zenith, where it disappears. Table 20A in Bowditch gives the correction for mean refraction. It is always subtracted from the altitude. In the higher altitudes, select the correction for the nearest degree.
You should avoid taking low altitudes (15 deg. or less) when the atmosphere is not perfectly clear. Haziness increases refraction.
Parallax is simply the difference in angular altitude of a heavenly body as measured from the center of the earth and as measured from the corresponding point on the surface of the earth. Parallax is greatest when the body is in the horizon, and disappears when it is at the zenith.
When the angular altitude of the sun in this diagram is 0, the parallax ABC is greatest. When the altitude is highest there is no parallax. The sun is so far away that its parallax never exceeds 9". The stars have practically none at all from the earth's surface. Parallax is always to be added in the case of the sun.
The semi-diameter of a heavenly body is half the angle subtended by the diameter of the visible disk at the eye of the observer. For the same body, the SD varies with the distance. Thus, the difference of the sun's SD at different times of the year is due to the change of the earth's distance from the sun.
The SD is to be added to the observed altitude in case the lower limb is brought in contact with the horizon, and subtracted if the upper limb is used. Probably most of the sights you take will be of the sun's lower limb, i.e., when the lower limb is brought in contact with the horizon, so all you need to remember is that in that event the SD is additive.
Now at first we will correct altitudes by applying each correction separately, but as soon as you get the idea, there is a short way to apply all four corrections at once. This is done in Table 46. However, disregard that for the moment. Put this in your Note-Book:
Dip is -. Table 14 Bowditch Refraction is -. " 20 A Bowditch Parallax is +. " 16 Bowditch S.D. is +. Nautical Almanac
Observed altitude of Sun's lower limb is expressed (_).
True altitude is expressed -(-)-.
Remember that before an observation is at all accurate, it must be corrected to make it a true altitude. Remember also that the IE must be applied, in addition to these other corrections, in order to make the observed altitude a -(-)- altitude. So there are really five corrections to make instead of four, providing, of course, your sextant has an IE.
Examples:
1. June 20th, 1919, observed altitude of (_) 69 deg. 25' 30". IE + 2' 30". HE 16 ft. Required -(-)-.
2. April 15th, 1919, observed altitude of (_) 58 deg. 29' 40". IE - 2' 30". HE 18 ft. Required -(-)-.
3. March 4th, 1919, observed altitude of (_) 44 deg. 44' 10". IE - 4' 20". HE 20 ft. Required -(-)-.
Etc.
WEEK IV--NAVIGATION
TUESDAY LECTURE
THE LINE OF POSITION
It is practically impossible to fix your position exactly by one observation of any celestial body. The most you can expect from one sight is to fix your line of position, i.e., the line somewhere along which you are. If, for instance, you can get a sight by sextant of the sun, you may be able to work out from this sight a very accurate calculation of what your latitude is. Say it is 50 deg. N. You are practically certain, then, that you are somewhere in latitude 50 deg. N, but just where you are you cannot tell until you get another sight for your longitude. Similarly, you may be able to fix your longitude, but not be able to fix your latitude until another sight is made. Celestial Navigation, then, reduces itself to securing lines of position and by manipulating these lines of position in a way to be described later, so that they intersect. If, for instance, you know you are on one line running North and South and on another line running East and West, the only spot where you _can_ be on _both_ lines is where they intersect. This diagram will make that clear:
Just what a line of position is will now be explained. Wherever the sun is, it must be perpendicularly above the same spot on the surface of the earth marked in the accompanying diagram by S and suppose a circle be drawn around this spot as ABCDE. Then if a man at A takes an altitude, he will get precisely the same one as men at B, C, D, and E, because they are all at equal distances from the sun, and hence on the circumference of a circle whose center is S. Conversely, if several observers situated at different parts of the earth's surface take simultaneous altitudes, and these altitudes are all the same, then the observers must all be on the circumference of a circle and _only one_ circle. If they are not on that circle, the altitude they take will be greater or less than the one in question.
Now such a circle on the surface of the earth would be very large--so large that a small arc of its circumference, say 25 or 30 miles, would be practically a straight line.
Suppose S to be the point over which the sun is vertical and GF part of the circumference of a circle drawn around the point. Suppose you were at B and from an altitude of the sun, taken by sextant, you worked out your position. You would find yourself on a little arc ABC which, for all purposes in Navigation, is a straight line at right angles to the true bearing of the sun from the point S. You can readily see this from the above diagram. Suppose your observer is at H. His line is GHI, which is again a straight line at right angles to the true bearing of the sun. He is not certain he is at H. He may be at G or I. He knows, however, he is somewhere on the line GHI, though where he is on that line he cannot tell exactly. That line GHI or ABC or DEF is the line of position and such a line is called a Sumner Line, after Capt. Thomas Sumner, who explained the theory some 45 years ago. Put in your Note-Book:
Any person taking an altitude of a celestial body must be, for all practical purposes, on a straight line which is at right angles to the true bearing of the body observed.
It should be perfectly clear now that if the sun bears due North or South of the observer, i.e., if the sun is on the observer's meridian, the resulting line of position _must_ run due East and West. In other words it is a parallel of latitude. And that explains why a noon observation is the best of the day for getting your latitude accurately. Again, if the sun bears due East or West the line of position must bear due North and South. And that explains why a morning or afternoon sight--about 8-9 A.M. or 3-4 P.M., if the sun bears either East or West respectively, is the best time for determining your North and South line, or longitude.
Now suppose you take an observation at 8 A.M. and you are not sure of your D.R. latitude. Your 8 A.M. position when the sun was nearly due East, will give, you an almost accurate North and South line and longitude. Suppose that from 8 A.M. to noon you sailed NE 60 miles. Suppose at noon you get another observation. That will give you an East and West line, for then the sun bears true North and South. An East and West line is your correct latitude. Now you have an 8 A.M. observation which is nearly correct for longitude and a noon position which is correct for latitude. How can you combine the two so as to get accurately both your latitude and longitude? Put in your Note-Book:
Through the 8 A.M. position, draw a line on the chart at right angles to the sun's true bearing. Suppose the sun bore true E 1/2 S. Then your line of position would run N 1/2 E. Mark it 1st Position Line.
Now draw a line running due East and West at right angles to the N-S noon bearing of the sun and mark this line Second Position Line. Advance your First Position Line the true course and distance sailed from 8 A.M. to noon, and through the extremity draw a third line exactly parallel to the first line of position. Where a third line (the First Position Line advanced) intersects the Second Position Line, will be your position at noon. It cannot be any other if your calculations are correct. You knew you were somewhere on your 8 A.M. line, you know you are somewhere on your noon line, and the only spot where you can be on both at once is the point where they intersect. You don't necessarily have to wait until noon to work two lines. You can do it at any time if a sufficient interval of time between sights is allowed. The whole matter simply resolves itself into getting your two lines of position, having them intersect and taking the point of intersection as the position of your ship.
There is one other way to get two lines to intersect and it is one of the best of all for fixing your position accurately. It is by getting lines of position by observation of two stars. If, for instance, you can get two stars, one East and the other West of you, you can take observations of both so closely together as to be practically simultaneous. Then your Easterly star would give you a line like AA' and the westerly star the line BB' and you would be at the intersection S.
Assign for reading: Articles in Bowditch 321-322-323-324. Spend the rest of the period in getting times from the N. A., getting true altitudes from observed altitudes, working examples in Mercator sailing, etc.
WEDNESDAY LECTURE
LATITUDE BY MERIDIAN ALTITUDE
A meridian altitude is an altitude taken when the sun or other celestial body observed bears true South or North of the observer or directly overhead. In other words, when the celestial body is on your meridian and you take an altitude of the body by sextant at that instant, the altitude you get is called a meridian altitude. In the case of the sun, such a meridian altitude is at apparent noon. Now latitude is always secured most accurately at noon by means of your meridian altitude. The reason for this was explained in yesterday's lecture. The general formula for latitude by meridian altitude is (Put in your Note-Book):
Latitude by meridian altitude = Zenith Distance (ZD) +- Declination (Dec).
Zenith distance is the distance in degrees, minutes and seconds from your zenith to the center of the observed body. For simplicity's sake, we will consider the sun only as the observed body. Then the zenith distance is the distance from your zenith to the center of the sun. Now suppose that you and the sun are both North of the equator and you are North of the sun. If you can determine exactly how far North you are of the sun and how far North the sun is of the equator, you will, by adding these two measurements together, know how far North of the equator you are, i.e., your latitude. As already explained, the declination of the sun is its distance in degrees, minutes and seconds from the equator and the exact amount of declination is, of course, corrected to the proper G.M.T. Your zenith distance is the distance in the celestial sphere you are from the sun. You know that it is 90 deg. from your zenith to the horizon. Your zenith distance, therefore, is the difference between the true meridian altitude of the sun, obtained by your sextant, and 90 deg.. Hence, having secured the true meridian altitude of the sun, you have only to subtract it from 90 deg. to find your zenith distance, i.e., how far you are from the sun. This diagram will make the whole matter clear:
The arc ABC measures 90 deg.. That is the distance from your zenith to the horizon. Now if BC is the true meridian altitude of the sun at noon, 90 deg.-BC or AB is your zenith distance. If BC measures by sextant 60 deg., AB measures 90 deg.-60 deg. or 30 deg.. This 30 deg. is your Zenith Distance. Now suppose that from the Nautical Almanac we find that the G.M.T. corresponding to the time at which we measured the meridian altitude of the sun shows the sun's declination to be 10 deg. N. Well, if you are 30 deg. North of the sun, and the sun is 10 deg. North of the equator, you must be 40 deg. North of the equator or in latitude 40 deg. N. For that is all latitude is, namely, the distance in degrees, minutes and seconds you are due North or South of the equator. That is the first and simplest case.
Another case is when you are somewhere in North latitude and the sun's declination is South. Then the situation would, roughly, look like this:
In this case, your distance North of the equator AD would be your zenith distance AB minus the sun's declination DB. This diagram is not strictly correct, for the observer's position on the earth 0 appears to be South of the equator instead of North of the equator. That is because the diagram is on a flat piece of paper instead of on a globe. So far as illustrating the Zenith Distance minus the Declination, however, the diagram is correct. The last case is where you are, say, 10 deg. N of the sun (your zenith Distance is 10 deg.) and the sun is in 20 deg. S declination. In that case you would have to subtract your zenith distance from the sun's declination to get your latitude, for the sun's latitude (its declination) is greater than yours.
Now from these three cases we deduce the following directions, which put in your Note-Book:
Begin to measure the altitude of the sun shortly before noon. By bringing its image down to the horizon, you can detect when its altitude stops increasing and starts to decrease. At that instant the sun is on your meridian, it is noon at the ship, and the angle you read from your sextant is the meridian altitude of the sun. To work out your latitude, name the meridian altitude S if the sun is south of you and N if north of you.
Correct the observed altitude to a true altitude by Table 46. If the altitude is S, the Zenith Distance is N or vice versa. (Note to Instructor: If the sun is South of you, you are North of the sun and vice versa.)
Correct the declination for the proper G.M.T. as shown by chronometer (corrected). If zenith distance and declination are both North or both South, add them and the sum will be the latitude, N or S as indicated. If one is N, and the other S, subtract the less from the greater and the result will be the latitude in, named N or S after the greater. Example:
At sea June 15th, observed altitude of (_) 71 deg. 15' S, IE--47', HE 25 ft. CT 3h--34m--15s P.M. Required latitude of ship.
(_) 71 deg. 15' 00 S IE -- 47' Corr. -- 36 24 HE + 10 36 ------------------- --------- -(-)- 70 deg. 38' 36" S Corr. -- 36' 24" -- 90 00 00 ------------------- ZD 19 deg. 21' 24" N Dec. 23 17 15 N (G.M.T. June 15--3h 34m 15s) ------------------- Lat. 42 deg. 38' 39" N -------------------
Assign for Night Work or to be worked in class room such examples as the following:
1. June 1st, 1919. (_) 33 deg. 50' 00" S. G.M.T. 8h 55m 44s. HE 20 ft. IE + 4' 3". Required latitude in at noon.
2. April 2nd, 1919. (_) 12 deg. 44' 30" N. CT was 2d 5h 14m 39s A.M., which was 1m 40s slow on March 1st (same CT) and 4m 29s fast on March 15th (same CT). IE -- 2' 20". HE 22 ft. Required latitude in at noon.
Assign for Night Work reading also, the following Articles in Bowditch: 344 and 223.
THURSDAY LECTURE
AZIMUTHS OF THE SUN
This is a peculiar word to spell and pronounce but its definition is really very simple. Put in your Note-Book:
The azimuth of a heavenly body is the angle at the zenith of the observer formed by the observer's meridian and a line drawn to the center of the body observed. Azimuths are named from the latitude in and toward the E in the A.M. and from the latitude in and toward the W in the P.M.
All this definition means is that, no matter where you are in N latitude, for instance, if you face N, the azimuth of the sun will be the true bearing of the sun from you. The same holds true for moon, star or planet, but in this lecture we will say nothing of the star azimuths for, in some other respects, they are found somewhat differently from the sun azimuths. Put this in your Note-Book:
To find an azimuth of the sun: Note the time of taking the azimuth by chronometer. Apply chronometer correction, if any, to get the G.M.T. Convert G.M.T. into G.A.T. by applying the equation of time. Convert G.A.T. into L.A.T. by applying the longitude in time. The result is L.A.T. or S.H.A. With the correct L.A.T., latitude and declination, enter the azimuth tables to get the sun's true bearing, i.e., its azimuth. Example:
March 15th, 1919. CT 10h -- 4m -- 32s. D.R. latitude 40 deg. 10' N, longitude 74 deg. W. Find the TZ.
G.M.T. 10h--04m--32s Eq. T. --09 --10
G.A.T. 9h--55m--22s
G.A.T. 9h--55m--22s Lo. in T. 4 --56 --00 (W--)
L.A.T. 4h--59m--22s Latitude and Declination opp. name. TZ = N 101 deg. 30'W
We will take up later a further use of azimuths to find the error of your compass. Right now all you have to keep in mind is what an azimuth is and how you apply the formulas already given you to get the information necessary to enter the Azimuth Tables for the sun's true bearing at any time of the astronomical day when the sun can be seen. In consulting these tables it must be remembered that if your L.A.T. or S.H.A. is, astronomically, 20h (A.M.), you must subtract 12 hours in order to bring the time within the scope of these tables which are arranged from apparent six o'clock A.M. to noon and from apparent noon to 6 P.M. respectively.
We are taking up sun azimuths today in order to get a thorough understanding of them before beginning a discussion of the Marc St. Hilaire Method which we will have tomorrow. You must get clearly in your minds just what a line of position is and how it is found. Yesterday I tried to explain what a line of position was, i.e., a line at right angles to the sun's or other celestial body's true bearing--in other words, a line at right angles to the sun's or other celestial body's azimuth. Today I tried to show you how to find your azimuth from the azimuth tables for any hour of the day. Tomorrow we will start to use azimuths in working out sights for lines of position by the Marc St. Hilaire Method.
Note to Instructor: Spend the rest of the time in finding sun azimuths in the tables by working out such examples as these:
1. April 29th, 1919. D.R. latitude 40 deg. 40' N, Longitude 74 deg. 55' 14" W. CT 10h--14m--24s. CC 4m--30s slow. Find TZ.