Chapter 8
Select in the table on p. 94 the R.A. of the star nearest in time to your R.A. just secured. Subtract the R.A. of the Mean Sun at local mean noon from the star's R.A. just found on p. 94 of the N.A. and the result will be the exact distance in sidereal time the star you have just identified is from your meridian, i.e., the time interval from local mean noon expressed in units of sidereal time. Convert this sidereal time interval into a mean time interval by always subtracting for the proper number of hours, minutes and seconds as per Table 8, Bowditch. You will then have secured the name of the star desired and the exact local mean time of the star's meridian passage.
Example No. 2: At sea Dec. 14, 1919. Desired to get a star on my meridian at 11 P.M. Lo. by D.R. 74 deg. W.
(.).R.A.G.M.N. 17h--28m--26s Corr. 74 deg. W. (4th - 56m W + ) + 0 --48.6 ___________________ (.).R.A. your M. 17h--29m--14.6s + 11 ___________________
R.A.M. 28h--29m--14.6s
-- 24 ___________________ 4h--29m--14.6s
R.A. of Star Aldebaran 4h--31m--18.5s Star R.A. 28h--31m--18.5 (.).R.A. your M. 17 -- 29--14.6 ---------------- Sid. Int. from L.M. Noon 11h--02m--03.9s Red for Sid. Int. (Table 8) -- 1 --48 ---------------- L.M.T. 11h--00m--15.9s
Aldebaran, then, is the star and the exact L.M.T. of its meridian passage will be 11h 00m 15.9s
Note: If the R.A.M. is more than 24 hours, deduct 24 hours. You will know whether the star is North or South of you by its declination. If you are in North latitude, the star will be south of you if its declination is South or if its declination is North and less than your latitude. If its declination and your latitude are both North and its declination is greater, the star will be north of you. The same principle applies if you are in South latitude.
Assign any of the following to be worked in the class room or at night:
1. At sea, November 1st, 1919. In Latitude 40 deg. N., Longitude 74 deg. W. WT 8h 30m P.M. Observed unknown star about 80 deg. east of my meridian and 25 deg. south of me. What was the star?
2. At sea, December 1st, 1919. CT 10h 45m 01s. CC 20m 16s slow. In D.R. Latitude 30 deg. N., Longitude 60 deg. 30' W. Observed unknown star about 60 deg. west of meridian and about 22 deg. S. What was the star?
3. March 15th, 1919. In D.R. Latitude 10 deg. 42' N, Longitude 150 deg. 14' 28" W. CT 5h 14m 28s. CC--2m 10s. Observed unknown star almost on my meridian and about 28 deg. north of me. What was the star?
4. Aug. 3, 1919, P.M. at ship. In D.R. Latitude 37 deg. 37' N. Longitude 38 deg. 37' W. At what local mean time will the Star Antares be on the meridian?
5. What star will transit at about 4:10 A.M. on Aug. 3rd, 1919? In D.R. position Latitude 38 deg. 10' N, Longitude 34 deg. 38' W.
6. At what local mean time will the Star Arcturus transit on July 17th, 1919, in Latitude 45 deg. 35' N., Longitude 28 deg. 06' W.?
WEDNESDAY LECTURE
LATITUDE BY MERIDIAN ALTITUDE OF A STAR--LATITUDE BY POLARIS (POLE OR NORTH STAR)
To find your latitude by taking an altitude of a star when it is on your meridian, is one of the quickest and easiest of calculations in all Navigation. The formula is exactly the same as for latitude by meridian altitude of the sun. In using a star, however, you do not have to consult your Nautical Almanac to get the G.M.T. and from that the declination. All you have to do is to turn to page 95 of the Nautical Almanac, on which is given the declination for every month of the year, of any star you desire. The rest of the computation is, as said before, the same as for latitude by the sun and follows the formula Lat. = Dec. +- Z.D. (90 deg. - true altitude). As when working latitude by the sun, you subtract the Z.D. and Dec. when of opposite name and add them when of the same name. Put in your Note-Book:
Formula: Lat. = Dec. +- Z.D. (90 deg. - true altitude).
At sea, Dec. 24th, 1919. Meridian altitude Star Aldebaran 52 deg. 36' S. HE 20 ft. Required latitude of ship.
Obs. Alt. 52 deg. 36' S Corr. - 5 08 ---------- True Alt. 52 deg. 30' 52" S - 90 00 00 ---------- Z.D. 37 deg. 29' 08" N Dec. 16 21 00 N ---------- Lat. 53 deg. 50' 08" N
Note to Instructor:
Have class work examples such as the following before taking up Latitude by Pole Star:
1. At sea, May 5th, 1919. Meridian altitude Star Capella, 70 deg. 29' S. HE 32 ft. Required latitude of ship.
2. At sea, August 14th, 1919. Meridian altitude Star Vega, 60 deg. 15' 45" N. HE 28 ft. Required latitude of ship.
Etc.
_Latitude by Polaris_ (_Pole or North Star_)
You remember we examined the formula in the N.A. for Lat. by the pole star when we were discussing sidereal time some weeks ago. We will now take up a practical case of securing your latitude by this method. Before doing so, however, it may be of benefit to understand how we can get our latitude by the pole star. In the first place, imagine that the Pole Star is directly over the N pole of the earth and is fixed. If that were so, and imagine for a minute that it is so, then it would be exactly 90 deg. from the Pole Star to the celestial equator. Now, no matter where you stand, it is 90 deg. from your zenith to your true horizon. Hence if you stood at the equator, your zenith would be in the celestial equator and your true horizon would exactly cut the Pole Star. Now, supposing you went 10 deg. N of the equator. Then your northerly horizon would drop by 10 deg. and the Pole Star would have an altitude of 10 deg.. In other words, when you were in 10 deg. N latitude, the pole star would measure 10 deg. high by sextant. And so on up to 90 deg., where the Pole Star would be directly over you and you would be at the North Pole. Now all this is based upon the Pole Star being in the celestial sphere exactly over the North Pole of the earth. It is not, however. Owing to the revolution of the earth, the star appears to move in an orbit of a maximum of 1 deg. 08'. Just what part of that 1 deg. 08' is to be applied to the true altitude of the star for any time of the sidereal day, has been figured out in the table on page 107 of the Nautical Almanac. What you have to get first is the L.S.T. Find from the table the correction corresponding to the L.S.T. and apply this correction with the proper sign to the true altitude of Polaris. The result is the latitude in. Put in your Note-Book:
To get latitude by pole star, first get L.S.T. This can be secured by using any one of the three formulas given you in Week III--Thursday's Lecture on Sidereal Time and Right Ascension. Then proceed as per formula in N.A.
* * * * *
Note to Instructor:
Spend rest of time in solving examples similar to the following:
1. At sea, Feb. 14th, 1919. CT 13d 21h 52m 33s. CC 1m 14s fast. In Lo. 72 deg. 49' 00" W. IE + 1' 10". HE 15 ft. Observed altitude Polaris 42 deg. 21' 30" N. Required latitude in.
2. At sea, March 31st, 1919. In Lo. 160 deg. 15' E. CT 7h 15m 19s. Observed altitude Polaris 38 deg. 18' N. IE + 3' 00". HE 17 ft. Required latitude in.
Etc.
THURSDAY LECTURE
MARC ST. HILAIRE METHOD BY A STAR SIGHT
You have already been given instructions for finding a Line of Position by the Marc St. Hilaire Method, using a sight of the sun. Today we will work out the same method by using a sight of a star. Put this in your Note-Book here and also under I(b) of the formula given you in Week IV--Friday's Lecture:
Get G.M.T. from corrected chronometer time. With your G.M.T. find the corresponding G.S.T. according to the formula already given you. With your G.S.T. apply the D.R. longitude
(- W. Lo.) ---------- (+ L. Lo.)
to get the L.S.T. With the L.S.T. and the star's R.A. subtract the less from the greater and the result is the star's H.A. at the ship or "t." In using Sun Azimuth tables always take "t" from the P.M. column. Mark Azimuth N or S according to the lat. in and E or W, according as to whether the Star is East or West of your meridian. Then proceed as in the case of a sun sight. Formula:
(-W. Lo.) G.M.T. + (.).R.A. + (+)CP = G.S.T. --------- = L.S.T.--Star's R.A. (+E. Lo.)
(or vice versa if Star's R.A. is greater) = Star's H.A. at ship (t). Then proceed as in case of sun sight.
Example:
On May 31st, 1919, in D.R. Lat. 50 deg. N, Lo. 45 deg. W, G.M.T. 31d 14h 33m 30s. What was Star's H.A. at ship?
G.M.T. 14h -- 33m -- 30s (.).R.A. 4 -- 31 -- 44.2 (+).C.P. 2 -- 23 -------------------- G.S.T. 19h -- 07m -- 37.2s W Lo.-- 3 -- 00 -- 00 -------------------- L.S.T. 16h -- 07m -- 37.2s Star's R.A.(Spica) 13 -- 20 -- 59 -------------------- Star's H.A. (t) 2h -- 46m -- 38.2s
Now let us work out some examples by this method:
1. Nov. 29th, 1919. CT 30d 2h 14m 39s A.M. CC 3m 14s fast. D.R. position Lat. 41 deg. 14' N, Lo. 68 deg. 46' W. Observed altitude Star Aldebaran East of meridian 50 deg. 29' 40". HE 29 ft. Required Line of Position by Marc St. Hilaire Method and most probable position of ship.
2. Jan. 23rd, 1919. P.M. at ship. CT 3h 45m 40s. Lat. by D.R. 38 deg. 44' 19" N. Lo. 121 deg. 16' 14" E. Observed altitude Star Rigel 28 deg. 59' 20" West of meridian. IE + 4' 30". HE 42 ft. Required Line of Position by Marc St. Hilaire Method and most probable position of ship.
Assign for Night Work one or two examples similar to the above.
FRIDAY LECTURE
EXAMPLES: LATITUDE BY MERIDIAN ALTITUDE OF A STAR, LATITUDE BY POLARIS, MARC ST. HILAIRE METHOD BY A STAR SIGHT
1. At sea, Dec. 5th, 1919. Observed meridian altitude Star Aldebaran 69 deg. 28' 40" S. No IE. HE 26 ft. Required latitude in.
2. At sea, Jan. 20th, 1919. CT 21d 2h 16m 48s A.M. In longitude 56 deg. 29' 46" W. Observed altitude of Star Polaris 48 deg. 44' 30" N. IE + 10' 20". HE 37 ft. Required latitude in.
3. At sea, June 4th, 1919. A.M. at ship. CT 10h 16m 32s. CC 5m 45s fast. Lat. by D.R. 42 deg. 44' N, Longitude 53 deg. 13' 44" E. Observed altitude of Star Altair East of meridian, 52 deg. 19' 30". IE--14' 00". HE 56 ft. Required line of position by Marc St. Hilaire Method and most probable position of ship.
Etc.
* * * * *
Assign for Night Work the following Articles in Bowditch: 336 through 341, disregarding the formulas.
SATURDAY LECTURE
LONGITUDE BY CHRONOMETER SIGHT OF THE SUN (TIME SIGHT)
You have now learned, first, how to get your latitude by a meridian altitude of the sun or a star and second, how to get your Line of Position and most probable fix, including both latitude and longitude, by the Marc St. Hilaire Method, using for your calculations either the sun or a star. We are now going to take up a method of getting your longitude only. This method requires as much, if not more, calculation than the Marc St. Hilaire Method. Its results, on the other hand, are far less complete, for while the Marc St. Hilaire Method will give you a fairly accurate idea of both your latitude and longitude, this method will, at best, only give you your longitude. Moreover, you can use it for accurate results only when the sun bears almost due East or West of you, for that is the best time, as you have already learned, to get a line of position running due North and South, which is nothing more than a meridian of longitude. The only reason we explain this method at all is because it is in common practice among merchantmen and may, therefore, be of assistance to you, if you go on a merchant ship. Remember, however, that it belongs to Old Navigation as distinguished from New Navigation, exemplified by the Marc St. Hilaire Method. It is undoubtedly being used less and less among progressive, up-to-date navigators, and will continue to be used less as time goes on. The fact remains, however, that at present many merchantmen practice it, and so it will do you no harm to become familiar with the method, too.
This method is based on securing your longitude by a time sight or longitude by chronometer sight, meaning that at the time the sun bears as near due East or West as possible, you take a sight of it by sextant and at the same instant note the time by chronometer. With this information you proceed to work out your problem and secure your longitude according to the following formula. Put in your Note-Book:
To find your longitude by chronometer (or time) sight.
1. Take sight by sextant only when the sun bears as near as possible due East or West. At exact time of taking sight, note chronometer time.
2. Get G.M.T. from corrected chronometer time. Apply Equation of Time to get the corresponding G.A.T.
3. Correct observed altitude to get T.C.A. Also have at hand Lat. by D. R. and Polar Distance. (Note: Secure P. D. by subtracting Dec. from 90 deg., if Lat. and Dec. are of same name. If Lat. and Dec. are of opposite name, secure P. D. by adding Dec. to 90 deg..)
4. Add together the T.C.A. the Lat. by D.R. and the P.D. Divide the sum by 2 and call the quotient Half Sum. From the Half Sum subtract the T.C.A. and call the answer the Difference.
5. Add together the secant of the Latitude, the cosecant of the P.D., the cosine of the Half Sum and the sine of the Difference (Table 44). The result will be the log haversine of the S.H.A. or L.A.T. It must always be less than 10. If greater than 10, subtract 10 or 20 to bring it less than 10.
6. From Table 45, take out the corresponding S.H.A. (L.A.T.), reading from the top of the page if P.M. at ship, or from bottom of page if A.M. at the ship.
7. Find the difference between L.A.T. and G.A.T. This difference is Lo. in Time which turns into degrees, minutes and seconds by Table 7. If G.A.T. is greater than L.A.T. longitude is West; if G.A.T. is less than L.A.T. longitude is East. Example:
August 26th, 1919, A.M. CT 26d 2h 29m 03s A.M. CC 16m 08s slow. (_) 44 deg. 57' 00". IE--1' 30". HE 32 ft. D.R. Lat. 4 deg. 55' 32" N. Required longitude in at time of observation.
26d--2h--29m--03s A.M. - 12 ------------------ CT 25d--14h--29m--03s --90 deg. 00' 00" CC+ +16 --08 Dec. 10 49 48 ------------------ -------------- G.M.T. 25d 14h 45m--11s P.D. 79 deg. 10' 12" Eq. T. -2 --05 ------------------ G.A.T. 25d 14h--43m--06s - 1' 30" + 9 27 (_) 44 deg. 57' 00" --------- Corr. + 7 57 Corr. + 7' 57" ----------- -(-)- 45 deg. 04' 57" Lat. 4 55 32 N sec. .00160 P.D. 79 10 12 cosec. .00781 ------------ 2) 129 deg. 10' 41" ------------ 1/2 S 64 deg. 35' 20" cos. 9.63266 - 9 -(-)- 45 04 57 ------------ Diff. 19 deg. 30' 23" sin. 9.5235O--14 ------------- 9.16557 + 5 + 5 ------------- log. hav. S.H.A. (L.A.T.) 9.16562
S.H.A. (L.A.T.) 25d--21h--00m--01s G.A.T. 25 --14 --43 --06 ----------------- Lo. in T. 6h--16m--55s E
Lo. (Table 7) 94 deg. 13' 45" E
I wish to caution you about confusing this method with the one Bowditch uses, and still another which Henderson uses in his book "Elements of Navigation." It is not exactly like either one. It requires one operation less than either, however, and it also requires the use of fewer parts of the various tables involved. For that reason it is given you.
Assign for work in class room and also for work at night examples similar to the following:
1. Oct. 1st, 1919. A.M. (_) 17 deg. 15' 00". G.M.T. 1d 11h 30m 00s A.M. D.R. Lat. 40 deg. 30' N. IE--2' 20". HE 25 ft. Required longitude in.
2. Oct. 10th, 1919. P.M. (_) 25 deg. 14' 30". CT 1h 15m 20s. CC 4m 39s slow. IE--3' 10". HE 26 ft. D.R. Lat. 41 deg. 29' 00" S. Required longitude in.
3. May 27, 1919. P.M. Lat. by D.R. 40 deg. 55' N. (_) 34 deg. 4' 00". IE + 1' 10". HE 10 ft. CT 8h 55m 42s. CC 2m 02s fast. Required longitude in.
4. May 18th, 1919. A.M. (_) 29 deg. 41' 15". WT 7h 20m 45s. C-W 2h 17m 06s. CC 4m 59s slow. Latitude by D.R. 41 deg. 33' N. IE--1' 30". HE 23 ft. Required longitude in.
5. August 24th, 1919. A.M. (_) 23 deg. 32' 10". IE--2'00". HE 16 ft. In latitude 39 deg. 04' N. CT 24d 2h 47m 28s A.M. CC + 4m 28s. Required longitude in.
6. June 26th, 1919. P.M. (_) 44 deg. 08' 20". IE--2' 20". HE 37 ft. CT 8h 18m 45s. CC 3m 20s fast. Latitude by D.R. 6 deg. 43' S. Required longitude in.
7. July 29th, 1919. A.M. CT 29d 11h 14m 39s A.M. CC 2m 18s slow. (_) 28 deg. 08' 30". IE + 0' 30". HE 38 ft. Latitude by D.R. 39 deg. 48' N. Required longitude in.
8. May 22nd, 1919. P.M. CT 9h 14m 38s. CC 5m 28s slow. (_) 21 deg. 07' 40". In latitude 41 deg. 26' N. IE + 3' 10". HE 40 ft. Required longitude in.
WEEK VI--NAVIGATION
TUESDAY LECTURE
LONGITUDE BY CHRONOMETER SIGHT OF A STAR
In getting your longitude by a time sight of a star, you proceed somewhat differently from the method used when observing the sun. What you wish to get first is G.S.T., i.e., the distance in time Greenwich is from the First Point of Aries. If you can then get the distance the ship is from the First Point of Aries, the difference between the two will be the longitude in, marked East or West according as to which is greater. By looking at the diagram furnished you when we were talking of Sidereal Time, all this becomes perfectly clear. The full rule for finding longitude by a star is as follows, which put in your Note-Book:
Correct your CT to get your G.M.T. From the G.M.T. get the G.S.T. From the observed altitude of the star, obtain the star's H.A. at the ship in the same way L.A.T. is secured in case of the sun. To or from the R.A. of the star add, if West of your meridian, subtract if East of your meridian, the star's H.A. at the ship, just obtained. The result is the R.A. of the ship's meridian or L.S.T.
Find the difference between G.S.T. and L.S.T. and the result is the longitude, marked East or West according as to whether G.S.T. is less or greater than L.S.T. Note: Always take the star's H.A. from the top of the page of Table 45.
Dec. 2, 1919. A.M. Observed altitude Star Sirius 2O deg. 05' 20", West of meridian. CT 11h--45m--29s P.M. CC 1m--28s slow. IE--1' 20". HE 21 ft. Latitude by D. R. 38 deg. 57' N. Required longitude in.
CT 11h--45m--29s CC + 1 --28 ------------------- G.M.T. 11h--46m--57s (.)RA 16 --37 --10.3 (+)CP 1 --56.1 ------------------- G.S.T. 28h--26m--03.4s IE -1' 20" --24 HE -7 08 ------------------- ------- G.S.T. 4h--26m--03.4s Corr. -8' 28"
Obs. Alt. 20 deg. 05' 20 Corr. -8 28 ---------- T.C.A. 19 deg. 56' 52" Lat. 38 57 sec. .10919 P.D. 106 36 24 cosec. .01849 + 1
2 ) 165 deg. 30' 16" ------------- 1/2 S 82 deg. 45' 08" cos. 9.10106 - 13 T.C.A. 19 56 52 ------------- Diff. 62 deg. 48' 16" sin. 9.94911 + 2 --------- 9.17785 - 11 --------- log. hav. Star's H.A. at ship 9.17774
Star's H.A. 3h--02m--40s Star's R.A. 6 --41 --39 -------------- L.S.T. 9h--44m--19s G.S.T. 4 --27 --01 -------------- Lo. in T. 5h--17m--18s E
Longitude in 79 deg. 19' 30" E
Assign for Night Work or work in the class room examples similar to the following:
1. April 16, 1919, in Latitude 11 deg. 47' S. Observed altitude of the Star Aldebaran, West of the meridian 23 deg. 13' 20". CT 6h 58m 29s. CC 2m 27s fast. IE--2' 00". HE 26 ft. Required longitude in.
2. Dec. 10th, 1919. Observed altitude of Star Sirius 20 deg. 05' 40" West of meridian. CT 11h 45m 29s. CC 1m 28s slow. IE--1' 20". HE 21 ft. D.R. latitude 38 deg. 57' N. Required longitude in.
Note to Instructor: If any time in the period is left or for Night Work assign examples to be worked by Marc St. Hilaire Method, changing slightly the D.R. Lat. and Longitude just obtained by the Time Sight Method.
WEDNESDAY LECTURE
EXAMPLES ON LONGITUDE BY CHRONOMETER SIGHT OF A STAR
1. Dec. 9th, 1919. In latitude 36 deg. 48' N. Observed altitude Star Capella, East of meridian 46 deg. 18' 30". IE 2' 50" off arc. HE 33 ft. CT 10d 3h 05m 05s A.M. CC 1m 18s slow. Declination of star is 45" 55' N. Required longitude in.
2. October 26th, 1919. In latitude 39 deg. 54' S. Observed altitude Star Rigel, West of meridian 42 deg. 18' 40". CT 27d 10h 32m 55s A.M. CC 2m 18s fast. IE 4' 20" off arc. HE 42 ft. Required longitude in.
3. April 11th, 1919. P.M. at ship. In latitude 43 deg. 16' 48" S. Observed altitude Star Spica 33 deg. 18' 20", East of meridian. CT 11h 08m 44s P.M. IE 3' 20" on arc. CC 4m 18s slow. HE 39 ft. Required longitude in.
4. September 15th, 1919. P.M. at ship. In latitude 49 deg. 38'N. Observed altitude Star Deneb, East of meridian, 36 deg. 16' 50". IE 3' 40" off arc. HE 40 ft. CC 6m 18s slow. CT 10h 00m 13s P.M. Declination of star is 44 deg. 59' 36" N. Required longitude in.
If any time is left, work same examples by Marc St. Hilaire Method assuming a position near the one found by Time Sight.
Assign for Night Work any of the above examples, to be worked either as Time Sights or by the Marc St. Hilaire Method, and also the following Arts. in Bowditch: 326-327-328-329.
THURSDAY LECTURE
LATITUDE BY EX-MERIDIAN ALTITUDE OF THE SUN
You have learned that when you calculate your latitude from a meridian altitude of the sun, one of the necessary requisites is to have the sun exactly on your meridian. In fact, that is just another way of expressing meridian altitude, i.e., an altitude taken when the sun is on your meridian. Now suppose that 10 or 15 minutes _before_ noon you fear that the sun will be clouded over _at_ noon so that a meridian altitude cannot be secured. There is a way to calculate your latitude, even though the altitude you secure is taken by sextant some minutes before or after noon. This is called latitude by an ex-meridian altitude. It must be kept in mind that this method can be used accurately only within 26 minutes of noon, either before or after, and only then when you know your longitude accurately. Put in your Note-Book:
1. Get your L.A.T. (S.H.A.).
2. Subtract it from 24h 00m 00s, or vice versa, according as to whether L.A.T. is just before or just after local apparent noon. Call the result "Time Interval from Meridian Passage."