Table of Contents

Class Prep

Class 1
Class 2
Class 3
Class 4
Class 5
Class 6
Class 7
Class 8
Class 9
Class 10

>>Topical Articles<<
Assumed Longitude
Bowditch
Bygrave
Casio fx-260 Solar II
Emergency Navigation
Making a Kamal
Noon Sight
Pub. 249 Vol. 1
Sextant Adjustment
Sextant Skills
Sight Averaging
Sight Planning,
  Error Ellipses,
  & Cocked Hats

Slide Rules
Standard Terminology
Star Chart
The Raft Book
Time
Worksheet Logic

BCOSA.ca

Sight Planning, Error Ellipses, and Cocked Hats
 

Error Ellipse*

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.


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.

  • A 90° cut of two LOPs where you think your error is likely ± 1 nm would be 2 nm x 2 nm x π = 12.6 nm2.
  • A 45° cut of two LOPs where you think your error is likely ± 1 nm would be 2 nm x 4.8 nm x π = 30.2 nm2.
  • A 20° cut of two LOPs where you think your error is likely ± 1 nm would be 2 nm x 11.3 nm x π = 80.0 nm2.
  • 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:

    Three LOPs are considered the minimum for a practical navigational fix.

    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:

    In the log for July 18 there is this entry: "Fine weather, wind south-southwest. Porpoises gamboling all about. The S.S. Olympia passed at 11:30 A.M., long. W. 34 degrees 50'."

    "It lacks now three minutes of the half-hour," shouted the captain, as he gave me the longitude and the time. I admired the businesslike air of the Olympia; but I have the feeling still that the captain was just a little too precise in his reckoning. That may be all well enough, however, where there is plenty of sea-room. But over-confidence, I believe, was the cause of the disaster to the liner Atlantic, and many more like her. The captain knew too well where he was.


    Analyzing the Cocked Hat

    If you are going to use a cocked hat, it is misleading to think that your "most probable position" is in the center of the triangle. Here is a diagram that will give you a sense of the probabilities that you lie outside the cocked hat, given a plus-or-minus uncertainty in each of your LOPs. Your probability is highest in the areas shaded the darkest.

    If you translate this into numerical probabilities, you can see that there is only a one-in-four chance that you are inside the area of the cocked hat at all.

    I find it a good bit easier to think in terms of error ellipses than I do to think in terms of cocked hats.

    Addendum

    A lot of digital ink has been spilled on the concepts of the cocked hat and the "most probable position" (MPP)...a discussion which can become quite technical at times with mathematical and graphical methods for determining the symmedian point of a cocked hat.

    Click here for an example of such a discussion. This one contains the helpful notion that "All that can be said about (your) position is that it is somewhere in the vicinity of where the lines cross, a patch covered by a broad thumbprint."

    If you have a software program available to you that will do all the sight reduction, plotting, and (if you are on a vessel) advancing the lines of position, it would be an interesting exercise to take 40 or 50 sights of a range of celestial objects, and see what kind of result you get. Compare it to the result you got from just the first two LOPs you plotted.

    Though I could never locate it again, I read one pre-GPS era publication from the US Navy that recommended 40 or 50 consecutive shots as a standard practice. Obviously they had people aboard with responsibilities such that they could take a couple of hours to do nothing other than shooting sextant sights and entering the data into a computer...and then do the same thing all over again a few hours later.

    Sun-Sun Sights

    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.


    New moon

    Timing:                                  2 weeks after last full moon
    Illumination:                            0% (it is not visible)
    Bearing spread between moon and sun:     0°
    Location of Moon:                        In line with the sun
    Moon in daytime sky at same time as sun: All day
    Usefulness in two-body fix:              Nil


    Waxing crescent moon

    Timing:                                  3½ days after new moon
    Illumination:                            25%
    Bearing spread between moon and sun:     45°
    Location of Moon:                        East of sun
    Moon in daytime sky at same time as sun: ¾ of day
    Usefulness in two-body fix:              Adequate


    First quarter moon (or half moon)

    Timing:                                  7 days after the new moon. 7 days before full moon.
    Illumination:                            50%
    Bearing spread between moon and sun:     90°
    Location of Moon:                        East of sun
    Moon in daytime sky at same time as sun: ½ of day
    Usefulness in two-body fix:              Excellent


    Waxing gibbous moon

    Timing:                                  3½ days before full moon
    Illumination:                            75%
    Bearing spread between moon and sun:     135°
    Location of Moon:                        East of sun
    Moon in daytime sky at same time as sun: ¼ of day
    Usefulness in two-body fix:              Adequate


    Full moon

    Timing:                                  2 weeks after new moon
    Illumination:                            100%
    Bearing spread between moon and sun:     180°
    Location of Moon:                        Opposite sun. As sundown occurs, moonrise happens.
    Moon in daytime sky at same time as sun: Not at all
    Usefulness in two-body fix:              Nil


    Waining gibbous moon

    Timing:                                  3½ days past the full moon
    Illumination:                            75%
    Bearing spread between moon and sun:     135°
    Location of Moon:                        West of sun
    Moon in daytime sky at same time as sun: ¼ of day
    Usefulness in two-body fix:              Adequate


    Third quarter moon (or a half moon)

    Timing:                                  7 days after the full moon. 7 days before new moon.
    Illumination:                            50%
    Bearing spread between moon and sun:     90°
    Location of Moon:                        West of sun
    Moon in daytime sky at same time as sun: ½ of day
    Usefulness in two-body fix:              Excellent


    Waning crescent moon

    Timing:                                  3½ days before new moon
    Illumination:                            25%
    Bearing spread between moon and sun:     45°
    Location of Moon:                        West of sun
    Moon in daytime sky at same time as sun: ¾ of day
    Usefulness in two-body fix:              Adequate


    Twilight Sights

    "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:

    • a two-object fix before dawn
    • a running fix at noon, using one of your twilight objects with the sun, and
    • another two-object fix after sunset.

    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.