Table of Contents
Sight Planning, Error Ellipses, and Cocked Hats
It is up to the navigator to decide on the size of his error circle if two LOPs cross at 90°, based on his assessment of the most likely accuracy of his sextant work under current conditions.
If the two LOPs cross at some angle θ (i.e. difference in azimuths), less than 90°, the circle becomes an ellipse. The ratio of the width of the ellipse (estimated by the navigator) and the length of it, which gets longer as the size of θ decreases, is:
Long Axis of Ellipse = Short Axis of Ellipse ÷ tan(θ x 0.5)
So then, if you take a sight where you think your error is likely ± 1 nm, if your LOPs cross at 90° you would draw your error ellipse as a perfect circle, 2 nm in diameter, centered on the point where your LOPs cross.
If the LOPs cross at 45° then the ratio becomes tan(45° x 0.5) = 0.414. Hence:
2 nm ÷ 0.414 = 4.8 nm.
2 nm ÷ 0.414 = 4.8 nm.
If you take a pair of sights of two objects where the cut is too small, even good quality sextant work can leave you with a large error ellipse. For instance, if the LOPs cross at 20°, then the ratio becomes tan(20° * 0.5) = 0.176. Hence:
2 nm ÷ 0.176 = 11.3 nm
θ should always be less than 90°. Treat a θ of 160° as equivalent to θ = 20°.
Here are a pair of practice sights which, individually, are pretty good. But the cut between them is only 4°. This would lead to an enormous error ellipse and a "celestial fix" that would be scores of miles to the northwest of the actual GPS location of the observer.
So fixes where the bearings/azimuths to the objects have a 90° spread: these are as good as you can get.
Fixes where the azimuth spread is 45° are pretty good.
If the objects you select have a spread of only 35°, you probably want to pick another pair of objects to use. Or, if you are planning a running fix, and will be using the sun for both sights, you should wait a little bit longer before taking your second observation.
The area of the ellipse = Short Axis x Long Axis x π.
It is clear, then, that the closer you can get the cut to 90°, the better off you are in terms of knowing where you are most likely to be.
The Cocked Hat
Generally speaking, I do not think a third sight adds enough useful information to warrant the amount of time it takes to reduce and plot it.
That is to say, in fixes that use more than two LOPs and generate a cocked hat, any additional LOPs less than the largest azimuth difference will not change the ellipse size but will only serve to shift the ellipse center position and rotate the length axis slightly.*This challenges the conventional wisdom about the value of the cocked hat. As the Wikipedia article on Position Fixing (retrieved July 21, 2019) says:
This is tue when it comes to GPS navigation, where multiple satellites generate 3-dimensional spheres of position which — at the point at which they intersect — can help dial in your position within inches. However, I think that the simple bit of trigonometry described above suggests that there is little additional value to be had in a third LOP for the celestial navigator with a sextant in his hand. Particularly where a sight can take 15 minutes to reduce — using Pub. 249 — and plot, one needs to know that he is getting genuine value from each additional sight he takes.
I believe a much more productive way of increasing one's confidence in a navigational fix is to use a mathematical technique — which I will teach in a subsequent class — called "sight averaging", that will allow you to take four sights in quick succession, then identify and throw out the sight(s) that are most likely to be inaccurate. After you use sight averaging to select the highest quality sights to use for a fix, and using a little trig to work out the size and orientation of your error ellipse, you pretty well have as much location data as you can get from your sextant.
The other problem with the cocked hat is that by NOT thinking about error ellipses, it can encourage the navigator to develop an unwarranted confidence that his true position is precisely where he marked a dot in the middle of the cocked hat. While that is not a problem if you are out in mid-ocean and deep water, if you are skirting the edge of a mid-ocean reef, it can encourage you to cut corners — since you "know where you are" and you think of it as a single point on the chart...rather than an ellipse that covers several square miles.
The running aground of the racing yacht "Vestas Wind" at high speed, which was caught on video, came as a complete surprise to the navigtor and crew.
Joshua Slocum, the first man to sail single-handed around the world (1895-98), alludes to this problem of over-confidence in one's position (which in a pre-GPS era was equivalent to an over-confidence in the time as shown one one's chronomter) in this portion of his book, Sailing Alone Around the World:
Analyzing the Cocked Hat
Imagine for a moment that it is the Spring Equinox, March 21. The sun rises at 6 AM, bearing at 90°. At noon, 6 hours later, the sun bears at 180° If you are going to take sun sights and have the resulting LOPs cross at 90°, the ideal time interval between them is 6 hours.
A time interval of 3 hours will give your LOPs a spread of 45°. A time interval of 9 hours between sights will give your LOPs a spread of 135°, which will yield the excact same error ellipse as a time interval of 3 hours.
If you are at sea, and are doing a running fix, because of uncertainties in your exact speed and course (due to helmsman inattention, vaariation in wind speed, cuttent and leeway), a 3 hour interval between sun sights is preferable to a 9 hour interval, even if the apparent geometry is the same.
The running fix, using sextant shots of the sun, is a mainstay of celestial navigation. Many navigators use this technique exclusively. But there are other combinations of celestial objects that will allow you to get a fix immediately, without the uncertainties that creep into the running fix.
Sun-Moon Daytime Sights
For roughly 2 weeks a month, the sun and moon will both be visible in the daytime sky. This gives you the opportunty to take a sight on two objects with a good cut (i.e. separation of bearings between them) during the day. For the alternate two weeks a month, the only option for a daytime fix is to use a running fix.
The moon's orbit wanders with respect to the ecliptic (the path that the sun and planets take). The orbit is sometimes on the ecliptic, in which case a first quarter moon could rise a predictable 6 hours after the sun rises. Other times, when the orbit is inclined 5° above or below the ecliptic, the rising/setting times could be offset. The best way to determine moonrise is to check the Nautical Almanac, or an app on your phone.
Waxing crescent moon
First quarter moon (or half moon)
Waxing gibbous moon
Waining gibbous moon
Third quarter moon (or a half moon)
Waning crescent moon
"Nautical twilight" is a period of time 30-40 minutes long before dawn, and again after sunset, when it is dark enough that the brightest stars are visible, and yet the oocean horizon is still visible. This means that you can get moon/planet/star shots...any objects that happen to be in view. If there are no clouds, it is generally easy to find two objects with a good cut.
You can plan for nautical twilight shots by getting the times from the nautical almanac, or from an app you load onto your phone.
During the two weeks each month when the moon is not usable for a daytime fix, the ideal times to get celestial fixes are:
While being up before dawn and after sunset gives you terrific location data...all you need to define the set and drift of any current you may be in, and to get some idea of the leeway you may be making...it can be exhausting for the navigator. For that reason, the navigator who uses celestial as his primary navigation method may not stand watch as do the other members of the crew.