Wednesday, January 15, 2020

Estimating Net Current Drift on a Long Ocean Passage

When planning a route across the equatorial current system from northern latitudes down into southern, or vice versa, it can be crucial to be aware of the net current set along the route. You are in trade winds the whole time—other than two hundred miles or so of doldrums between them. The trades will vary from the NE to E in the north and from SE to the E in the south, typically at some 10 to 18 kts. You will be crossing trade wind waves, which are larger than we might guess due to the long fetch and duration of the trades. There will most likely be long stretches of a close reach in these big waves, so it might be hard to maintain a due south course.

On top of that, the net current drift to the west could be large enough that a due south heading could not make the target destination. This note shows how to estimate how much that set will be, so you have a good reminder that one navigational goal, throughout much of the route, will be to make miles to the east whenever you can to overcome the anticipated set and potentially strong SE wind.

Such long range sets can also enter into emergency route selection, and as such we have in our book Emergency Navigation, a section of estimating the net E-W drift on a voyage from San Diego to the Marquesas. In this passage the main set to the west is reduced by a section of the route, about several hundred miles long, where the drift is to the east in the countercurrent, a system we discuss in another article.

The method we used then, which is still feasible for planning, is the use of Pilot Charts as they give climatic current drifts in knots or nmi per day. Pilot Charts are available as PDF from the NGA site (google "MSI NGA" to find the link), or better still use the digital RNC versions from OpenCPN.org under chart resources.  These can be viewed in any enav program, including of course their own free OpenCPN.

In our online course on Emergency Navigation we have a practice exercise on this process, and this note is essentially an outline of the answer to that exercise (Quiz 9, Exercise 1). This exercise crosses the same basic current system, but instead goes from Hawaii to Tahiti in July, about 2900 nmi on rhumbline 170T, but the punchline is if you sailed that heading the whole way with a speed made good of 6.0 kts you will miss the Society Islands.

To work with with Pilot Charts, you can set up plots like the following, either on printed charts or all done digitally, as this one was.


Pilot Charts for July, N Pac and S Pac. with 3.3º  (~200 nmi) steps marked along the route. 

There will not be current data (shown as green vectors) in each of the boxes where we want it, so we have to the best we can to estimate this based on nearby data. We also need to keep in mind we are working on estimates of the estimates.  We won't know ahead of time what the wind speed is, but generally when the charts specify a range such as 10 to 20 miles a day, we would anticipate the stronger drift with strong trades (approaching or over 20 kts) and the lower values with lighter trades of say below 10 kts.

This then, we leave as an exercise to fill out the list and sum up for a net drift.

Solution by Model Forecasts

Another approach to this analysis is to use ocean model forecasts of the surface currents. We have several options in this part of the world. We can use the global RTOFS model, HYCOM model, OSCAR data, or the French data from Copernicus. We have an overview of ocean currents at www.starpath.com/currents, with these references.

If you are evaluating this in about real time, such as just before departure, then the process is very easy, because the data are contemporary, and easy to come by. You can get the data, usually RTOFS model, from many sources, including Saildocs or LuckGrib, both of which are explained in Modern Marine Weather.

A twist to this solution is the date of the exercise. It is now January, and we want this analysis for July. These currents change somewhat from month to month, as noted in the countercurrent article linked above. An interesting fact about the popular RTOFS data is that, unlike other model data,  it is not archived, so we cannot use that. The HYCOM model,  on the other hand, gives very similar results, and it is archived.

We list sources for all archived weather data in the Archives Atlas of the Weather Trainer program, which is just one of its extensive ongoing resources. Find the archived HYCOM data along with other model data at the NCEI web page (the live ocean data, in contrast, is found on the NOMADS page, or more conveniently from one of the several GRIB viewers, such as LuckGrib).

However, finding the data for this retrospective analysis is not the end of the challenge, because the digital data are not in GRIB format. The are in NetCDF format (.nc), which must be read in a program like Panoply. We have a video on the use of Panoply for viewing archived ASCAT data.

I am not proposing this approach as a standard procedure, but for now, I am just using it to analyze these July currents so they can be compared to the Pilot Chart solution.

Below are the HYCOM currents viewed in Panoply for the route in question. These are the ones I used to read off the currents every 2º going south.

HYCOM currents for July 2019 viewed in Panoply, used to evaluate E-W set to compare with Pilot Chart results for the same study. Keep in  mind, Pilot Charts are climatic values, and the above is one specific year.

Below is the summary of currents, where the E-W component has been pulled out. In other words, when you do this, and you find a current of 1.0 kts to the NW, we have to break that into N component and W components, which for this 45º is just x 0.7.  So 1.0 to the NW adds 0.7 to the West.  For this study we do not care about the N-S components.  The N-S component primarily determines how long it will take, but not the set to the West. Technically it has some effect, in that it slows us down or speeds us up, so the time spent in that current segment is not the same as the 6.0 kts we are assuming. But these currents are small, so that is a relatively smaller effect... i.e., 6.0 vs say 6.4.

Here is first the raw data, followed by a transcription—remember this is the answer to a training exercise.


We are assuming a SMG of 6.0 kts, so time in the current for each 2º of Lat change is 120/6 = 20 hr, then we multiply: 20 hr x the drift to get the set, i.e., 0.4 kts W for 20 hr = net 8 nmi to the west.


HYCOM forecasts for net drift to the west in July 2019. We estimate the missing counter current below which is shown to reduce the set from 252 to 176 nmi.

We ended up in July of 2019 with a total westerly set of 252 nmi, but notice that this year there was no notable counter current forecasted, usually seen in July in the range of about 5N to 10N. That result is not common, and likely unique to this model for this year, so we look into ways to estimate the countercurrent.

Below are the OSCAR currents for this same time period, and they do show the countercurrent in about the right place. It is these data that were the basis of the net drifts shown above.

The OSCAR data shows the same strong patch to the west just below the countercurrent, but it is weaker. The OSCAR currents are 8-day averages of changing values so they are always going to be slower than any instantaneous forecast from another model.  In fact, the estimates above for the counter currents are likely low, but our earlier study showed these as fairly close in this particular system.

Another way we get evidence that there was indeed a countercurrent in July of 2019—and likely to be one when we get there in some July of the future—can be seen in the French ocean currents forecasts from the Copernicus viewer. This data can then be exported to Google Earth, which we have done below.


Ocean current forecasts for July 15, 2019 from the Copernicus program in France, exported to Google Earth

We did not do the current summary from this data, but the countercurrent is clearly visible. We will leave this as part of the net drift exercise to find the total set to the west using this data. It is fairly high resolution in Google earth and the drift speed scale color bar is shown top left in units of meters/sec.  Multiply by 2 (1.93) to get kts.

So the new wording of the exercise is use the Pilot Chart plots given above to find the net drift and then compare that to what you get from the Copernicus forecasts obtained from the links above.

For best understanding of this process, do a Pilot Chart study for the present month using live RTOFS data. Then the digital data is easy to get.


Saturday, January 11, 2020

Finding UTC of LAN using an Analemma

Finding latitude at noon—when the sun is at its peak height in the sky as it crosses our meridian bearing either due south or due north—is the most basic technique in celestial navigation, but it is a relatively long process. We can minimized this time by estimating when it will occur based on our DR longitude. Then we can be on deck and ready to go maybe just 10 minutes earlier, and not waste a lot of time, often in extremely hot temperatures under the high sun.

In another post we show how to figure the time of LAN from our compact perpetual sun almanac by just looking up the time that the GHA of the sun is equal to our DR-Lon. In this note we look at another alternative to the traditional procedures that requires a Nautical Almanac.

The traditional procedure is to look up the local mean time (LMT) of meridian passage (mer pass) from the daily pages of the Nautical Almanac and then correct that for our longitude, and then convert that UTC to watch time WT.

The Nautical Almanac gives this time accurate to the second, but we have no need for that precision, not to mention that our DR-Lon could be off enough to shift this time a few minutes—in the tropics, a 15 mile DR error would be a 1 minute time error.

A quick way to get this LMT of mer pass without an almanac is to have at hand the unique figure shown below called an analemma.

These figures are usually drawn slightly differently to match the track of the sun in the sky throughout the year, but we made this one years ago specifically to be used to find real values of the sun's declination and the Equation of Time (EqT) based on the day of the month. The link above tells more about the origin of this famous figure—well known to those who know it well.

To use this drawing, estimate the date of interest along the curve that marks the first of each month, then the sun's declination is on the left scale and the EqT is on the bottom. Each dot is 1 minute of time or 1 degree of angle.

The Equation of Time is not the relativistic secret to the universe that it might sound like, but rather the more humble difference between 1200 UTC and the actual UTC that the sun crosses the Greenwich meridian. In other words:

LMT of mer pass = 1200 UTC ± EqT

This is the practicable interpretation of LMT. If you want a moment of celestial navigation entertainment, you can read the official definition of LMT.

Thus if the Nautical Almanac tells us that the LMT of mer pass is 1207 on some specific date, it means the EqT is +7 min.  If the Almanac says LMT mer pass is 1144, it means EqT is -16 min.

We want to use this the other way around. We know a date we care about, then we use the diagram to find the EqT and apply it to 1200 to find the LMT of LAN on that date.

Once we know the time the sun goes by Greenwich, we can figure when it will get to us. The earth turns 360º in 24 hr beneath the sun, which means the geographical position (GP) of the sun  moves west at the rate of 360º/24h = 15º/1h = 15'/1m. These can be further rewritten at 1º = 4 min and 1' = 4 sec.

An example: The date is July 19 and my DR-Lon is 138º 25', what time do I expect the sun to be at its peak height in the sky, bearing due south?  My watch is set to PDT, zone description (ZD) +7.  The analemma tells me the LMT is 1207 (meaning at Greenwich this happens at 1207 UTC), so we are just left with converting 138º 25' to time and adding that to 1207.

The usual solution here is to refer to the Arc to Time Table from the Nautical Almanac, where we find that 138º = 9h 12m, and 25' = 1m 40s, so our DR-Lon is equivalent to about 9h 14m. We add this to 1207 to get 21h 26m UTC, and for watch time we undo the ZD  to get 14h 26m.

Without such a table, we can figure it manually. One shortcut is just divide by 15 to get the hours and then figure the minutes. In this case 138/15 = 9.2 hr = 9h 12m and then add on the arc minutes part of the Lon: 25' x 4s/1m = 100s = 1m 40s or about 2 min, so the answer is 9h and 14m, which is what we get from the tables.

In eastern longitudes, we subtract our DR-Lon time from the Mer Pass time at Greenwich, because the sun moving west goes by us first before reaching Greenwich.

Note the above example did not have a date. These times do vary slightly over the leap year cycle, then repeat every 4 years, but this variation is just a minute or so, which is not crucial to our planning needs.

That is the end of the procedure discussion. Below is a bit more on the motion that causes this.

___________

There are two reasons the sun's GP does not circle the earth at an exactly constant rate throughout the year, which leads to the varying times of LAN. One is the earth's orbit is not a circle, but rather it is slightly elliptical, and orbital speed changes slightly at various parts of the ellipse. The other reason is the tilt of the earth's axis relative to the plane of its orbit, about 23.4º. This adds a N-S component to its actual path across the earth, leading to a varying westerly speed. This also leads to the tropics band on earth (23.4N to 23.4S) that covers all latitudes where the sun might be directly overhead.

Both of these effects are regular cyclic patterns, but they are not in phase, which leads to the unusual shape of the EqT shown below as well as to the odd shape of the analemma.







Friday, January 10, 2020

Progress to Weather

The key to long term success in navigation is good DR (dead reckoning). In its broadest sense it means figuring the best estimate of your present and future positions (without piloting or electronic aids) using all other information available to you.

Your log and compass readings are the starting points, but then there are many corrections and adjustments to make, not the least of which are tied to the strength and direction of the wind.

Strong persistent head winds bring a new twist to navigation that has a serious affect on DR if not accounted for. Three factors that don’t matter much in light air now matter a lot. These are wind-driven current, helm bias, and leeway. They are each fairly small effects, even in strong winds, but they all cause error in the same direction, so their sum is not small. They cause navigational error because they are invisible—they take us off course and we have no way of knowing it until the next position fix. In short, we must simply make an educated guess of their individual sets and drifts and correct our DR accordingly.


Figure 1. On a reach or in moderate wind, the wake is straight aft indicating no leeway. When sailing to weather in stronger winds leeway sets in and the wake appears to shift to windward as the boat slips to leeward. Schematic drawing adopted from the Starpath Online course on Inland and Coastal Navigation (www.starpath.com/courses). In the golden age of sailing when sailing ships slipped very much more than today it was common advice to new helmsmen to “Keep your wake right astern.”

To read one treatment of this from that era see Lecky’s, Wrinkles in Practical Navigation, page 664. We have made this link to get you a free copy of this great book: www.tinyurl.com/1918Lecky.

Leeway is how much a boat slips to leeward on a windward course. It is a function of the boat’s draft, the point of sail, and the wind speed. It’s only a navigational factor when going to weather close hauled in strong winds—or very light wind, but that is not the subject here. The effect is distinctly different from current set because we can measure leeway underway (without electronics in some cases), so it is not strictly invisible as implied above. A typical keel boat of 6-foot draft might slip as much as 10° to leeward in a solid 15 knots of true wind. Yacht design specs might have this number as just a few degrees, but here we are discussing the reality of navigation, not a design parameter that may have a more complex interpretation.

Leeway can be discerned in your GPS derived data in special cases. For example, if the wind has just started to blow (so it has not had time to generate any surface current) and there are no other sources of current in the waterway, then when close hauled you will find your average COG to be some degrees to leeward of your average compass heading when steering a steady course on the wind, whereas your average SOG will match your average knotmeter speed. When this happens symmetrically on both tacks, you have a nice snapshot of your leeway. Sometimes you can actually see your wake bent to windward, which is the same effect. Yacht designers have developed underwater gimbaled vanes that measure the angle of motion through the water relative to the centerline for an accurate measurement of leeway vs wind speed and heeling angle.

With differential or WAAS enabled GPS, leeway shows up very nicely on units that directly compute current based on SOG and COG compared with accurate inputs of knotmeter speed and compass course. Going to weather in still water, you will have current on your windward beam regardless of your tack.

From a practical point of view, you can ignore leeway as soon as you fall off the wind. Above some 45° apparent you can forget it unless you are still well heeled over or have other evidence that you might be slipping. Remember that leeway, unlike current, adds uncertainty only to your course direction, not your speed. In slack water, your knotmeter speed is your SOG even as you are slipping with leeway.


Figure 2. This graph assumes a maximum wind drift current of about 2.5% of the wind speed when fully developed. Shown on the left are the times required to develop this current.

The blue example marked shows a 30-knot wind producing a maximum of 0.75 knots, which would take some 19 hours to develop. When the wind has blown only 8 hours, the wind drift would be more like 0.4 knots.

Notice on the data that below about 30 kts, a required duration of half the wind speed is a good guess for maximum development, but at higher wind speeds it is more like hours = knots to get the water moving at max speed.

Adapted from the Starpath Weather Trainer Live software program.

Leeway depends on wind speed. If your optimum wind speed is 10 knots true, then leeway increases going both up and down from there. In one sense, this is how optimum wind speeds are defined for sailing vessels. If it is, say 6° at optimum, then by the time you get to 20 knots it might be as high as 15°. In practice, however, it doesn’t get much higher than this in normal operations because by then you start to fall off—except in some well designed race boats, it just doesn’t pay to pound into the waves in very strong winds. And once you fall off, the slipping stops. Likewise at very low winds (a knot or two) you will also slip a lot, but again at some point you fall off and it goes away.

Leeway also depends on keel depth—the depth is much more important than the shape. Sailing a kayak, for example, just a paddle down in the water makes a world of difference. Likewise, to first approximation, a high tech racing keel and a full length cruising keel are about the same in this regard for a given depth. The high tech fin keel, however, can make up a lot by the actual lift it adds as water flows over it as wind does on airplane wings.

Leeway occurs in all waters, regardless of actual current flow. In strong winds, however, no water (ocean or lake) will stay still for long. When the wind blows steadily for half a day or longer it generates a surface current in any body of water. This new current must be added vectorially to the prevailing ocean or tidal current, or treated as a new issue in areas with no natural currents.

As a rule of thumb the strength of the wind drift is some 2.5% of the wind strength, directed some 20° to the right of the wind in higher northern latitudes and to the left of the wind in higher southern latitudes. In central latitudes the set is more in line with the wind. In Puget Sound or Juan de Fuca Strait and similar confined waterways where the land constrains the wind to flow along the waterway, wind drift here can be figured as essentially parallel to the waterways—with or against the tidal flow. In any waters, though, a 25-knot steady wind for a day or so will generate a current flowing downwind of some 0.6 knots.

In long heavy rains the wind driven current tends to be larger, since the brackish surface layer slips more easily over the denser salt water below. In extreme cases you might expect surface wind drift of over 3% in long, strong winds with much rain.

Helm bias is even more evasive in our navigation reckoning. Strong winds bring high seas (at least in the ocean) and with them the problem of steering over them. It is usually possible in these circumstances to detect a persistent course alteration at each wave. A common tendency going to weather is to fall off or get pushed off slightly at each wave. The only way to gauge this effect on the average course manually is to stand and watch the helm and compass for some time. Then make a guess at an average offset.

Or simply look at the track of a GPS plot of positions, which is what one would do in a normal situation—although it would still be difficult to pull out the helm bias from leeway and wind-driven current in some cases.

We are discussing DR here, however, and that means we are assuming we don’t have these aids to look at. But it does bring up the important point that the best use of such wonderful nav aids is to use them to teach us about DR. In other words, when the waves start to build, confirm with the helmsman what course is being steered, and then watch the plot of positions to see what is being made good. There can of course be other influences (the subject at hand), but by noting what is being made good and then just standing there and watching the compass and the helm for a while, you can see what is taking place.

We are looking at going to weather here, which won’t happen much in strong winds unless you are racing or trying to claw off a coast after getting in too close, but this same helm bias occurs going downwind as well. The bias sailing downwind tends to be to leeward (the right way) in big waves and fresh air, but in light air it might tend to be upwind in big waves as you try to keep the boat moving. So the summary is sailing to weather helm bias will most likely be to leeward, but sailing downwind it could be either way.

Also if you are navigating a race boat there can be numerous types of helm bias to watch out for, and they might be personality driven. Some drivers like to go fast regardless of what the course is supposed to be. Others might choose a more conservative course than called for to not risk a round up.

Here’s an example to sum up these invisible problems. Going to weather across a 0.6 knot wind drift at 6 knots would set you off course some 5°, your leeway might be some 10°, and a helm bias account for another 5°. In this case the overall off-course set is about 20°. In the ocean, after 100 miles you would be some 30 to 40 miles off course to leeward if you did not figure these factors in your DR. Summarized another way, going to weather in a steady 20 to 25 kts of wind, will most likely cost you at least 20° of course made good.

When sailing to weather in strong winds, you will always be slipping down wind more than you might guess. While the GPS is still working, keep a record of what angles you actually tack through in big waves and strong winds—not compass headings, but the actual angle between the two track lines on the GPS plot. If you end up having to DR in these conditions without electronics, then it is usually a reasonable first guess to assume you did indeed cover the miles your log indicates, but then simply set the course made good some 20° to leeward.

When racing, there is an obvious advantage to tacking at the right time. If the electronics take a hike when you need them, these are the basics to fall back on. The compass headings alone won’t help. In the above example, it would mean overstanding the windward mark by 20°.

Accuracy evaluation and statistical errors in DR are covered in detail in the book Emergency Navigation, because in an emergency without the aid of our accustomed instrumentation we are left with little but DR to go by.


Figure 3. How to use electronic charting display of your track to choose the time to tack when you are not making good the course you are steering. Here the boat is steering 045 on port tack in a northerly wind, but only making good 055. On a starboard tack, steering 315, you only make good 305. Once you have your actual tracks established on the chart on both tacks, you can then draw a range and bearing line parallel to the CMG of the final starboard tack, and move it to the windward mark. Then where that line crosses your last port tack line (projected from your COG), is when you tack to lay the mark. This would be the same maneuver to make if it were tidal current setting you off course rather than just wind. 

Thursday, January 9, 2020

The Regiment of the North Star

The principle of finding your latitude in the Northern Hemisphere from the angular height of the North Star, Polaris, was known in 250 BC, when an application of the principle was used to make an accurate measurement of the earth's circumference. There is no written correlation of these two applications of the same principle that I know of, but it is indeed the same, and my guess is it was surely on the mind of countless monks, tucked away somewhere thinking on such things across the middle ages.

The principle is almost forced upon us. If we think of Polaris as located at the hub of the sky, directly over the North Pole on earth, if I am standing at that point (Lat = 90º N), the star is directly overhead, with its height above the horizon equal to 90º.

If I then travel due south, I leave the point where the star is overhead, so it descends in the sky. If I travel 60 nmi south, then my new latitude will be 89º N, because 60 nautical miles is defined as the distance spanned by 1º of latitude. At this point, I can look up and confirm that the star is no longer over head; it is lower in the sky, and indeed by exactly the 1º I moved away from the pole. If I travel on down to 87º N, I have moved down the globe by 3º, and the star will then have descended by 3º, and its sextant height above the horizon will now be 87º. And this behavior continues as I head down into more likely cruising waters; at 30º N, the height of Polaris will be 30º above the horizon. The height of Polaris above the horizon observed from anywhere in the Northern Hemisphere is always equal to the latitude of the observer.

We could prove that mathematically with geometry and some discussion of parallel light rays from distant stars, but wouldn't this be our first guess without any math: I go 2º away; it goes 2º down. The sketch below summarizes the geometry.


Light rays from stars (millions of millions of miles away) arrive parallel, across the entire span of the earth's orbit (<200 million miles) about the sun. At the pole, the star is overhead; at latitude Lat, it is at angle H above the horizon. Angle sides in red and blue are mutually perpendicular, so the angles are equal, H = Lat.

The problem with applying this simple principle to navigation is Polaris is not in fact located at the hub of the sky. In modern times, it is located about 0º 40' off of the hub, so it, like all other stars, circles the true pole of the sky once a day. This one just happens to be on a very tiny radius of 40' so we cannot detect that motion with the human eye; we do not see it move throughout the night. In the days of Columbus, it was 3.5º off of the hub.

So the principle needs clarification: It is the height above the horizon of the true pole of the sky that matches our latitude, not the height of Polaris itself, which is moving around it.



Polaris circles the true pole at a radius of 0º 40' today; it was off by 3º 30' in the time of Columbus.

When the star is below the pole, we need to add something (the angular radius of its circulation) to that height to get our latitude, and when the star is above the pole, we need to subtract that correction.

In the earliest days of Portuguese navigation, the height of Polaris was not used for a direct measure of latitude, which was not even marked on charts in those days, but rather to keep track of distance traveled from Lisbon, but this circular motion of the star (3.5º off the pole at the time) led to complex procedures. These procedures were later simplified by various sets of rules for making the correction, called regiments.

According to EGR Taylor in her "Navigating Manual of Columbus", the first regiments were too complex for the average marnier—she puts Columbus in that category—but then a breakthrough came by describing the location of Polaris relative to the pole in terms of the two stars on the tip of the cup of the Little Dipper, thought of then as a horn. These stars are called The Guards; one of them (Kochab) and Polaris define a clock hand, as the stars circle the pole. Indeed, that was the trick. The Guards had been used "for centuries" earlier on for time keeping, by both mariners and shepherds, relative to an imagined human figure in the sky with outstretched arms to specify the quadrants.  See our own prescription for a modern star clock.


The location of the Guards had been used for telling time at night since the 13th century, so it offered a familiar way to estimate the Polaris correction for latitude. This graphic of the stars is not to scale.



Example of Guards marking the effect of the rotation of Polaris around the Pole in late 1400s. The "line" is between Polaris and Kochab. The numbers mark the height of Polaris in degrees viewed from Lisbon with the Guards in those positions. Lisbon is at about 38.5º N, so Guards in the left shoulder meant Polaris was directly below the pole (see previous graphic), and Guards on the right hip meant Polaris directly over the Pole. A sailor farther south would then measure the height of polaris and note the location of the Guards to compare with this diagram (or printed Regiment) to learn how far south of Lisbon they were.

There is record of a Portuguese observation of Polaris in 1462 by Diego Gomez Cintra, who used a quadrant to measure the height of Polaris to determine latitude.  He went on to "discover" Sierra Leone in 1465. It is not known what Regiment he used.

Eventually the reference to Lisbon was replaced with a more generic reference to Latitude based on the location of the Guards relative to this imaginary figure. The earliest such references were:


"Guards in the head, North Star 3° under the Pole, 
Guards in the feet, Star 3° above the Pole"


The first detailed documentation of a device to make this measurement and a tabulated set of more sophisticated corrections is found in Arte de Navegar by Martin Cortes written in 1545. The original illustration is shown below.


The Guards stars (tips of the horn, above the word GVARDAS) are strangely not marked as the other stars in this original sketch, and it is not entirely clear what was meant to be the POLE—though we can piece this together since we know where these stars really are.

Cortes states that he uses 4.9º for the maximum offset, because that is what the astronomers told him, and they know the most about stars, but he acknowledges that mariners believe it is 3.5º, which is in fact what it was in the late 1400s, which probably reflects the actual time of its common usage. This was some 60 or 70 years earlier than the Cortes book. At the time of Cortes is was just about 3º.

This classic Spanish text was translated to English by Richard Eden in 1561. The full text is online.


A rendition of the Cortes instrument presented by David Waters in The Art of Navigation in England in Elizabethan and Early Stuart Times (1958). A handle presumably there based on nocturnal designs is not shown. The center hole is open to be aligned with Polaris, while the horn is rotated to match the orientation of the Guards. The correction is read from the mouthpiece of the horn. This one is interpreted as 3º above the Pole and NW of it, but based on a faulty scale.

I have not tracked down how this method finally faded away. We had good star almacs by early 1700s which would rule out such needs. It was used at least into the early 1600s. Here is an example from a 1595 voyage of Thomas Hariot who had made an improvement to the earlier Regiments, but it seems to have been lost. The process seems a bit complex (from David Water's book).



With modern almanacs and super tools like Stellarium, we can now investigate these early procedures and indeed make up our own rules, such as this Modern Regiment of the North Star.  As celestial navigation itself fades into the horizon for land based navigation, these earlier, simpler tools may find a place again in the backup kit of prudent navigators.















Wednesday, January 8, 2020

A Modern Regiment of the North Star

The height of Polaris, the North Star, can be used to find latitude, whenever the star is visible, which is typically above latitudes of about 5º N.  The star is too faint to be seen very often at lower latitudes. The method dates back to the mid 1400s, as noted in our short review of the history of this method, which describes how early mariners read the necessary correction directly from the relative positions of other stars, in particular Kochab, the brighter of the two Guards stars in the Little Dipper.

In modern cel nav we do not use such methods, but rather look up the corrections in the Nautical Almanac, which change from year to year. The total correction is made up of three parts (a0, a1, and a2) which are applied to the observed altitude Ho of Polaris. This process is unique in that it is the only navigation technique that is fully described in the Nautical Almanac, whose instructions otherwise deal only with the astronomical data it presents.

In our recent book called GPS Backup with a Mark 3 Sextant we distill celestial navigation down to the easy-to-learn-and-apply basics needed to find your position should other methods (notably GPS) not be available for any reason. This leaves us with Lat and Lon from a noon sight, globally, plus Latitude from Polaris in the Northern Hemisphere.

As we are dealing with a backup solution, and indeed using a sextant with an accuracy of just a couple miles (the Davis Mark 3 for about $49), we do not need the precision of an almanac solution of the Polaris measurement. Instead, we can go right back to the fifteenth century basics that went on to be used for the discovery of most of the world as we know it.

This method is time and date independent, which is just what we want for a backup method. To this end we have created a modern regiment of the North Star based on the location of the star Alkaid (trailing star of the Big Dipper) or we can use Segin, the trailing star of Cassiopeia on the other side of the pole. Having options on opposite sides of the pole is required for this method because we use convenient reference stars that are well separated from Polaris, which means one might be below the horizon. The key star Kochab used in the traditional solution is 16º away from Polaris, which means that method works down to about 16º N; our method works anytime Polaris is visible.

The geometry of the new rules is shown below.
Our task is to determine the bearing of Alkaid or Segin as if they are laid onto a compass card.

The corrections are then read from this table, which is the one in the book cited above. We add one with finer steps below.


Here are several examples.


You can get a rough guess of the angle S or A by just looking at the stars.  Here is a sample.


First identify the stars, then estimate the angles as if you were looking at Polaris through a hole in the center of a compass rose, with the North-South line perpendicular to the horizon.

When I did this test by eye alone, I called S = 20º to the right of vertical (i.e., S = 020), and A = 30º to the left of vertical (A = 210). I know they should be 180 apart, meaning 20º to the left, but to my eyes on this picture, A looks bigger without any further aids.

 In our GPS Backup Kit we include a Douglas protractor and a roll up ruler that together can be used for alignment. To optimize this viewing at night, we found that a white tape along the bottom edge of the clear protractor made it easier to hold this bottom edge parallel to the horizon. In the kit we use a short piece of a plastic page binding clip that can be slid on and off so the protractor can still be used for other navigation.

We have learned that it is relatively easy to measure these angles fairly accurately using a tool of this type.



This is a schematic view. It will not look quite as nice as this, and indeed having a ruler to hold

 up to it will help. 



With such a tool we can read off better values, as noted below



Angles measured to determine the Polaris correction with the Backup regiment.

Now we could interpolate the Q-Table in the book, or make a new one with expanded scale, as shown below.
Our A = 24 would imply a Q= +31. It is plenty accurate to round these, or take the one in the middle if closer.  [Technical note to the careful observer: this formulation of the table did not reproduce exactly the values of all entrants of the original tables, though still well within expected uncertainties. As soon as possible, we will track down the best values and make the needed updates.]

If in this example we were using a sextant with an index correction IC = 2' OFF from an eye height HE = 10 ft, and we measured the sextant height of polaris to be Hs = 48º 56.9', we would make the corrections for IC (+2.0'), dip (-3.1'), and Refraction (-0.8') to get an  Ho = 48º 55.0', and then

Lat = Ho ± Q = 48º 55' +31' = 49º 26' N.

Here are two practice exercises with answers. Remembering in looking for the key stars, that the Big Dipper, home of Alkaid, is about the same distance from Polaris as Cassiopeia (home of Segin) is on the other side.  These two constellations are thus spread across a large span of the sky.  When the S-A line is roughly parallel to the horizon, the two stars span the northern quadrant of the sky, near 90º apart.

Example 1.


July 19, 2021, UTC = 05:56:28
DR: 44N, 141W
Hs Polaris = 43º 23.4’
IC=0
HE=10ft
Find Lat

Example 2.


July 14, 2020, UTC = 14:42:44
DR: 25N, 151W
Hs Polaris = 25º 29.4'
IC=0
HE=10ft
Find Lat

Answers

Pros and Cons
Pros: The regiment method does not need almanac, nor time, nor any dated tables. Even no tables work if you refer to the figure at the top and remember that Polaris is 15º forward of the Alkaid to Segin line, with the Polaris on the Cassiopeia side. Then draw it out and you can create the Q tables knowing the max offset is 40'.

Cons: we have to make the S or A angle estimate, which takes a bit of practice,  and in rare cases we might see Polaris for a good altitude, but not either of the two needed stars.

The traditional almanac procedure with known UTC has the advantage of being pure cookbook, no such star angle estimates needed, and indeed the results will be more accurate. With the Regiment we consider 3 or 4 miles accuracy as good. With timed sights and almanac reduction we expect 1 mile accuracy as good.

How to practice reading the angle
(1) Go outside at night and with low background lights study the north sky. Then by eye alone, estimate the angles S and A and write them down with the time.  Then use what tools you find best to estimate the angle of Segin or Alkaid, usually relative to horizon or the vertical is easiest. Write those values down and note the time.

This angle practice can be done anytime of night, even though we must eventually take the sights at twilight when we can see the horizon. For this practice you just need some way to discern a vertical line or horizontal one, even though you may not have a sharp enough horizon for the sight. Indeed, you can practice this when the horizon is blocked by buildings. You can often use their roofs for a reference level.

(2) Back at home or at work, on the computer, set up Stellarium (free super program, Mac or PC) with your time and date, Lat and Lon, and see the same sky you were looking at.  Then measure S and A from the screen in some manner, or print and use a protractor.  Mac users can use the excellent PixelStick app for this.