7 PLOTTING POSITION LINES
21 THE POSITION LINE PLOTTING PROBLEM
78. INTRODUCTION: DOUBLE AND SIMULTANEOUS SIGHTS.
In the previous chapters, we showed how an Astronomical position line could be plotted as a straight line at right angles to the line of true bearing of the observed celestial body at the appropriate intercept (either towards or away from) the chosen or assumed position used for calculating the sight. We also stressed that as in terrestrial navigation, a single position does no more than tell the navigator that his position is somewhere on it. A navigational fix or as it is termed in Astro-navigation an observed position requires at least two position lines, each derived from separate observations.
At those times in the day when two or more celestial bodies are visible in the sly at the same time, a navigator can take several observations in fairly rapid succession, calculate the intercepts and true bearings of each observed body and plot their position lines. These observations are called simultaneous sights even though in practice several minutes my have elapsed between observing and recording the first body and observing and recording the last body. Unless the observer’s vessel is extremely fast, the distance it will have travelled in these few minutes is unlikely to affect the accuracy of the observed position. Providing the observer measures his altitudes, records the times, and performs his calculations and plotting with reasonable care. the accuracy of the data in the N.A. and the Sight Reduction Tables is such that his plotted position lines will intersect at a common point or at worst with only a small cocked hat which will establish his position.
Astronomical position lines, just like their terrestrial counterparts, can be transferred in the same manner as a running fix is employed when necessary in coastal navigation. A double sight is one in which the position line obtained from the observation of a celestial body is run-on along the vessel’s estimated course and distance made good and combined with a later observation of either of the same (when its azimuth has changed through at least 45º) or with the observation of another body whose azimuth is suitable. A variety of double sights may be used: –
c. Sun – run – Sun.
d. Sun – run – Sun’s meridian altitude (or Sun’s Mer. Alt. – run – Sun)
e. Sun – run – Moon (or Moon – run – Sun)
f. Venus ♀ or Jupiter♃- run – Sun ( or, Sun – run – Venus ♀or Jupiter♃)
g. Venus ♀or Jupiter♃are frequently visible after sunrise and before sunset)
h. Terrestrial position line – run – Astronomical position line (or vice versa)
The reliance which can be placed upon the position obtained by a particular combination depends of course, on any error involved in the estimation of the run between sights. By taking simultaneous sights of stars and planets during morning or evening twilight, the navigator clearly avoids the error that a long run between sights may introduce, and in addition, he has the satisfaction of obtaining his position without delay. It should be appreciated, however, that in practice at sea when cloudy conditions exist, sights might have to be taken singly whenever opportunity offers.
In coastal navigation, the plotting of courses, position lines, fixes and running fixes is normally performed on the navigational working chart in use at the time. It would be very convenient if the same procedure could be adopted for the plotting of Astronomical position lines, and indeed, providing the scale of the navigational chart in use is sufficiently large, this can be done. When out of sight of land, however, the sea or ocean chart in use is usually of a very small scale and consequently an accurate position cannot be found by the above method. Besides, if the D.R. position, assumed positions for several sights, intercepts, courses steered and transferred position lines are all ruled in on a navigation chart, the net result is a complex of lines all of which can add up to a certain amount of confusion which may contribute to the possibility of errors.
For the above reasons, the plotting of intercepts, position lines, runs between sights, etc, is normally performed on a separate plotting chart or sheet, marking only the final result on the navigation chart.
Further reflection will reveal that although no problem will arise, plotting Lat. and distance (which both use the same scale on a Mercator chart), finding the Longitude is going to create a problem because whereas a degree of Longitude measures the same as a degree of Lat. at the equator (60 miles), a degree of Longitude gets progressively smaller the further N. or S. of the equator the observer is, so that it measures only about 39 miles in Lat. 50º and only about 30 miles in Lat. 60º. The Longitude scale on a plotting chart must therefore not only be different from the Lat. scale but will also vary according to the mean Lat. of the plotting chart.
79. PLOTTING CHARTS, PLOTTING SHEETS AND DIAGRAMS
At first glance, the use of a separate plotting chart for plotting position lines, etc., may seem a comparatively simple matter. A sheet of plain paper, a blank page in the logbook, and the back of an old chart all suggest themselves as suitable media on which to plot. But a little reflection will show that you will need a scale of Lat. and Longitude and at least one parallel of Lat. and one meridian in order to plot and locate positions, and that although protractors could be used, a compass rose would be infinitely more convenient for the drawing of true bearings and courses.
Providing the area to be covered by a plotting chart does not embrace a belt of Lat. greater than about two degrees, the length of a degree of Longitude in inches along the top and bottom borders can be determined by multiplying the length of a degree of Lat. in inches by the cosine of the mean Lat. of the chart (or conversely by multiplying a fixed Longitude scale by the secant of the mean Lat. in order to find the Lat. spacing). if the chart embraces a belt of Lat. greater than two degrees it is necessary that for a given Longitude scale, the Lat. scale should be constructed according to a difference of Meridional Parts table so that the length in inches of each division of the Lat. scale increases as the Lat. increases. These methods of constructing an accurate plotting chart are outside the scope of this study and are only mentioned to emphasise the problems involved in attempting to plot on plain sheets.
Very few professional navigators, and almost no amateur navigators, will have either the time or the inclination to construct their own plotting charts, particularly when it in realised a different chart will be needed every time the vessel changes her Lat. by a few degrees. Nor is there any necessity to do so when for a few pence an Admiralty Chart Agent can supply a variety of professionally constructed and printed plotting charts, sheets and diagrams which are not only simple to use but which, by plotting lightly in pencil so that each plot can be erased after use, can be used time and time again.
One such chart is the Sumner Plotting Chart (Admiralty Chart No. 5015), which in a blank chart showing no coast line, or indeed any information other than an enormous compass rose from which one can measure off degrees with great accuracy, and scales of latitudes at one side which range from 0º to 60º, all of which are in correct proportion to the existing Longitude scale boldly shown across the top and bottom of the chart. The Longitude scale, although drawn in, is not numbered, so that the user can designate the meridians according to his E.P., and he can also designate the central parallel of Lat. according to the assumed Lat. used for reducing his sights.
To plot on the Sumner chart, the measurement of intercepts, distance and Lat. is taken from the appropriate Lat. scale at the side of the chart corresponding to the assumed Lat. used for the control parallel of Lat., while Longitude is taken directly from the top or bottom borders. This chart can be used in any location in the world between Latitudes 60º N. and 60º S. and is particularly useful for plotting simultaneous sights.
The Plotting Sheets provided for use with this Astro-navigation study are similar to the Sumner Plotting Chart except that they only cover Latitudes 45º to 60º, N. or S., (the same Lat. range as Volume 4 of N.P. 401), and are smaller in scale. Other plotting charts available include Admiralty Chart No. 5004, which consists of a large blank sheet with two large compass roses, no Lat. or Longitude scales are printed on the chart, but there is a diagram for finding the Lat. corresponding to a given Lat., with which the navigator can construct his own scales by following the detailed instructions printed on the chart.
Admiralty Chart 5004C is a smaller plotting chart with one compass rose and a fixed Lat. scale; with this chart, the navigator must construct his own Longitude scale corresponding to the mean Lat. chosen for the chart. Probably the easiest plotting sheets to use are those in the Quarter Million plotting sheet series (Admiralty Charts 5339-5349) each chart of which covers a 6º band of Lat. between the equator and Lat. 66º against a fixed Longitude scale and has two large compass roses. Each chart in the quarter million series has four separate Lat. scales in the left and right-hand borders, each scale covering 1½º of Lat.
For instance, Chart 5347 has scales for Lat. (48º – 49º 30.0), (49º 30.0 – 51º), and 51º – 52º 30.0), and (52º 30.0 – 54º), and the fixed Longitude scale in the top and bottom borders covers a range of 3º. To use these charts, it is only necessary to select the E. and W. Lat. borders appropriate to the area of operation, rule the parallels of Lat. to terminate at these borders. Insert Longitude figures against the top and bottom scales and rule in any required meridians.
For those who do not wish to invest in a properly constructed plotting chart or diagram, or: in an emergency when the plotting chart has been damaged or lost, squared paper can be used as a substitute-plotting sheet. The paper represents a plane chart on which the scale is equal throughout its area, the squares being the scale of nautical miles and Latitude. Angles for courses, bearings and intercepts must be measured with a protractor, but the length of intercepts and the run between sights can be measured directly from the squares on the paper, the length of the side of one small square representing one mile. The problem of picking off the Longitude must, however, be solved by construction, preferably on a separate piece of squared paper so as not to confuse the main plotting.
The E.- W. distance in nautical miles between any two points is called the Departure. In fig.21-2 the departure between A and C is the distance AB = 9.8, and the departure between A and F is the distance AE = 20.5. The departure between any two points on a plot on squared paper can therefore be found simply by counting the horizontal number of squares between the two points.
To convert departure into Difference of Long (d.long), at point A (fig. 21-2) make an angle equal to the mean Lat. of the area represented by the squared paper. If the mean Lat. is say, Lat. 49º 30.0 N., then lay off an angle of 49½ at A.
On the horizontal line through A mark off the length of the departure which it is required to convert into d.long. Suppose a dep. of 9.8 is measured from the main plot then 9.8 would be measured from A to point B, and a line BC run vertically upwards to meet the Lat. line at C. Then AC is the d.long. required. The length of AC can be measured with a pair of compasses or dividers, sweeping the arc down the AC1 to count the number of squares; in this case AC1 measures 15.2 so in Lat. 49½ a departure of 9.8 is equal to a d.long. of 15.2. If point A represents Longitude 10º 00.0 W., then point C must be in Longitude 09º 44.8 W. (10º 00.0 – 15.2).
Similarly, if the d.long corresponding to a departure of 20.5 miles is required (AE in fig. 21-2), E would be raised vertically to F and the distance AF measured by swinging compasses or dividers down to F1 and measuring AF1, giving the required d.long. of 31.4. Again if point A is in Long 10º 00.0 W., then point F would be in Longitude 09º 28.6 W.
Plotting on squared paper in this manner is a little cumbersome and not as convenient as using a properly constructed plotting chart, but providing the range or Lat. covered does in exceed about 1º or 2º at most, is reasonably accurate and quite satisfactory for those who prefer this method.
80. THE SYMBOLS USED IN PLOTTING
The conventional symbols and markings used in Astronomical plotting are almost identical to those used in coastal navigation. as shown in fig. 21-3 above.