A meridian transit observation allows the determination of latitude by simple arithmetic - spherical trig is not needed in this case.  For example, a noon altitude of 40º (i.e, zenith distance of 50º) of the sun with declination of S 20º observed due south from the northern hemisphere translates into latitude of N 30º (= -20º + 50º).

If, however, this altitude was observed not quite at the time of local apparent noon (LAN) but, say, 10 minutes before or after LAN, then this observed altitude is slightly less compared to what it otherwise would have been, had it been measured at LAN.  As a result, in our example, the LAN zenith distance is lower than 50º by a certain small amount, which means that the latitude is N 30º minus that value.  In cases like these it is still possible to avoid a more complicated calculation with the use of Bowditch Tables 24 and 25 to compute this small correction.  Alternatively, one might use the ex_meridian.xls spreadsheet to find that, in this example, the correction amounts to 3.5', resulting in latitude of N 29º 56.5'.

ex_meridian.xls

In a recent NavList posting Jeremy provided a set of high-quality real-life observations that can be reduced to a celestial fix for his vessel at a specified moment in time.  The application of Navigation Spreadsheets to this data set results in a very good fix, both by direct computation as well as by plotting.

The following table shows the computed ephemerides, corrected sextant altitudes, line-of position (LOP) characteristics for two choices of an assumed position (AP), and the accounting of vessel motion through dead reckoning (where the mini-spreadsheet time.xls was used to express the time intervals in decimal hours).



The fix at 19:00:00 local time (= 11:00:00 UT, since ZD = -8) is computed to be:
Latitude: N 8º 49.0'
Longitude: E 109º 45.2'

from the many_body_fix.xls spreadsheet:



This location is marked by a black square on the two subsequent plots.


The first plot uses the VSOP plotting sheet (scale 20 nautical miles per inch) with AP at N 9º and E 110º.  The LOPs were drawn with a T-Plotter.  The LOPs in this plot are not shifted by DR.




The second plot "zooms in" with the scale of 1 NM per centimeter to get a more accurate look.  The reference AP is N 8º 50' and E 109º 40', which has been individually shifted for each observed celestial body along the vessel's track (037) in order to account for the motion of the vessel.



The results are excellent, with all LOPs running within a mile of the computed fix.


In the above procedures all LOPs were treated as equally valid.  The fact that there are pairs of LOPs that run nearly parallel to each other complicates matters somewhat.  The GPS fix given by Jeremy:

Latitude: N 8º 48.4'
Longitude: 109º 45.0'

is recovered almost exactly (both by computation and by plotting) if only Jupiter, Rigel, and Fomalhaut LOPs are used.

Finally, here is Jeremy's position marked by the square in Google Earth:

The recent addition of a protractor to the T-Plotter makes it a fully self-contained tool for plotting of celestial LOPs, as is shown in this demonstration video.  It is important to note, that the use of the T-Plotter is not limited to the scale of 20 nautical miles per inch printed on the instrument.  It is always possible to read off the equivalent number of tics from the latitude scale of any chart according to the picture below.

In his NavList posting Greg Rudzinski explains how you can construct your own plotting sheet using an updated version of the T-Plotter, that is equipped with a protractor.  The orientation of the T-Plotter in the following image is for latitude of 34º.

Some time ago Antoine "Kermit" Couëtte kindly responded to my question about chart scales by stating that he used 1 cm per nautical mile.  In this post I show how that can be achieved using metric graph paper.  As an illustration I pick the Antares line of position (LOP) from the "many-body fix" example (vessel motion is ignored in this post).

Here we really need to "zoom in" in order to use this scale, so I changed the latitude of the assumed position AP by half a degree to N 31º 30'.  The AP longitude stayed the same at W 15º.

intercept.xls:




The spreadsheet intercept.xls calculates several characteristics of this LOP.  Besides the usual intercept and azimuth, it also gives its intersections with the AP's parallel (cell F12) and meridian (cell F15).  This information allows the quick plotting of the LOP simply by marking and connecting these two points.  This technique will not work for LOPs that run close to the cardinal directions, but I believe that it is a good option to have available nonetheless.



After the two points F12 and F15 were connected thus forming the LOP, several measurements were made on the chart to show internal consistency of all the other data computed by the intercept.xls spreadsheet.  The intercept is indeed AWAY (cell E6) with azimuth (cell F6) 152º and it is 3.4 nm (cell D6, allowing about 1 mm for plotting imperfections that only translate to 0.1 nm, which does not degrade the precision achievable in celestial navigation).

intercept: 3.4 nm away from the geographical position (GP) of the observed body




azimuth: 152º toward the GP




LOP direction: 242º and 62º - both perpendicular to the azimuth



This last piece of information allows for yet another way of plotting this LOP using parallel rules on a chart or a plotting sheet with a preprinted compass rose, as it is shown here.

Comparisons with the 2013 Nautical Almanac Commercial Edition show that our spreadsheets remain sufficiently accurate for the year 2013 without the need for any changes or updates.






Later in the year the Venus spreadsheet develops a difference of 0.4' in its GHA value compared to the published almanac.  This is still within the limits of acceptability for practical celestial navigation.

The sextant is a device for measuring angles between two lines of sight.  As such, its use is not limited to observations of celestial bodies.  For instance, the angle between two landmarks can be observed, which (via the central angle theorem), confines the vessel to a circular line of position.  In two separate NavList postings Greg Rudzinski recently published his technique of using the T-Plotter in constructing such an LOP. (post 1, post 2)

The recent transit of Venus prompted a thread on the NavList discussion group (postings by Chief Byron Franklin and Greg Rudzinski, see here) about historical almanac data.  The following two screenshots show that indeed on June 3, 1769, when Captain James Cook observed a Venus transit in Tahiti, the geographical positions (GP) of the Sun and Venus were nearly the same.  This observation is consistent with the notion of Venus traversing between the Sun and the Earth at that time.


sun.xls:



venus.xls:


Furthermore, the computed altitude of the Sun from the given assumed position of Tahiti (next screenshot, with Ho arbitrarily set to Hc) identifies that archipelago as a possible viewing location of the transit.

intercept.xls:



This is an indication of the wide range of validity of the almanac data generating algorithms that are encoded in the spreadsheets.  For future predictions, which is essential for practical navigation, the only "wildcard" is the value of ΔT, which we can forecast with a smaller degree of confidence due to the somewhat unpredictable irregularity of Earth's rotation rate.  We plan to continue to annually check our almanac spreadsheets predictions against more official Nautical Almanac sources, and, if necessary, release updates.  In the three years since the launch of the Navigation Spreadsheets project (and some initial refinements, thanks, Frank Reed ) such changes have not been required.

In a recent posting to NavList, Greg Rudzinski has shared his novel idea of using the T-Plotter in conjunction with a square protractor.  He illustrated the steps of his alternative procedure of plotting a celestial LOP obtained by the intercept method with the following photographs:

The selection of available T-Plotters now includes smaller, "Compact" models.  For additional info visit:
http://www.t-plotter.com


T-Plotter Compact



T-Plotter Compact Blank