Finding UTC of LAN with the Starpath Custom Sun Almanac

We published a short book last year that is intended to be the bare minimum celestial navigation training that will serve as a backup to a loss of GPS. In keeping with the backup concept, the book is presented in such a way that it is totally self-contained, which means the book also covers how to use the simple Davis Mark 3 sextant. This is a small device, for under $50, that could be used to safely circumnavigate the world.

Our promise is that this book could be opened up and read for the first time when it was actually needed, and it would be adequate to teach how to take the sights, and then find your position from them using only data in this small book—a large part of which is a custom Sun Almanac designed to make the position solutions especially easy.

GPS Backup with a Mark 3 Sextant

The techniques taught in this book are finding Lat and Lon from "noon sights" (local apparent noon,  LAN) as well as Lat by Polaris in the Northern Hemisphere. What we forgot to include in this first printing was a way to determine an efficient time to start taking the sights near midday. LAN occurs when the sun crosses our meridian, bearing due north or due south at its peak height in the sky.

When the sun is less than halfway up the sky at noon, we can approximate its motion along the horizon as a bearing change of 15º per hour.  So if we want to start about 30 min before the sun reaches the horizon, we would start taking sights when the sun was bearing about 173 T.  After that first round of sights, we will know the time of LAN and the next day we can fine tune the starting time.

When the noon sun is much higher in the sky it is more difficult to predict its bearing change rate as it approaches noon, and as we head off toward the tropics the sun will indeed be much higher at noon, so it is valuable to have a systematic way to predict the time of LAN to plan around. The custom Sun Almanac offers an easy way to do this. In fact, it is easier than the standard methods we use when teaching the "full cel nav" course.  The full theory is in the picture below from our textbook Celestial Navigation: A Complete Home Study Course.

As the earth rotates toward the east, the position on earth directly below the sun (its geographical position, GP) moves west at the rate we are rotating, namely 360º of Lon in 24h = 15º of Lon per hour.

The custom Sun Almanac tells us the longitude of the GP (called its Greenwich hour angle, GHA) every hour of every day. So we can go into Sun Almanac on the right day to see which whole hour of UTC has the sun's GHA just east of our location.  Then we subtract that GHA from our DR-Lon to see how far it has to go (as an angle) to get to us, and then we just covert this angle to time at the rate we are turning, which can be derived from 15º = 1 hr, 1º = 60 min/15 = 4 min, etc.

                      1º = 4 min

                      1' = 4 sec.

Here are two practice problems from our textbook (page 33.)

Example 1.  July 25 from DR-Lon = 122º 18'W.

DR  = 122º 18'   =  121º 78.0'
20h = 118º 21.7' = -118º 21.7'
            diff =    3º 56.3'

3º = 3º x 4m/1º                 = 12m
56.3' = 56.3' x 4s/1' = 225.2 s =  3m 45s
                            SUM = 15m 45s

Final time is 20h 15m 45s, which we would round to 2016 UTC, as we never need to know these times more precise than that, not to mention that the DR position that it is based on has some uncertainty.

This is the UTC of LAN observed from 122º 18' W. If we wanted the watch time (WT) of the event for a watch set to zone description (ZD) + 7, then we would have to back out that ZD.

Recall the definition of ZD, which comes from this equation UTC = WT + ZD, where WT is the watch time being used for navigation. Thus if the ZD = +7 and the WT = 1200, then the UTC = 1900. Likewise if we know the UTC is 1900 on a watch with ZD = +7, then the WT = 1200. 

In the above example, LAN = 2016 UTC would correspond to WT = 1316 for a watch set to ZD = +7.

Example 2. Find  UTC of LAN on October 27 viewed from 136º 10.5' E.

This is an eastern longitude. In east longitudes the GP of the sun is still moving west with increasing time, so the longitude it crosses gets smaller as it passes across the eastern half of the globe. But that does not matter to us, because, unlike longitude, GHA does not decrease on the eastern half of the globe. GHA is defined as 0º to 360º, measured west from  Greenwich.  Our job is to determine what GHA does 136º 10.5' E correspond to. Starting at Greenwich we head west to 180º W (the dateline), and then proceed from 180º E to 136º 10.5' E, or we cover 180º - 136º 10.5'  =  179 60 - 136 10.5 = 43º 49.5', which is what we must add to 180º to get the GHA equivalent of this eastern longitude. The answer is 223º 49.5'.

Now we are back to solving the problem just as we did for western longitudes.

 "DR" = 223º 49.5'E 
  02h = 214º  1.7'  
 diff =   9º 47.8'

9º = 9º x 4m/1º                  =  36m 00s
47.8' = 47.8' x 4s/1' =    191.2s =  3m 11s
                              SUM = 39m 11s

We add this to 02h to get 02h 39m = 0239 UTC,

If we need to convert back to WT, we refer to the definition: UTC = WT + ZD, or with ZD = -9, we WT = UTC +9h = 1139 WT. 

For those who might like extra practice, you can use the USNO sun data computer (Form B) to randomly select locations and dates to compute UTC of LAN (they call it "sun transit"), and compare that with what you get from the custom Sun Almanac.

In principle the answer depends on the year, but our custom Sun Almanac averages out yearly differences, but even with this, the time of LAN found from the these tables will always be right to well within one minute.

Notes: 
• the USNO site is down until April 30, 2020
• the new printing of this backup book now includes an arc to time table as well as an analemma as an alternative means of finding time of LAN.

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 UTC of meridian passage (mer pass) at Greenwich 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 the UTC of mer pass at Greenwich 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:

UTC of mer pass at Greenwich = 1200 UTC ± EqT

Thus if the Nautical Almanac tells us that the UTC of mer pass at Greenwich is 1207 on some specific date, it means the EqT is +7 min.  If the Almanac says UTC 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 UTC of mer pass at Greenwich 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 UTC at Greenwich is 1207, 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 21m UTC, and for watch time we undo the ZD  to get 14h 21m.

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. (See section of an Arc to Time table at the end here.)

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.

_________________

Lunar Distance by Calculator — The Letcher Method

The lunar distance method is a way to find UTC by measuring the angular distance between the moon and an adjacent body (sun, star or planet).  The measurements are called "lunars." And finding UTC is equivalent to finding your longitude. Recall we can find an accurate latitude without knowing the correct time, but standard procedures requires accurate time to find longitude. Lunars are thus a way to fill in your position if you have lost accurate time—or, starting from scratch, you never had accurate time.

It is an advanced skill in cel nav, because it requires very accurate sights and special analysis. Finding UTC to an accuracy of ± 30 sec or so is considered good work. We have notes on the history and practice of the method (starpath.com/lunars), which shows that there are essentially three approaches to the solution.

The first solution is the easy one. There are numerous apps online and for sale that solve lunars. Our own StarPilot was one of the first ones.  With these apps, you can just enter your DR position, the date and best guess of the right time, the name of the body you are using, and the lunar distance. There is usually an optional input for the measured altitudes of the moon and body, but these are not needed accurately, so they can be computed from the DR position. 

Everything else is computed and the output is the right time and right longitude, keeping in mind the uncertainty of the process: namely, every 0.1' of error in the  measured distance or clearing process leads to about 12 sec of time error, which corresponds to 3' for longitude error. 

The second solution is just the opposite: do it all by books and tables, like it was done in the late 1700s to early 1800s. The only math required is adding and subtracting longish numbers, but the tables and procedures can be complicated. With that said, this is the only logical solution, in that it is waterproof and no batteries are required.

Luckily, we have a modern all-paper solution created some years ago by Starpath friend and associate the late Bruce Stark. His book Stark Tables for Clearing the Lunar Distance — And Finding Universal Time by Sextant Observation has become a modern classic in cel nav studies. Experts consider his solution superior to some of those actually used historically (see discussion in the lunars link above.) But like the historic versions, doing this all with tables takes some practice before it becomes routine.

The third solution, which is the subject at hand, is a compromise of sorts between the first two. Namely we use a set of four trig equations that we can solve on any simple trig calculator along with standard sight reduction tables and a Nautical Almanac, and we compute the solution "by hand."

To my knowledge, this method was first put together by John S. Letcher, Jr in his excellent 1977 book on cel nav called Self-Contained Celestial Navigation with H.O. 208.  He was also a major technical authority and innovator of self steering devices, as presented in his 1974 book Self Steering for Sailing Craft. Both books are rare, but still found periodically in used book sites.

Below is an outline of his method. It has been elaborated upon in the 1980 book by Shufeldt and Newcomer called The Calculator Afloat, which is online in full. 

To help learn this process, we made an app (Letcher Lunar Distance Calculator)  and spreadsheet that solves the Letcher method that can be used to double check that you are computing the terms correctly. The app includes the examples presented in both books above. There is no logic to using this for actual solutions as there are other apps that are equally, if not a bit more, accurate that require much less input. The app and spreadsheet are purely training tools.

The idea here is you practice this a few times then write this prescription in your logbook to fall back on as needed, keeping in mind that the StarPilot, which covers all aspects of ocean and inland navigation computations, includes a lunar solution if needed.

It is important to skim over the discussion in the lunars link above. The summary is:

1) We measure the distance between moon and another body , which will be edge to edge. Then we use the Almanac to find the semi diameter of the moon (and sun if using it) so we can correct the edge to edge to get D, the lunar distance center to center, at time Ts which is our best guess of the UTC. In practice we need to take several sights and plot them D vs T then do a best fit to the line and find our best estimate of D and its corresponding Ts.

2) Then we "clear" that distance, which in this case means applying two corrections to D, refraction (R) and parallax (P). These corrections require us to compute the Hc of the moon (Hm) and of the body (Hb) at time Ts, which we do by looking up their GHAs and Decs, and then compute the distances with the formulas below. 

Note if you are underway, or have good horizon for the bodies on land, you can measure these heights in the normal way, and plot them to find a fix. The Lat of that fix will be correct, but the Lon will be off by the error in Ts, which we are finding from the lunars.

3) Once we have the cleared lunar (Dc) we need to see what time the two bodies where that far apart. We do this by computing the distance a the whole hour before Ts and at the whole hour after Ts.

Note that the lunar distance is just the zenith distance (z) to the moon assuming you are standing at the GP of the body being used. Thus D = z = 90º - Hc, which we can compute using the navigation triangle formula below (we just change our a-Lat to the dec of the body and our a-Lon to the GHA of the body).

Here is how the app input looks, showing also the Almanac data we need to enter.

Here is the sample solution from the Shufeldt book.

Once we have our measured Dc, we need to find out what time were these two bodies that far apart, keeping in mind the main premise of this measurement being that the moon moves eastward relative to the other bodies in the sky at a rate of about 12º per day, which is about 2' per minute. Not much, but right at the edge of our being able to measure this with standard sextant.

We figure this by interpolation as shown below:

Then within the accuracy of this method, Tc is he right time and Tc - Ts is our watch error.  The corresponding Lon error is 15'/1 min of time error.

The app is essentially two worked out examples that you can test with your own calculator.  The spreadsheets are there for those who want to look at that to see how these equations are entered into Excel.

Note that both examples given in the app are from May 27, 1973, so to follow though each step, here is the almanac data for that date that shows where the input came from.

These data are also needed if you want to check any of the online apps, as they typically want input of raw sextant data and then they compute everything, so to use them we have to unfold the two heights to Hs values and the LD they want is edge to edge, then they make the corrections in the apps.  Convenient for use underway, but not convenient for checking the Letcher method, so let me just add that the Letcher method does indeed work (within limits noted below), and can be improved a bit more by adding easy corrections for augmentation of the moon's SD and earth oblateness.

Letcher streses that his procedure for the important refraction correction is not valid for moon or body altitudes or lunar distances less than 10º and also the method is not valid for Venus and Mars since parallax and phase corrections are not addressed. These are not severe limitations for practical use, but we should keep them in mind.

 

USCG Chronometer Time — Bless Their Hearts

We have often had occasion to point out how the USCG supports navigation schools, but one of the contenders for the top price is their use of chronometer time (CT). Someone making up the tests must have read the fine print of the Bowditch definition and had an aha moment on how they could support navigation schools.

“chronometer time. The hour of the day as indicated by a chronometer. Shipboard chronometers are generally set to Greenwich mean time. Unless the chronometer has a 24-hour dial, chronometer time is usually expressed on a 12-hour cycle and labeled AM or PM.”

In other words, CT is the same as UTC, except for a possible chronometer error (CE) correction, but it can be kept on a 12-hr watch dial.  The insidious step in the USCG exam preparation was to use this concept of CT on a 12-hr dial, and then not tell the test taker if the chronometer time is AM or PM! That way, every single USCG cel nav exam question must start out with the candidate having to figure out what time they meant using other information in the question—it brings to mind the old computer game called Myst. 

Granted, candidates do need broader cel nav knowledge to figure out the time, but it runs the risk of implying this is, in some universe, a viable way to keep time, which of course it is not. No navigator in the world would use such a system.

So the first step of each USCG cel nav question is to, for example, read the given time of the event as CT = 10h 13m 20s and then use other information in the question to determine if this is 10h 13m 20s UTC (CT was AM) or is it 22h 13m 20s UTC (CT was PM).

Sometimes the process is straightforward, in that we get at least one zone time (ZT) and a DR-lon.  We can figure the zone description (ZD) and from this convert the ZT given to UTC.

The rule for finding ZD is round the Lon to nearest degree, divide by 15, and round the result to nearest whole number. West is + and East is -.  ZD is defined by this equation:

UTC = ZT + ZD.

Below are a few examples of how candidates must figure out the time.

Here is an easy one:
On 15 August your 0512 zone time position was LAT 29°18.0’N, LON57/G 57°24.0’W. Your vessel was steaming on course 262°T at a speed of 20.0 knots. An observation of the Sun’s lower limb was made at 0824 ZT. The chronometer read 00h 22m 24s and was slow 01m 34s.

Find ZD: 57/15 = 3.8, so ZD = +4.  Then UTC = 0824 + 04 = 1224, so CT must be PM, and we get the right UTC = 12h 22m 24s + 01m 34s = 12h 23m 57s.  The chronometer error correction (CE) is normal. If slow you add it; if fast you subtract it.

Another easy one:
On 10 March in DR position LAT 21°42.0’S, LONG 57°28.0’E, you take an ex-meridian observation of the Sun’s lower limb. The chronometer time of the sight is 08h 28m 17s, and the chronometer error is 00m 00s.

No zone time given, and it does not say upper or lower transit, but it has to be upper for the sun viewed from 21S. We also know it must be near midday on ZT.   Find ZD = 57/15 = 3.8, so ZD = -4 and use UTC = ZT + ZD to find ZT = 0828 - (-4) = 1228.  Or it could be ZT = 2028 -(-4) = 2408,  which is near local midnight, so this can’t be right. The first is correct and UTC = 08h 28m 17s   (CE = 0).

Another (sort of) easy one:
On 24 August in DR position LAT 26°49.4’N, LONG 146°19.4’E, you observe an amplitude of the Sun. The Sun’s center is on the celestial horizon and bears 084°psc. The chronometer reads 07h 55m 06s and is 01m 11s fast. Variation in the area is 15°W. What is the deviation of the magnetic compass?
o (A) 8.0°E
• (B) 8.3°E
o (C) 8.5°E
o (D) 8.7°E°.

This is an amplitude problem, so we know the sun is rising or setting. With bearing given as about 069T (084-15), it must be rising, so we know local time (ZT) is early morning.  The ZD = 146/15 = 9.7 or ZD = -10.  So we have two choices for the local ZT, which in turn tells us if the CT is AM or PM. If CT is AM, then ZT = 0755 + 10 = 1755 ZT, or if PM we have ZT = 1955 +10 = 0555 ZT the previous day. That is, going back to the definition of ZD, we have UTC = ZT + ZD = 0555 ZT (Aug 24) + (-10h) = 1955 Aug 23.

Without looking up the sunrise times, we know this must be 0555 ZT (ie CT was PM) and therefore the right UTC of the sight is 19h 55m 06s -(01m 11s) = 19h 53m 55s on Aug 23.

This beauty brings out a couple nuances for test takers to note. For one, the phrase “On 24 August in DR position...” implies that the date is the zone time date.  DR positions are always given in ZT—at least on all the test questions where the date is not ambiguous, and fortunately that is most of them. So we then learn that the CT given could indeed be on a different date. One way to check this is to compute the actual Zn of the sun at that time on each of the two days by standard sight reduction. The result would be:

19 53 55 Aug 23   GHA 117 51.5   dec N11 16.2   Hc = - 0 01.8    Zn = 077.3 (= 092.3 M) -> dev 8.3E
19 53 55 Aug 24,  GHA 117 55.5   dec N10 55.7   Hc = - 0 07.8    Zn = 077.7 (= 092.7 M) -> dev 8.7E

The other point we are reminded of is the wrong answers in USCG exams are not random. It is probably not fair to call the problems “trick questions,” but there are certainly trick answers. Namely, the wrong answers for most USCG exam questions are the result you would get from making a typical error. We see this here by noting the wrong date leads to one of the wrong answers, but the distracting answers in most amplitude questions are more subtle.  The wrong answers are mostly tied to making an error in the conventional amplitude solution (using Bowditch tables 22 and 23), which is the intended way to solve the problem. I have more to say about these amplitude questions elsewhere, but it is beyond the subject of figuring out what CT means.

Here is one a little more involved:
On 16 June in DR position LAT 50°57.0’S, LONG 53°03.9’W (ZD+4), you take an ex-meridian observation of Acrux at lower transit. The chronometer time of the sight is 10h 08m 18s, and the chronometer error is 02m 12s fast.

Again, the ZT of the sight was not given, but with ZD = +4h we can check CT. That is, CT = 1008, means either UTC = 1008 or 2208. The first gives sight time = 0608 ZT; the latter gives 1808ZT.  Under some circumstances we might be able to judge from this much information alone, but not in this case— either one could be within twilight without further knowledge. So we need to check which one of these happens to be in twilight when stars and horizon can both be seen. 

Checking the 1981 Nautical Almanac, and noting that DR-Lon 53º 3.9’ = 3h 32m (Arc to Time table), we find that sight time in the morning (ie nautical to civil twilight) = 0646 to 0812 LMT, which corresponds to 1018 to 1144 UTC.  We find the LMTs in the Almanac, and then correct for the longitude to get the UTC, ie 0646 LMT + 0332 (Lon correction) = 0978 = 1018 UTC.

Evening sights are taken between civil and nautical twilight, which in this case would be 1630 to 1715 LMT, which corresponds to 2002 to 2047 UTC. 

We see that an AM CT is very close to morning twilight time, but the PM CT is not close at all. Thus we know then that the CT given must have been AM, and that the UTC we need for the problem is 10h 08m 18s - (2m 12s) = 10h 06m 06s.

And one final example:
On 30 March in DR position LAT 20°26.2’N, LONG 131°17.9’E, you take an ex-meridian observation of the Moon’s lower limb at upper transit. The chronometer time of the sight is 10h 36m 02s, and the chronometer error is 02m 06s slow.

This one is a more challenging CT puzzle, meaning we simply have to know they ask such questions and be prepared with how to solve them.  There are no zone times given for anything, which always makes the solution more interesting.  We can assume ZD is -9 (from 131º/15 = 8.7, which rounds to -9.

So we ask (after applying CE) does UTC = 10 38 08 or UTC = 22 38 08?

Checking to see what the ZT of the sight  would have been for each of these interpretations, we have to choose between: 1938 ZT Mar 30 or 0737 ZT on Mar 31?  But since this is a moon sight, we cannot get any hints from the twilight times—moon sights could occur in either twilight or throughout the day.  So they are handing us a bit more than the average amount of sleuth work. Unlike the sun, when we know mer pass is near midday on ZT,  we have no such hint about when the moon might cross the meridian.

At this point, we might resort to a nuance mentioned in an earlier example, namely the statement  that your sight is from a DR position on a given date essentially implies the ZT date of the sight time is the one given.   There is no zone time given, just the CT, and the CT is UTC (either AM or PM), but without other information, we have to assume the UTC date is the one given. In other words, this is either 1038 UTC on Mar 30 or it is 2238 on Mar 30.  These two different interpretations of the CT would in fact lead to different days on the ZT clock, which could be all we need to know to choose the right UTC. In other words, from the DR date argument alone we would have to choose the PM interpretation of CT leading to UTC = 22 38 08 Mar 30.

But these arguments about the DR and CT dates are just what we have discerned from looking at a lot of USCG problems. That interpretation of the dates is not spelled out in any official definition.  It would be reassuring to have some physical evidence to back this up—navigators do not like to  rely on just one source of information for crucial decisions.  So we should look into this a bit further.

It is an upper transit, so that means the GHA of the moon must be near the DR-Lon at the sight time. The Lon of 131º E corresponds to a GHA of (360 -131) = 229º.  So we can look up the GHA at 1038 UTC on Mar 30 and at 2238  UTC on Mar 30 to see which one puts the moon on or near the meridian, ie has a GHA of about this value.

Using the Nautical Almanac, we find that on Mar 30, 1981:

1038 UTC,  GHA moon =   46º 39.5’ 
2238 UTC,  GHA moon = 220º 21.5’

At 1038z the moon is no where near the meridian (actually not even above the horizon), so the CT was clearly set to PM, and we have the correct UTC = 10h 38m 08s. 

To save time, you could learn this faster by just checking the GHA at the whole hours in the Almanac (1000 and 2200) since one is not even close. 

Looking ahead, with a GHA of 220W and a Lon of “229W” (ie 131 E), when we finish this problem, we should expect the moon to be to the left of the meridian. Its GP (moving west around the globe) has another 9º to go before it gets to the DR meridian.

Finding Longitude From the Time of Sunset

With accurate time available, there are several ways to find your Lat and Lon without a sextant if you ever lose GPS data. We cover each of these in the book Emergency Navigation. In this note we work through one example, which is just noting the time of sunrise or sunset. The principle behind this sunset method is easy enough to understand, but as we show here there are details to executing it, and a couple different approaches.

The measurement was kindly provided by meteorologist Angeline Pendergrass  during a research voyage on the RV Thomas G Thompson in the Strait of Juan de Fuca. She took the data from the aft deck at a height of eye estimated to be 10 ft above the water.

This type of sight requires seeing the top of the sun (upper limb, UL) disappear below the visible sea horizon. This observation is not quite as common as we might guess, even at sea. More often than not, there is a low layer of clouds on the horizon so we do not get to see a nice clean crossing of true sea horizon.

Since she did not have a watch at hand, the procedure was to take a cell phone picture of the sunset, just as the upper limb dropped below the visible horizon, which marked that time in her phone. Then she proceeded to the wheelhouse and took a picture of the GPS screen, which showed the UTC and the location of the vessel, which was drifting at the time. The cell phone time showed the delay was 1m 7s, so she could then figure an accurate time of sunset with the associated position.

The results were:

Sunset (upper limb crossing the visible horizon)
03:17:49  April 23, 2013 UTC
Lat 48º  16’  19.5493’’ N
Lon 123º  58’  57.5673’’  W
Height of eye 10 ft 

[I just found this article on my desktop, started 5 years ago, and finishing it now for our cel nav and emergency nav courses.]

Converting to decimal minutes, we will round this to 48º 16.3’ N, 123º 59’ W, which represents the true position of the vessel at the time of the sight.

We can solve the sunset method several ways. Some are easier in principle; others are faster to implement. We start with one that does not require any knowledge of celestial navigation, and follow it with a much faster method for those trained in cel nav.

SOLUTION 1. NAUTICAL ALMANAC SUNSET TABLES

We can find our Lon using Sunrise-Sunset Tables, but these tables come in several formats. A common type lists the times for the specific standard meridian (the longitude center of a time zone) of a specific place, using that specific time zone, such sunrise and set times for Seattle. Generalizing that type of table to our needs adds another layer of complexity to the process. Luckily, there are sunrise and set data in Tide Tables and the Nautical Almanac that are more generalized. These tables rely on the definition:

UTC sunset (Lat 1, Lon 1)  =  UTC sunset (Lat 1, Lon 0) + (Lon 1) x (1 hr/15º).

In words: the UTC (once called GMT) of sunset observed at (Lat 1, Lon1) is equal to the UTC of sunset at Lat 1 and Lon = 0 (Greenwich meridian) plus the time it takes the sunset to get to us as it moves west at a rate of 15º of Lon each hour.  (In East Lon, the last term is negative.)

We can rewrite that to solve for Lon:

Lon  = (15º/1 hr) x (UTC sunset observed – UTC sunset at Greenwich)

Our Lon is just the difference between observed sunset time and the sunset time at Greenwich at the same Lat converted to degrees at the rate of 1 hr = 15º, which means that 1 min = 15', and 4 sec = 1'

We observe the first term by timing the sunset, so the whole process boils down to looking up the time of sunset at Greenwich on the date and Lat of interest.

You can go online and ask for that time (http://aa.usno.navy.mil/data/docs/RS_OneDay.php) and you will get 1902 UTC for this example, which is given rounded to nearest whole minute, but assuming we do not have internet when we need to find our Lon, we have to use the standard tables, which always requires some interpolation, followed by the time-to-angle conversion. Sample USNO online screens are below. They also offer options to print out various tables.

Top is input page; bottom is output page from USNO site.

The virtue of using the sunrise sunset tables in the Nautical Almanac is they provide just what we want, namely the values of sunrise and sunset in UTC as observed at Greenwich.  Normally, the navigator has to then apply a Lon correction to learn what to expect at their location, but now we are working this backwards. Below is a sample of a full daily page of the Almanac with the sunrise/set data marked.

The Almanac lists only one set of sunrise/sunset data on each page that covers 3 days. The data given are always for the middle date, so we are lucky here in that we do not have to interpolate for the date. If we wanted this on Apr 22, we would have to interpolate between Apr 20 and Apr 23 before we interpolate for Lat. Since we have the right day in this case, we just need to interpolate for the Lat.

50º 00’ N 19:06

48º 16.33 N hh:mm

45º 00’ N 18:56

At this place and time of year, the sunset time increases with Lat by 10 min (1906 - 1856) per 5º of Lat (50 - 45).

We are 3º 16.33’ (3.27º) above 45º, so we find  sunset time at Greenwich at this Lat as:

hh:mm = 18h 56m +(10m/5º) x 3.27º = 18 56 + 6.54m = 18h 62.54m = 19h 02m 32s.

Our Lon is then ( 03h 17m 49s - 19h 02m 32s) converted to degrees.

This is a negative number, which might pose some confusion, but the solution in practice is easy; just add 24 hr. You can think of this (Figure below) as the time it takes the sunset to move west from 1902 to midnight (24:00:00 - 19:02:32) + time it takes to get to us from there (03:17:49 - 00:00:00).

Schematic depiction of the sunset moving west at the rate of 15º of Lon per hour.

Thus we have this sum to make of hr, min, and sec,  done as separate columns. The answer is our "Lon" is 8h 15m 19s, which we then convert to time at the rate of 1h = 15º, 1m = 15', and every 4s = 1'.

Thus the Lon we find from the time of sunset using sunset tables is 123º 50' W, which is to be compared to our actual location of 123º 59' W.

This is 9' of Lon wrong, which at 48 N corresponds to 6 nmi (9 x Cos48). This is well within the expected uncertainty of this sunset tables solution. The cel nav sight reduction in Solution 2 below is more accurate, but even using that method we have to accept an uncertainty of about  ± 5 nmi, based on many measurements of our own and others.  This is mainly due to uncertainties in atmospheric refraction, plus the sunset tables method has rounded base times and no correction for height of eye—someone high in the rigging will see the sunset a bit later than someone on deck.

Before we do the cel nav solution, just a note that using sunset tables included in official NOAA Tide Tables (figure below) there is usually an additional interpolation required. Also note that the sunset times do vary slightly (±1 min or so) over the leap year cycle, so you either have to have the right year or one that is exactly 4 years different from the right year.

Sunrise sunset tables from the official NOAA Tide Tables.  This is the only set we have around at the moment, which is not for the right year. It is just intended to show the format.  (This is a 2018 write up of an article started in 2013!) 

SOLUTION 2 CELESTIAL SIGHT REDUCTION USING COMPUTATION

If you know celestial navigation, an easy solution is to just treat this timing of sunset as if it were a normal sextant sight of the upper limb of the sun, and then do the sight reduction with a trig formula or with a cel nav program that computes the sight reduction. This approach has to be distinguished from solving the sight reduction using Sight Reduction Tables and plotting. It is possible to solve it that way, but that traditional approach takes more time and more graphic interpolation.

In this computed sight reduction approach, we have sextant height Hs = 0º 0.0’ (UL) at UTC = time the upper limb crossed the visible horizon. In this application we would have index correction = 0.0’ (no actual sextant involved) and height of eye (HE) = 10 ft in this particular sighting.  That is all a celestial navigator needs to know to find a line of position (LOP) from that sight. Since we are looking roughly westward at the time, this would be a roughly vertical LOP on the chart, and the place our known Lat crosses that LOP marks the Lon we are seeking.

A key point to keep in mind for this, and other direct determinations of Lon, is we are free to assume we know our Lat precisely. We have many ways to find accurate Lat, even without accurate time. Finding Lon is always the more interesting challenge.  We can get Lat very easily from a noon sight, or any two star sights, even if the watch used is wrong—we just need to know the time difference between the two star sights. A hack watch with unknown error will give us that.  In this type of star sightings, the Lat will be right, but the Lon will be wrong by an amount directly proportional to the watch error  at the rate of 15’ Lon error for each 1 min of time error.

Thus we will assume we know our Lat, and figure our Lon from the time of sunset.  So for now we just choose some random value of what we might have thought our Lon was before we did the sight.  Let us say our DR Lon before the sight was 124º 05’ W. In this case we actually know our true Lon, so our goal now is to assume some wrong Lon and then discover how wrong it was.

Next we do a normal computed sight reduction using HE = 10 ft and index correction IC = 0 and  Hs (UL) sun = 0º 0',  UTC = 03h 17m 49s on April 23, 2013, and we do this from a DR position of 48º 16.3’ N, 124º 05.0’ W. Also, looking ahead, we will use an air temp 49º F and a pressure of 1033 mb.

You can compute this sight reduction various ways. A convenient free solution for Windows computers is the Celestial Tools program of Stan Klein.  Or solve it directly with a trig calculator using the solutions we present in our online glossary under Navigator’s Triangle... although when you go that route we lose a lot of the efficiency. I want to stress that even though it is taking me some long discussion here to explain the procedure, once you have a good cel nav program in hand, the entire process takes seconds, not minutes!

The result of your computed sight reduction will be an intercept very close to:  a = 4.5’ A 290.1. Various programs may differ by a few tenths.  To check your  computed solution by hand you can use the USNO value of Hc (below), and apply the dip (-3.1') and altitude correction (-50.2') from the Nautical Almanac. Plus when doing it by hand we have to apply the additional altitude correction that depends on temp and pressure. This is done automatically in most computer solutions.

Angie's data did not include air temp and pressure, but we can look these up for nearby buoys (NDBC) at the time of the sight to find they were 1032 mb and 48º F.  Sample below

Since the standard is 50º F and 1010 mb, we have a small correction due to pressure alone, as shown in the Nautical Almanac table below.

Temp and pressure corrections from the Nautical Almanac. 

In our example, there is a -1.1' correction but notice that at more extreme values the correction on the horizon (Apparent altitude = 0) can be notable. These corrections are made automatically in most computer solutions, so you just enter temp and pressure. They might not even inform you of how much correction they made. And again I stress that these corrections have an uncertainty in them that is at least as large as the correction itself. These timed horizon sights have, overall, an uncertainty of about ± 5', with maybe slightly better average if done carefully in normal conditions.  Below is the raw data for a manual check of your favorite program.

Data from USNO site that we link to at www.starpath.com/usno.

Once we have our sight reduction done and found a = 4.5’ A 290.1 (again, this is seconds to enter the data and get the result), we can plot to get the answer very quickly. Below is a plot of the LOP using OpenCPN.

Plotting a celestial LOP as a route segment from a plotted waypoint in OpenCPN.

Here we plot a waypoint at the DR position that we used for the sight reduction, then create a one-leg route in the direction Away from 290 T (290-180=110) with a length of 4.5 nmi. Then we draw another route leg perpendicular to that (ie in direction 290+90=020) and that is our LOP.

Then draw in the known Lat line and measure the Lon where it intersects the LOP. In this case, we are very close. Just 0.75 nmi off the true position.  This is a satisfying result (clearly better than the sunset tables solution), but we still have to consider it fortuitous. Nevertheless, it is a superior solution, because it accounts for the height of eye and the additional altitude corrections for temp and pressure. Also we keep in mind that even though a procedure does have a known statistical uncertainty based on many measurements from various sources, it does not mean it will be wrong by that much.

Regardless of how you solve it, this method of finding Lon belongs in the navigator's bag of tricks and it also stresses the value of wearing a watch with known rate so you can always figure accurate UTC.  See our recent note on how to rate a watch for accurate time.

A passing note: Load lines on the vessel indicate that the height of eye may have been a bit higher than estimated, but this factor enters the sight reduction as a square root. So even if it were 13 ft instead of 10 ft, the intercept value would only change by 0.4'. Nevertheless, this is a reminder that in this type of sight and indeed all cel nav sights, we are better off using data as accurate as possible. In this example, this potential difference is not crucial. If the HE was 13 and not 10, then the error was just over 1 nmi, not 0.75 nmi, but if you are taking sights routinely from a higher deck, at some point it is valuable to just drop a line over the side and measure it. Only has to be done once.

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Role of SCAMIN on ENC (September 2021)

Online ENC Object Catalogs (September 2021)

Landmarks on ENC: The Nuts and Bolts (September 2021)

Where Have All the Towers Gone? — Getting Into ENC... (September 2021)

MHW on ENC (August 2021)

Online Global Chart Viewers (June 2021)

Sharing Routes Between OpenCPN and qtVlm (June 2021)

Say Goodbye to the First Paper Chart (February 2021)

How to Report Nautical Chart Corrections (November 2020)

Into the Weeds with Abbreviations (November 2020)

MHW on ENC (August 2020)

Free eChart Viewers (July 2020)
"Swirls" — An Introduction to ENCs, and Why We Lik... (July 2020)
Introduction to NMEA 2000, with a refresher on NMEA 0183 (February 2020)
Who Is Ahead in an Ocean Race? (December 2019)
How to Make an RNC eChart From a Chart Image (December 2019)
Where to Get Nautical Charts (November 2019)
Assigning O-SENC Charts to a Dongle (August 2019)
O-Charts.org—Revolutionizing Nautical Chart Distri... (August 2019)
Best International Chart Deal Ever! (June 2019)
Moving Files Around on an iPhone and iPad (February 2019)
How to Wirelessly Transfer Files Among Computers a... (February 2019)
Chart Symbols: Rock or Coral on RNCs and ENCs (August 2018)
Paper Charts vs. Electronic Charts — Some Thoughts... (May 2018)
OpenCPN, Quick Start, One Chart (April 15, 2018)
Introduction to Charts in OpenCPN (April, 2018)
Minor Light Symbols on ENC in OpenCPN (September 2017)
Forecasting Local Winds with Meteograms (August 2017) 
How to Add a Draw Style to LuckGrib (August 2017) 
How to Combine Grib Files (August 2017) 
Simulate Instrument Data with OpenCPN's NMEA ... (August 2017)
Converting a Google Earth Path to a GPX Route (April 2017)
Rock Talk 2: RNC to ENC (November 2016)
Rock Talk — Is it all awash, or not? (November 2016)
History of Electronic Charting (November 2016)
How to Master Electronic Chart Navigation with the... (October 2016)
Network Connections to Navigation Software (October 2016)
Great App Idea from UKHO (October 2016)
How to Buy, Register, and Download Canadian Echarts (March 12, 2016)
Naming and Boundary Conventions on ENC eCharts (March 6, 2016)
The Strange World of Electronic Chart Boundaries (February 26, 2016)
Viewing and Downloading Nautical Chart with Google Earth (December 14, 2015)
Commercial Echart Ad is Food for Thought (August 26, 2015)
Don't Blame eCharts for Anything (December 8, 2014)
ECS Without GPS (January 20, 2014)
Free or Demo eChart Viewers (September 1, 2013)
Canadian eCharts for Electronic Chart Systems (August 4, 2012)

Coastal

WA State Nautical Charts in Flux (October 2025)

Steering in a following current (September 2025)

Predicting Tidal Currents in the Swinomish Channel (February 2025)

USCG License Exams Come of Age — Tide Wise (October 2024)

How to Get a Copy of a Discontinued NOAA Chart (July 2024)

Nautical Chart Carriage Requirements When Traditio... (July 2024)

Great Circle Distance — The Three Options (December 2023)

Speed and Heading — Basic, But Not Always Simple (June 2023)

Slow Water Rule (November 2022)

Danger Bearings (April 2022)

NOAA Chart Catalogs—A Thing of the Past...Not Really (December 2021)

NOAA Discontinues Tide and Current Books — What Do... (July 2021)

No More Tide and Currents Table 2 — Navigation Stu... (July 2021)

Notes on Bank, Cushion, and Squat (July 2021)

US Coast Pilot Interactive Index (July 2021)

Correcting Add East—An Overview of Compass Convers... (August 2020)

Introduction to NMEA 2000, with a refresher on NMEA 0183 (Feb 2020)

Where to Get Nautical Charts (November 2019)
How to Fold a Chart (March 2019)
Countercurrent Crossing Nav Problem Solutions (January 2019)
Florida Gulf Stream: An Exercise in Sources (January 2019)
Navigation Exercise: Crossing Currents (January 2019)
Race to Alaska (R2AK) Navigation (June 2018)
Weems & Plath Expanding Square SAR Course Identifi... (June 2018)
Paper Charts vs. Electronic Charts — Some Thoughts... (May 2018)
OpenCPN, Quick Start, One Chart (April 15, 2018)
Introduction to Charts in OpenCPN (April, 2018)
Compass Bearing Fix — An Overview (November 2017)
UW Soundings Generator (August 2017)
Light Lists—A Modern Look (July 206)
Navigation Lights—The R2AK Short Course (July 2016)
Chart Plotting Tutorials (May 28, 2016)
Introduction to AIS (February 20, 2016)
Geographic Range — Time to Get Trivial (October 1, 2015)
Boxing the Compass (February 27, 2015)
Rotary Currents (September 1, 2014)
Piloting Practice with Geocaching — You Can Do It Online (March 24, 2014)
Google Earth as an Aid to Chart Reading (December 27, 2013)
Print-on-Demand Charts and pdf Charts (October 26, 2013)
Checking your Compass with the Sun (October 14, 2013)
Tides in Puget Sound (August 8, 2013)
How to Plot with Triangles (April 25, 2013)
Small Craft Wave Danger as a Function of Wave Steepness (April 9, 2013)
High Definition (HD) and Broadband Radar by Larry Brandt (April 8, 2013)
How to make current stations into waypoints in a batch process (April 6, 2012)
Natural Ranges—How accurate are they? (April 4, 2012)
GPS Accuracy in Tight Quarters (April 12, 2012)
Route Planning and Sharing with gpx Files (March 31, 2012)

Nav Rules

Introduction to Vessel Running Lights (Navigation ... (June 2020)
Social Distancing and the Navigation Rules (April 2020)
USCG Deck Exam Questions at Starpath.com (July 2018)
Note on USCG Exam Questions (July 2018)
Showing the Right Day Shape (August 2017)
Can exceptional behavior be considered ordinary? (June 13, 2016)
Right of Way in the Traffic Lanes (March 14, 2016)
Quick Way to Know if you are in the Traffic Lanes (February 12, 2016)
The New Inland Navigation Rules (July 28, 2015)
Lobster Pot...my foot. (July 29, 2014)

Miscellaneous

Symmedian point in a triangle (August 2025)

Battery Life of Phones Running Location Services (May 2023)

Bruce Stark, Navigation Historian, Dies at 95 (October 2021)

How to Activate a Color Filter on iPhones for Nigh... (September 2021)

How to Delete Browser Cookies (May 2020)
Navigator's Library for Extended Ocean Voyaging (November 2019)
Lat-Lon from Street Address (July 2019)
Compact Day of the Year Calendar (May 2019)
USCG Deck Exam Questions at Starpath.com (July 2018)
Note on USCG Exam Questions (July 2018)
Christopher Columbus and the Spherical Earth (May 2018)
Ocean Rowboat Polar Diagram (May 2018)
Effect of Leeway on Knotmeter Speed (April 18, 2018)
Shortcut to NGA Publications (January 2018)
Magnetic Dip and Zone Balancing (October 2017)
Viewing Eclipse by Boat May Bring Surprises (August 2017)
UW Soundings Generator (August 2017)
Adobe Digital Editions (ADE) User's Guide (March 2017)
NPR lets slip poor reporting of the El Faro case s... (December 2016)
Network Connections to Navigation Software (October 2016)
Using Polarized Sun Glasses as a Viking Sunstone (July 2016)
How to Verify Computer File Integrity (January 30, 2016)

AP Release on Tragic Loss of El Faro is Misleading. (October 5, 2015)
Ways to Get Accurate GMT (UTC) (April 1, 2015)
Telling Time by the Stars (March 2, 2015)
How a big, well-run, high-tech race boat, can go aground... almost (December 4, 2014)
Tricky Terms in Navigation (May 20, 2014)
Worldwide Shipping Lanes (December 6, 2013)
One Rombe Wrong, and What Do You Get? (November 22, 2013)
Notes on Loran (November 10, 2013)
Open Letter to the America's Cup Rules Committee (August 27, 2013)
Answers to Geocaching Exercise (July 14, 2013)
New Starpath Kindle and iBooks now online (July 3, 2013)
Ebooks Onboard (June 8, 2013)
A Few Nuts and Bolts of Iridium SatPhone Usage (May 4, 2013)
Two New Books from Starpath (April 2, 2013)
Course Boxes - Still Valuable in Modern Sailing (March 12, 2013)
More mainstream media on Viking sunstones (March 7, 2013)
Obtaining Ocean Current Data Underway (February 10, 2013)
State of the Art Ocean Current Presentation (January 31, 2013)
Atlantic Ocean Current Data and 6-day Forecasts (January 12, 2013)
Ocean Currents (January 8, 2013)
A Local Mini Surge (December 17, 2012)
NOAA Chief Misspeaks on Sandy Surge (December 14, 2012)
OPC maps for Google Earth (November 2, 2012)
Leif Karlsen's work on Viking Sunstones (August 23, 2012)
How to Print a Degree Symbol (June 11, 2012)
Robin Hood Complex in Book Publishing (March 22, 2012)