19 THE CALCULATED ALTITUDE BY INSPECTION TABLES
73. SIGHT REDUCTION TABLES FOR MARINE NAVIGATION (N.P. 401)
The quickest and easiest way to obtain the calculated altitude and azimuth in by means of already prepared inspection tables, which can be entered with the L.H.A. Lat. and Dec. for an assumed position and from which the required information can be extracted directly. Such tables consist in essence of a great number of ready-solved, tabulated, standard spherical (PZX Astro-navigational) triangles which differ from one another by small amounts.
Since the only requirement of the assumed position is that it Shall be reasonably near the true position it is possible to choose an assumed position which fits one of the ready-solved triangles in the tables and thus get the calculated altitude and azimuth for that position.
A brief history of the evolution of Inspection Tables for navigational sight reduction was given in the first chapter of this study, in which we introduced the latest finest and most comprehensive set of tables for marine navigation currently available. SIGHT REDUCTION TABLES FOR MARINE NAVIGATION N.P. 401 (H.0. 229 IN THE UNITED STATES), and explained our choice of Volume 4 (Latitudes 45º to 60º) as the most appropriate volume on which to base the text, examples and problems of the Astro-navigation study. Students are recommended to refresh their memory on Inspection Tables and the Tabular Method of Sight Reduction by reading again this section in Ch. 1.
The precision of the N.P. 401 tables was achieved by performing the calculations on an IBM electronic computer using nine significant figures in order to ensure the accuracy of the altitude to a tenth of a minute of arc and the azimuth angle to a tenth of a degree. The results of the computations for any one volume were produced on magnetic tapes, were automatically differenced for checking; manual recalculations were also used in the verification of several quantities on each page.
After the computations were checked, programmed editing instructions were added and the combined data were transcribed on to a 15 channel paper tape which was used to operate photo composite equipment. The data were photoset, character-by-character, to a photographic positive of the final pages; photo prints were then examined for completeness and imperfections. All data on each page were punched on to cards, one line to a card, and these cards were compared automatically with the data originally produced by the computer. Before final printing, a systematic examination of the proof was made and numerous independent checks applied.
Beyond doubt N.P. 401 (H.O. 229) are the beet sight reduction tables currently available for marine navigation, and are likely to remain so for many years. They are also beautifully easy to use. An assumed position is chosen, the Longitude of which is selected to be that Longitude, nearest to the D.R. Longitude, which results in a whole degree L.H.A., and the Lat. of which is the nearest whole degree of Lat. nearest to the D.R. Lat. The tables are entered with these two arguments and a tabulated value of Dec. near the actual Dec. of the body observed, and tabulated values of altitude and azimuth corresponding to these values of L.H.A., Lat. and Dec. are taken out by direct inspection. Any difference between the actual Dec. and the tabulated Dec. with which the tables are entered is taken care of by applying corrections to the altitude extracted from the tables, these corrections also being obtainable by direct inspection.
Thus no mental interpolation is required at all in extracting the calculated altitude from the tables and such interpolation as may be necessary to obtain the azimuth to accuracy adequate for navigation (about ½º) can be done easily by eye and without calculation.
If tables were constructed for the actual sides of the PZX triangle (fig. 19-1) it would be necessary to enter them with PZ (= Co-Lat.), PX (= 90º-Dec.), and ZPX (= L.H.A.). It is much simpler for the navigator to use the Lat., Dec. and L.H.A., and the arguments in N.P. 401 are therefore for these values and not for the actual sides of the PZX triangle. For the same reason, simplicity, it is the calculated altitude, which is extracted from the tables and not the zenith distance (ZX), since the intercept can be as easily obtained from the difference between the true and calculated altitudes as from the difference between the T.Z.D. and C.Z.D.
74. CHOOSING THE ASSUMED POSITION
The calculated altitude and the azimuth can be extracted directly from the tables if they are entered with a whole degree of L.H.A., Lat. and Dec., and if it could be arranged that this was possible, no additional correction would be necessary. By working the sight from an assumed position chosen so that the Lat. is an integral degree, and the Longitude is such that the L.H.A. of the celestial body observed becomes an integral degree, this eliminates interpolation for the L.H.A. and Lat. arguments, but nothing can be done about the Dec. unless this happens by chance to be an integral degree. In the majority of cases, therefore, an additional correction is necessary to take account of the difference between the body’s actual Dec. and the nearest integral degree of Dec. with which the tables were entered.
Choosing the Assumed Position is done in the following manner: –
Choice of Lat. The assumed Lat. should be the nearest whole degree to the vessel’s D.R. or E.P. E.g., if the D.R. Lat. is 49º 45.0, then the Assumed Lat. should be 50º 00.0.
Choice of Longitude The assumed Longitude must be such that when applied to the G.H.A. of the body as tabulated in the N.A. the resulting L.H.A. is a whole degree. This is achieved when the Longitude is westerly by making the minutes of the assumed Longitude the same as the minutes of the G.H.A. and when the Longitude is easterly by making the minutes of the assumed Longitude (60 – minutes of the G.H.A.).
(Note from this example that since the sum of the G.H.A. and the chosen Longitude exceeds 360º, then 360º must be subtracted to obtain the L.H.A.)
(Note from this example that since the sum of the G.H.A. and the chosen Longitude exceeds 360o, then 360o must be subtracted to obtain the L.H.A.)
A disadvantage of the method of using an Assumed Position as described above, is that each sight in worked from a different Longitude and thus, when plotting several sights observed (nearly) simultaneously, for a fix, three or more intercepts have to be plotted for different positions. The use of the Assumed Position, however. lessens the work involved in reducing a sight, and the chances of error are consequently reduced.
If it is desired to plot simultaneous sights from the same position, say the D.R. or E.P. this can be done, but the extra time and labour involved in using the Special Techniques described in section C6 of the Introduction to N.P. 401 will have to be accepted. Except as regards convenience and simplicity of plotting, there are no advantages to be gained by using these special techniques employing the auxiliary plotting sheets in N.P. 401, and the errors that exist in the use of an Assumed Position with long intercepts are similar to the errors introduced by the interpolations in these special techniques. Students are therefore recommended to adopt the normal usage method employing an Assumed Position, and no attempt will be made in this study to explain the special techniques of altitude interpolation for Lat. and L.H.A.
Those students who wish to study these special techniques will find a full explanation on pages xvii to xxi of Volume 4 of N.P. 401.
The possible errors arising from the use of an Assumed Position are almost always acceptable because the simplicity of the method reduces the probability of errors arising through interpolation. The errors arising, from the normal usage of N.P. 401 ( as described in these Chapters) can generally be neglected in relation to the normal accuracy of sextant observations at sea.
In general, to avoid the possibility of error arising from the use of N.P. 401, altitudes in excess of about 60º, should be avoided, but errors in altitudes greater than this are unlikely to exceed 0.24, which for most navigational purposes is negligible.
75. LAYOUT OF N.P. 401 (H.O. 229)
Students could have a copy of Volume 4 of N.P. 401 (H.0. 229) in front of them for reference while reading this description.
The opening pages of each volume (numbered with Roman numerals) consist of a Preface Explanatory Note, a Glossary of Terms and an Introduction. The main body of tables lies between P.2-365 but is divided into two sections; the first section (pp. 2-.183) relates to the first zone of latitudes covered by the volume (latitudes 45º to 52º in Vol. 4). and the second section (pp. 184-365) relating to the second zone of latitudes (latitudes 53º to 60º in Vol. 4). Thus if the assumed Lat. is 50º N., the first section of the volume would be used, but if the assumed Lat. is 55º N., the second section of the volume would be used. The two sections are separated (between pages 183 & 184) by Diagrams A, B, and C., which are used only for the special techniques referred to above.
The Page required for the reduction of a sight taken within a particular zone of Lat., either N. or S., is determined by the value of the L.H.A. Whatever the L.H.A. tabulations for that entry and for all declinations and all latitudes within the zone are immediately available at the one opening. The required L.H.A. is printed at the top and bottom of each page in large bold figures. Each left-hand page is for two whole degree values of L.H.A., which together add up to 360º (Pº and 360º-Pº). e.g., for L.H.A. 58º & L.H.A. 302º (360º-13º) turn to P. 300,
Each right-hand page consists of two portions of which the upper portion covers the same values of L.H.A. (Pº and 360º-Pº) as the facing left-hand page, while the lower portion (separated by the stepped horizontal lines across the page) is for the two L.H.A. values 180º – Pº and 180º +Pº, e.g. L.H.A. 122º (180º-58º) and L.H.A. 238º (180º + 58º) on P.301. In other words, if the required L.H.A. lies between 0º ~ 90º, and the Lat. is between 53º ~ 60º N. or S., the second section of the tables (pp. 184 to 365) would be entered and the values of L.H.A. between 0º – 90º will be found to increase page by page from the top of P.184 to the top of P. 365.
If the required L.H.A. lies between 91º ~ 180º these values will be found to increase if the pages are turned backwards from P.365 towards P.184, reading the figures at the bottom of the right-hand pages.
If the required L.H.A. lies between 181º ~ 270º, these values will be found to increase page by page from the bottom of P.183 forwards to the bottom of P.365, using the right-hand pages only. If the required L.H.A. lies between 271º ~ 360º these values will be found to increase across the top of each page turned backwards from page 365 towards P.183 The same layout will be found in the first section of the tables (between P.2~183) for latitudes between 45º ~ 52º N. or S.
Having found the appropriate page opening according to the L.H.A., entry to the columns is by way of the assumed Latitudes tabulated horizontally across the top of each page.
It will be noticed that all left-hand pages are headed Lat. SAME Name as Dec., while the top of all right-hand pages is headed Lat. CONTRARY Name to Dec., and the bottom of all right-hand pages Lat. SAME Name as Dec..
Students must realise the significance of the terms Lat. and Dec. SAME name and Lat. and Dec. CONTRARY name. If an observer is in N. Lat. and the observed body has NORTHERLY Dec., or if the observer is in S. Lat. and the observed body has SOUTHERLY Dec., then the SAME name tables are used, but if the observer is in N. Lat. and the observed body has SOUTHERLY Dec., or if the observer is in S. Lat. and the observed body has NORTHERLY Dec., then the CONTRARY name tables are used. It is most important that the correct page or portion of a page of the tables is entered in this respect.
The two portions of each right-hand page are separated, in each column by a horizontal rule, which, together with the vertical lines separating each degree of Lat., form a configuration across the page resembling the profile of a staircase. Hereafter this line separating data of Contrary and Same Name will be referred to as the C-S line. The horizontal segments of this line indicate the degree of Dec. in which the horizon occurs. An observer in N. Lat. can imagine the celestial equator arching across the sky to the southward of him so that celestial bodies with N. Dec. will be higher in the sky than the celestial equator, and the greater the northerly Dec. then so much higher in the sky will the body be found. Conversely, celestial bodies with S. Dec. will be lower in the sky than the celestial equator and the greater the southerly Dec. then the lower in the sky will the body be found until a point is reached where it will not even rise above the horizon. Thus it will be seen that an increase in Dec. of SAME NAME will result in an increase in altitude, whilst an increase in the Dec. of CONTRARY NAME will result in a lesser altitude.
Having found the appropriate page opening according to the L.H.A., the appropriate column according to the assumed Lat. and the names of the Lat. and Declinations the required altitude and azimuth will be found abreast of the whole degree of Dec. tabulated vertically down the sides of each page. If the Dec. is not a whole degree, then the next lower whole degree of Dec. should be used. In the correct Lat. column abreast of the Dec. will be found the required calculated altitude under the heading Hc (height calculated), and the azimuth angle under the heading Z.
Between these two values, under the heading d is the change in altitude expressed in minutes of arc between one whole degree of Dec. and the next higher (the use of the d value will be explained a little later).
The tabulated azimuths (Z) are in values which will readily convert into true bearings (Zn) measured clockwise in three-figure notation from N., by following the simple rules printed at the top of each left-hand page and the bottom of each right-hand page. The tabulated azimuth (Z) should be interpolated mentally to the actual Dec.
76. ALTITUDE INTERPOLATION FOR MINUTES OF DEC.
We have shown how interpolation for Lat. and for L.H.A. in sight reduction tables can be avoided by the adoption of an Assumed Position, which results in whole-degree figures for these arguments. Odd minutes of Dec. cannot, however, be eliminated in this way and the tables accordingly provide for interpolating the calculated altitude for odd minutes of Dec.
No mental interpolation is required for this. Correction of Tab. Alt. for odd minutes and decimals of a minute of Dec. is effected by means of First Altitude Difference, i.e. the figure in smaller type in the column headed d between Hc the Tab. Alt. and Z the tabulated azimuth angle, in conjunction with a special Interpolation Table to be found inside the front cover and back cover of each volume of N.P. 401.
If this Interpolation Table in now inspected it will be seen that the main, vertica1 argument in the table is headed Dec, Inc – the Dec. Increment. This is the amount by which the actual Dec. exceeds the whole degree tabulated with which the main tables were entered. For example, using fig. 19-2. if the actual Dec. is 31º 52.4 the main tables would be entered with Dec. 31º and the Interpolation Table would be entered with Dec. Increment 52.4.
Similarly, if the actual Dec. is 10º 52.9 the main tables would be entered with Dec. 10º and the Interpolation Table would be entered with Dec. Inc. 52.9. Note that for the altitude correction to be valid the main tables must be entered with the next lowest whole degree of Dec. (unless there are no odd. minutes in the Dec.), since the Dec. Inc. in the Interpolation Table is the excess of the actua1 Dec. over the whole degree used as the main table entry.
The Interpolation Table is arranged so that the inside front cover and facing page of the volume provide for the range 0.0 to 31.9 of the Dec. Increment, while the inside back cover and facing page provide for the range 28.0 to 59.9 of Dec. Inc. The horizontal argument in the Interpolation Table is the first altitude difference figure (d) taken from the main table. For convenience and simplicity, this argument is divided into two parts, one for the Tens value of d and the other for the remaining Units and Decimals.
The tens value of d is always taken from the horizontal line abreast of the Dec. Inc, for example, if the Dec. Inc. is 52.3 and the d value is 40.0 then the altitude correction would be 34.9, similarly if the Dec. Inc. is 52.8 and the d value is 50.0 then the altitude correction would be 44.0.
The Units and Decimals values of ‘d’ are taken together from the right-hand sub-table which is given opposite each range of one minute (10 entries) of the Dec. Inc, the units being used as the horizontal argument and the decimals as the vertical argument. For example, if the Dec. Inc. is 52.1 and the ‘d’ value is 46.7 then the altitude correction would be 40.7.
This was determined by moving horizontally across from the Dec. Inc. 52.1 and extracting 34.7 from the 40 columns. then continuing horizontally to the 6th column, and moving vertically down this column until abreast of the .7 decimal figure separating the Tens and Units parts of the table, and there extracting the figure for 6.7 = 5.9. the figure for 40 (34.8) added to the figure for 6.7 (5.9) gives the total altitude correction of 40.7.
Similarly, if the Dec. Inc. is 52.6 and the ‘d’ value is 35.4. the altitude correction would be 31.0. The value for 30 is found in the third column abreast the Dec. Inc. 52.6 and is found to be 26.3. Moving across to the 2 units column and then down opposite the .4 decimal gives the value for the 5.4 to be 4.7 and this added to the value for 30 (26.3) gives the total altitude correction 31.0. (This example is illustrated by fig. 19-2 above).
The altitude correction always takes the same sign ( + or -) as the tabulated d value in the main table. If this sign is + the altitude correction is added to the tabulated value of Hc, and if it is – the altitude correction is subtracted from the tabulated value of Hc to give the required Calculated Altitude.
In the majority of cases, the foregoing is all the altitude correction, which is required. It will be noticed, however, in the main tables, that in some cases when the altitude exceeds 60º the figures in the d column are printed in italics and followed by a dot.
When this occurs and only where this occurs, it is recommended that an additional correction be made in the interpolation of altitude for Dec., this correction being called the Second Difference Correction.
In the few cases where it is necessary to apply the Second Difference Correction when extracting the d value from the Reduction Tables, also take out the values of d immediately above and below the d which corresponds to the whole decree of the actual Dec., and subtract one from the other. The difference thus found is called the Double-Second Difference (D.S.D.) and the value of the Second Difference Correction in obtained directly from the critical table on the right-hand side of the Interpolation Table headed Double Second Diff. and Corrn. Again no mental interpolation is required for this – merely select the two figures in the D.S.D. column between which the actual D.S.D. lies and read off the correction in the adjacent column. When the D.S.D. corresponds exactly to one of the tabulated D.S.D. values, always use the upper of the two possible corrections. Always use the compartment of the D.S.D. table opposite the block of the Interpolation Table wherein was found the Dec. Inc. entry. The Second Difference Correction is always added irrespective of the sign of d.