Wednesday, August 28, 2019

Great Circle Sailing with the 2102-D Star Finder

The 2102-D Star Finder is used to identify stars after sighting them or to choose the optimum set of stars before sighting them. It has other uses as well, as we describe in The Star Finder Book: A Complete Guide to the Many uses of the 2102-D Star Finder.

It turns out that this title is not quite true for the 1st and 2nd editions of the book, because the 3rd ed (Sept, 2019) includes a new application—namely great circle sailing solutions; the subject at hand.

We did not invent this new application, and indeed did not think of it ourselves over the years working with this star finder. We learned of it by chance because one of the original 1921 versions  (HO 2102-A) was found by Mike Walker in a garage sale in Rowley, MA. He recognized its historical value and kindly donated it to Starpath. We are in the process of documenting the device, after which we will post it on the Institute of Navigation's  Virtual Museum, and then donate it to a real maritime museum.

The original patent for the device by Capt. Gilbert Thomas Rude (pronounced Roo dee) did not mention this new application (solving for great circle initial heading and total distance), but the instruction sheet included with the actual Navy versions did include a couple sentences outlining the procedure. It seems those instructions are not quite right, but the principle is clear, so we can work around them. The method will definitely work better on the original 2102-A (star disk diameter 14") than on the present 2102-D (disk diameter 8"), but as we show below, and in an accompanying video, the current version that thousands of mariners own does indeed still provide a useable initial heading and total distance for great circle sailing. This technique was not mentioned on the later versions 2102-C (1932) and 2102-D. Obviously, it is not as precise as we get from an app in our phones, but it is one more tool in our bag of tricks that does not require power and can be dropped into water, stepped on, and kicked around, and still work fine.

Let's consider a hypothetical—but not at all random!—case of being located at a waypoint called Deneb and we want the great circle distance and initial heading to a waypoint called Hamal, as shown below.

For comparisons in the following, accurate great circle (GC) and rhumb line (RL) solutions can be computed online at www.starpath.com/calc.


These waypoints happen to be the fixed positions of two navigational stars that are plotted on the white disk of the 2102-D star finder. The GC solution by star finder comes about because the arcs on the blue templates are great circles plotted on the same projection used for the white star baseplate. A sample is below.


We chose Deneb for this example because its declination (N 45º 21.2') nearly matches the Lat 45 N blue template of the star finder. There is a template for every 10º of Lat, up and down from 45. We chose Hamal more or less randomly.

Thus we imagine the earth not rotating and we are at the geographical position of Deneb, meaning it is directly overhead, 90º above the horizon, and we want the initial GC heading and distance to Hamal. The blue lines are all great circles, so we just find the one that goes from Deneb to Hamal, and read that true bearing on the rim of the blue template, which corresponds to the horizon as viewed from Deneb, or more generally from the center of the template, wherever it is located.

In this case, we see the bearing to Hamal is about 078 or 079 T, and the altitude (Hc) of Hamal is about halfway between 20º and 25º above the horizon. We know from cel nav that the distance between them is the zenith distance, or  90º - 22.5º = 67.5º, and each degree is 60 nmi, so the GC distance we read is 4050 nmi.

Thus in this example we get from the star finder disk an initial heading of 079 ±0.5 compared to correct value of 078.5 and a star finder distance of 4050 compared to a correct value of 4063.5.

This GC heading is a whopping 30º north of the RL route in this example,  so this could have a major impact on navigation decisions.  We don't care so much that one route is shorter than the other, even when this difference is large as in this case, because we are dominated by wind and rarely can make good such routes. But knowing that 078 is just as good or even better than 108 gives us some freedom in planning what to do in local winds.

In the real world, our initial latitude will not coincide with a template value, so we have to improvise the process.  We will work two examples.

Example 1. West Coast of US at 45N, 125W to Japan 38N, 142E. For this route the GC distance  is 3962.1 nmi, with an initial heading of 300.6º T. This heading is 36.3º north of the RL heading of 264.3.  The GC distance is 247.1 nmi shorter than the RL distance of 4209.2 nmi.

Example 2. Exit of the Strait of Juan de Fuca at 48N, 125W to HI at 21.5N, 157W.  This is a GC distance of 2210.3, which is just 14.7 nmi shorter than the RL distance of 2225.0. The initial GC heading of 235.3º T is 10.9º north of the RL heading of 224.4.

The question is, how close can we get to these GC values using the 8" disk of the 2102-D star finder?  We see the answers in the video below which works these two examples from scratch.

The procedure 
Illustrated in two videos below for Northern Lat departures

(1) On the N side of the white disk, draw a thin line to mark the departure meridian going through 0º on the rim to the center of the centerpin.  Or perhaps easier, draw in two meridians, one from the precise rim location (0º) to the left of the centerpin, and one to the right of the centerpin, and know the proper location is between those two lines.... all done carefully, with a sharp pencil.

(2) Use the red template scales to plot your departure point on the white disk on the departure meridian.  The celestial equator on the white disk is equivalent to the equator for this plotting. It is likely best to use dividers on the red template to get the right lat spacing, and then transfer that to the white disk. All of this plotting should be done as carefully as possible as the scales involved are all compressed.

(3) Figure the Lon difference (dLon) between departure and arrival, and note if arrival is west or east of departure.  If arrival is to the east, the arrival meridian is just dLon to the right of 0º on the rim. If arrival is to the west, the arrival meridian is located at 360 - dLon to the left of 0º. Again, draw in the arrival meridian carefully as noted in (1).

(4) Set dividers to arrival Lat on the red disk and then plot it on the arrival meridian.

(5) Find the blue template with the closest Lat to your departure Lat. Do not put it on the centerpin, but instead move it above or below the pin (keeping the blue arrowed line on the template coinciding with the departure meridian on the white disk) until the crosshair at the center of the blue template is precisely over your departure point.

(6) Then hold the template in place and carefully read the altitude (Hc) and bearing (Zn) to your arrival point, interpolating as best you can. The initial GC heading is Zn; the total GC distance is approximately (90 - Hc) x 60 nmi.

The answer is in Example 1 we get from the Star Finder initial heading of 302 T (correct is 300.6) and distance of 3900, whereas correct is 3962.

The answer is in Example 2 we get from the Star Finder initial heading of 235 T (correct is 235.3) and distance of 2100, whereas correct is 2225.



The two examples above worked on the star finder.




This method of solving great circle sailings basic data (range and initial heading) is not accurate enough for the USCG exam for unlimited masters. Indeed, we show in another article that the only method that works dependably for all exam questions is to compute the solution directly. See Great Circle Sailing by Sight Reduction.





No comments: