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line of Position Worksheets

This section has the step by step instructions of how to construct a line-of-Position starting from a Sextant (Altitude) measurement. This process is rather tedious and prone to making errors. This is why the calculations involved should be organized in some kind of logical scheme, which is fixed in a worksheet that can serve as a guide through the process of elaborating the data to construct the line-of-Position.

Over the last decade I've been optimizing the worksheets presented here, with the goal to minimize processing time without compromising on reliableness. As you get more familiar with the process of line-of-Position elaboration you may most probably want to develop your own work scheme. Therefore, the worksheets discussed here present only one out of many possible ways of organizing the process of elaborating a celestial position fix.

Working out a line-of-Position from an Altitude measurement is roughly done along the following scheme:

  1. Record the Sextant and Chronometer data
  2. Adjust the recorded Sextant and Chronometer data
  3. Get the coordinates for the sighted celestial object at the time of observation
  4. Do the Sight-Reduction process
  5. Determine the Altitude difference between calculated and measured Altitude
  6. Create a Mercator Plotting Sheet or find an appropriate nautical chart
  7. Construct the line-of-Position in the prepared chart

Some worksheets are available to go through the above process in a structured and intuitive way. The worksheets are an attempt to avoid errors while dealing with tables and doing simple manual computations:

While on deck to do an altitude measurement with the sextant, it is useful to carry a pen, some scratch paper and a watch synchronized to UTC time shortly before (e.g. with a precision chronometer). Immediately after the Altitude (object on the Horizon) is fixed on the sextant, the time is recorded. To check the results of the sight reduction it's not a bad idea to record the estimated compass direction in which the sight was done. Also useful is making a small sketch of the view seen through the Sextant, which documents which limb was used and for the Moon it should show the orientation of the visible part of the Moon.
Then the Altitude is carefully read from the Sextant, and written down with a double check of the recorded value. After this, the Sextant is set to 0° to find the index error. The index error is recorded on the scratch paper and - if available - also the current GPS position. This way it is possible to verify the fix at the end of the process.
The next step (usually below deck) is to transfer the data from the scratch paper, in the appropriate fields of the Altitude-Intercept Worksheet.

Altitude-Intercept Worksheet Instruction Guide

0.Section 0
Record your Estimated Position in the first lines of the worksheet.
This is the position at the moment of the sextant measurement, obtained e.g. from dead reckoning.
1.Section 1
Note the date in the first line of the "Time of Observation" section. Be careful to use the UTC date at the moment of the observation.
From the scratch paper calculate the exact time of observation and enter this value in the line Chronometer.
This value is the sum of the time at which the chronometer was started plus the time the chronometer showed when it was stopped at the moment of the measurement.
The watch used for the celestial observations will be checked in regular intervals and the value of the "chronometer error" will be accordingly maintained.
This value is required to calculate the exact time of observation.
If the watch used is set to local time, the next correction will be the for the time zone to obtain the exact UTC time of the observation.
5.Section 2
Record the Sextant data for Altitude (Hs) and Index Error (IE) in the first lines of the "Observed Altitude" section.
The name of the observed Object comes in the cel.obj. field. I also record the approximate compass direction in which the observation was made. This can be used later to verify the calculated Azimuth value, which can be wrong by 180° if the wrong table data is read. But this can be easily detected by comparing the obtained value with the recorded value from the appr.dir. field.
Look up Dip correction and calculate Apparent Altitude (Ha). The corrections for "dip of the horizon" can be found in the Correction Tables for Sextant Altitudes (page 9).
Look up Semi-Diameter (SD) and Refraction and make the appropriate corrections. The SD values come from the Nautical Almanac. The correction for Refraction can be found in the Correction Tables for Sextant Altitudes (page 11).

The entry for the refraction correction table is the previously calculated Apparent Altitude (Ha). Check for the correct sign (±) of the SD correction (add the SD for the lower limb; subtract the SD fot the upper limb). The correction for Refraction is always negative.
Look up Parallax Correction and calculate the Observed Altitude (Ho). The correction for the Parallax Error can be found in the Correction Tables for Sextant Altitudes (pages 13-32). The entry for the parallax correction table is the previously calculated altitude (previous line). The parallax correction is always positive.
10.Section 3
In section 3 the celestial coordinates of the observed object are worked out from the Nautical Almanac.
Go to the correct page according to the UTC Date (of section 1) and find the integral UT line in the Almanac for the observed object and write the GHA, ddGHA, Dec and dDec values in the appropriate fields.
Find the 15°/h interpolated value for the mean increase of the GHA (dHa). This value can be found in the Interpolation Tables ("dHA" column at the corresponding Min/Sec entry).
Calculate the interpolated fractions of the ddGHA and dDec increments for the Minutes / Seconds fraction of the UT time of observation. Again use the Interpolation Tables for this purpose.
Finalize the calculation of GHA and Dec for the exact UTC time of observation.
14.Section 4
With the celestial coordinates from section 3 and the Estimated Position of section 0, the Altitude and Azimuth of the observed celestial object can be calculated. This calculation is called "Sight Reduction". There are different options to do this and therefore it is performed on a separate worksheet.
The results of this process are values for the calculated Altitude (Hc) and calculated Azimuth (Zc) for the estimated position. Depending on the choosen process, also an Assumed Position different from the Estimated Position of section 0 may be part of the result.
15.Section 5
The final step to obtain the required data for constructing the line-of-Position is to calculate the Altitude difference (Hd) for the Assumed Position and recall the calculated Azimuth value (Zc).

This finishes the "calculation" part of finding a line-of-Position. To continue the graphical evaluation for the LoP in the Mercator Plot, the following information is needed:

  • Assumed Latitude (AssLat)
  • Assumed Longitude (AssLon)
  • Altitude Difference (Hd)
  • Azimuth Angle (Zc)

Sight-Reduction Worksheet Instruction Guide

The following instruction are for the Sight-Reduction Worksheet using the Altitude-Azimuth Tables. This method requires the specification of an Assumed Position, which has to be chosen close to the Estimated Position but such, that the resulting Latitude and Local Hour Angle are integral degree values.

1.Section 4
Recall the GHA value (from section 3) and an Assumed Longitude (AssLon) in the first lines of the worksheet. Enter east longitudes with positive sign and west longitudes with negative sign and choose the Assumed Longitude such that the resulting LHA is an integral degree value.
Get the Assumed Latitude (AssLat) by rounding the Estimated Latitude to the nearest integral degree value.
Reenter the Declination (from section 3) and select the "SAME/CONTRARY" case.
Mark "SAME" if both Declination and your Assumed Latitude are north or if both are south. Mark "CONTRARY" if Declination and your Assumed Latitude have contrary names (north/south or south/north).
Use the integral values AssLat, LHA and Dec as entry arguments for the Sight Reduction Tables.
Record the appropriate values for Tabulated Altitude (H), Altitude Increment (dH) and Tabulated Azimuth Angle (Z) in the worksheet.
Calculate the final value for the Azimuth (Zc) according to the rules printed on the right hand side of the section.
NOTICE: For better accuracy the value of Z should be interpolated for the fractional part of the Declination. Azimuth lines are drawn in a chart with an accuracy of about 0.5° so it suffices when the interpolation result is in this range of accuracy. Therefore interpolation for the Azimuth Angle can be done by an estimation of the interpolation fraction by simple inspection of the Sight Reduction Tables.
Interpolate the Altitude for the fractional part of the Declination using H, Hd and the (decimal) Minutes of the Dec value (previous line). The same Interpolation Tables used for interpolating the Nautical Almanac data can be used for this purpose.

Mercator Plot Worksheet Instruction Guide

From the Altitude-Intercept Worksheet the following information to construct a line-of-Position on an chart is obtained:

  • Assumed Latitude (AssLat)
  • Assumed Longitude (AssLon)
  • Altitude Difference (Hd)
  • Azimuth Angle (Zc)

In order to plot a line-of-Position an appropriate chart must be used or a sheet of paper with a suitable Mercator Grid valid for the Assumed Latitude of the observer must be prepared (use the proper page from the Mercator Plotting Sheets Book or follow the section on Plotting Sheets). If a nautical chart is used, it must be large enough so that both the Estimated Position and the Assumed Position can be plotted in it.

The following instructions are given for usage of a scratch Mercator Grid Plotting Sheet.

Label both the Latitude and Longitude scale in the plotting sheet. Draw full grid lines for the main Meridian and main Parallel on the chart.
Mark the Assumed Position (label it "AP") as well as the Estimated Position (label it "EP") in the plotting sheet.
Plot an Azimuth line through the Assumed Position in direction of the Azimuth Angle Zc. This line should extend in both directions from the point "AP". Mark with an arrow on this line in which direction the "Geographical Point" (GP) of the sighted object is. Also write the name of the object along with the UT time of observation (e.g. SUN1345) and the Zc value near the arrow-marked end of the line.
Use the latitude scale (meridian) to measure the distance Hd (remember Hd = Hc - Ho; from the Altitude-Intercept Worksheet). Plot this distance along the azimuth line. If Hd > 0 measure Hd away from the GP of the sighted object (marked with the arrow); if Hd < 0 measure the distance Hd toward the GP of the sighted object. Mark this spot on the azimuth line.
Draw a line-of-Position (LoP) perpendicular to the azimuth line that passes through the point marked in the previous step. Label this line as "LoP" and include the name of the object sighted and the UT time of observation (e.g. LOP_SUN1345). Your position at that time is somewhere along this line.
The LoP constructed in the previous step is actually not a straight line but a "circle of equal altitude". The radius of this circle is equal to the Zenith Distance and is related to the Observed Altitude by:
Zenith Distance = 90° - Observed Altitude

If the radius of this circle is large enough (for altitudes below 60°) the straight line of position is a good the approximation of this circle in the vicinity of the Assumed Position. However for larger altitudes the "circle of equal altitude" should be approximated by the adjustment technique described in the Table of Offsets.
In order to get a position you will have to construct a second LoP: wait for about 3 hours, record the displacement vector (direction and distance traveled during this time), do a new observation with the sextant and get a new data set (AssLon, AssLat, Hd and Zc).
The value for AssLat in the new data set should be the same as in the old one (you should explicitly use the same value for the Sight Reduction process, or scale the plotting sheet such that both Parallels for the current and previous Assumed Latitude can be plotted on the same sheet).
Plot the new Assumed Position in the plotting sheet and repeat the steps 2 to 6 to get a second LoP. Move the first LoP over the displacement vector and label this new line as "LoP" including the UT time of the second(!) observation. The intersection point between the new LoP and the "advanced" old LoP is your position at the moment of the second observation (both lines are valid LoPs at identical times so the intersection must be the position where you are).

In the following section some examples are worked out according to the above instructions.

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