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Author
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Topic: Lunars and lunar distance
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David Burch
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posted April 07, 2004 05:47 PM
A "lunar" is a sextant sight that measures the diagonal distance between and edge of the moon and another celestial body. From this data one can use special sight reduction procedures to figure the GMT of the sight and from that your longitude. It is a 19th century technique for finding longitude without time that did not last very long, being soon replaced by accurate clocks.
The practice of this technique had more or less fallen off the face of the earth in about 1917 when Bowditch and other navigation references stopped prublishing the necessary tables and instructions on the procedure. It has been revitalized a few years ago primarily becuase of the work of Bruce Stark and his publication of a modern set of Lunar Distance Tables. With this in mind, as well as the value of the technique in emergency navigation, we included a computational solution to the problem in the StarPilot calculator and PC programs. With the latter products, once you have the sights in hand, it takes just seconds to find your GMT. Reduction by tables will do the job, and is more representative of the way it was originally done, but it is of course more work. These two independent methods should in principle agree even though the formalisms are rather different—providing one uses the same almanac data.
The main point is, the technique does indeed work and is a fascinating study. Accurate time can often be found this way to well within a minute, and with practice and good luck, the precision can be down to 10 or 20 seconds or still more rarely, even better. We cannot overemphasize, however, that the sights must be accurate, including precise index correction measurements(use the solar method, we describe elsewhere).
If you care to push your skills in cel nav to the limits, there is no better way.
We are in contact with numerous navigators who are actively doing lunars these days, either solving the sight reductions by hand using the Stark Tables, or digitally using the StarPilot software or calculator. It is a most wonderful exericse and certainly qualifies the navigator as master in cel nav, becuase the sights themselves require special care and practice.
We will use this location to post and discuss the subject and to share data if you care to. We have one example online already, which includes some further discussion of the technique.
From: Starpath, Seattle, WA
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David Burch
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posted April 07, 2004 06:15 PM
Here is one set of data taken on 01 March 2004 sent in by JH at DR = true position = 53° 12'N, 2° 22'W, taken from inner edge of the moon to Venus.
code:
GMT (hack) Measured Distance corrected for IC
19 02 00 74 44.0'
19 03 55 74 45.0'
19 06 00 74 46.5'
19 09 35 74 48.3'
19 11 25 74 50.0'
19 15 30 74 51.8'
Avergage chosen for the analysis =
19 08 04 74 47.6'
(not stated if numeric average or from plotting)
using Stark Tables he finds the actual sight time was 19h 09m 00s, or the hack GMT was 56s slow. Using corrected time, this implied 14' longitude error, or roughly 8 nmi at this Lat.
When we do a quick solution of the same data with the StarPilot we get 19:09:19, which is essentially in agreement — until we would have time to think on the data more.
thank you JH for sharing this data.
From: Starpath, Seattle, WA
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Wendel Brunner
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posted April 10, 2004 12:21 PM
Dear Starpath,
As a celestial navigation enthusiast (I suppose now in this century it is really just a hobby), I enjoy visiting your web site, and especially looking at the articles you post periodically on various topics. Your article on plastic sextants, for example, inspired me to pull out my old Davis Mark 15, and see if I could, with care, get some decent results out of it.
On a recent perusal of your site, I came across the short article on lunar distance ("Lunar Distance with StarPath Star Pilot") and downloaded the LD observation data to use as a practice problem. While the observations were good, I had difficulty following the logic of your data analysis. Your procedure is presumably valid, for the process eventually did converge on a solution, although only after a large number (eight) of what appeared to be complicated iterations. I would like to suggest below an alternative procedure for determining GMT and Longitude from lunar distance observations. This procedure is more straightforward, and converges generally in two iterations to a more accurate result.
Averaging the Multiple Lunar Distance Observations
I use the term "Observed Lunar Distance" (OLD) to describe the actual angular distance in the sky between the inner rims of the sun and the moon. OLD can be either measured with a sextant, or calculated from a known time and position by a process that is essentially the inverse of the familiar "clearing the lunar distance." I replotted your OLD data points, and drew the "best" straight line through them, much as you did in your example. I then threw out the worst point, and calculated the best linear regression line through the remaining points. By whatever means a regression line is selected, that line then represents the averaging process. Any single point on the regression line represents the average of all the data, and there is no further need to deal with multiple observations in the analysis. I selected the observation at GMT 23:24:00, OLD 51°43.6' as the data average, as most of the reasonable regression lines go pretty much through that point. (I also cheated, and did a little side calculation unfairly using your known GMT and Longitude. I found that this observation was very good, only off by 0.2' arc. This is better than the stated accuracy of my Astra sextant, and better than I can usually make real measurements.)
Calculating GMT and Longitude from Lunar Distance Observations
The basic procedure for calculating GMT starts off as you do in your example. Assume a GMT and a corresponding assumed Longitude. I use the same assumed time you do - 14 minutes fast on actual GMT - and the corresponding assumed Longitude of 3° 30' to the west. The Latitude 47°40.5', of course, we know from the meridian passage of some body (or else we peek at the GPS, but only for the Latitude!). The assumed Longitude comes from the intersection of the parallel of known Latitude with an LOP from a sight using the assumed GMT. The procedure then is as follows:
1) From the assumed position at the assumed time, calculate the corresponding OLD. This is the inverse process of clearing the lunar distance.
2) Compare the calculated OLD with the actual measured OLD. From the difference in these Lunar Distances, calculate the time correction to the assumed time. In this example, I use the rate of 106 seconds of time/minute of arc (106 sec/1') that you calculated to make this correction, so that the details of our calculations would be comparable. As you point out, the rate of lunar distance change with time can vary from 106sec/1 to 136sec/1'. The actual value used for this correction doesn't matter much; any value around 12°/day will work fine and converge to the correct answer. However, you must use a rate similar to one calculated from the Almanac; using the slope of the OLD observations regression line (in your example, 156sec/1' ) can converge very poorly. This procedure is analogous to comparing HC with HO, and using the difference to determine the intercept.
3) Apply the time correction to the assumed time, and make the corresponding correction to the assumed Longitude. The assumed Longitude, of course, is increased to the west by 1° for every four minutes increase in assumed time. This produces a new assumed time, and a corresponding new assumed position.
4) Go back to step #1. When the difference between the calculated OLD and observed OLD is less than the measurement error (I generally use 0.2'), you are through, and have found both GMT and the Longitude.
We work out the example below. The actual measured OLD we are using is 51° 43' 36". The corresponding assumed time (14 minutes fast on actual GMT) for that observation is 23:38:00. As there are some rounding errors, we carry out the calculations to the nearest second:
Time Longitude Calculated OLD
1) 23:38:00 125° 53' 54" 51° 51' 26"
The difference between calculated and measure OLD (D OLD) is 7' 50", leading to a time correction (D T) of 13' 52" and a corresponding longitude correction (D L) of 3° 27' 58" . Applying the time and longitude corrections and calculating the new OLD at the new time and longitude we get:
2) 23:24:08 122° 25' 56" 51° 43' 29"
This time the D OLD is 0' 7" , the corresponding D T is 12" , and the DL is 3'. Applying these corrections and calculating the new OLD, we get:
3) 23:24:20 122° 28' 56" 51° 43' 36"
At this point the calculated OLD is the same as the measured OLD, and the calculation is complete. We find a GMT 20 secs fast, and a corresponding longitude 5' to far west. The posted calculations on your site required 8 iterations, and found the GMT to within 29 secs. Somewhere in those posted calculations there developed a disconnect between the time and the longitude, as a 29 sec time error should correspond to a 7' longitude error, not 4' as suggested.
The method describe here converges very fast and accurately. The error here of 20 sec corresponds almost exactly to the observational error of 0.2 . If an assumed time with one hour error is used, with a corresponding longitude error of 15 degrees (up to 900 miles), this procedure still converges in two iterations to the same solution.
These calculations can be done with a Nautical Almanac and a simple scientific calculator. Indeed, I did it at sea that way once, and got a result about 7 miles from my actual position, but it took me a number of hours. I suppose one could use the six place log trig tables in the back of Bowdich too, but you would really have to want to know your longitude pretty badly to do it that way. I have programmed a pocket Casio calculator with the very accurate Meese algorithms for the sun and moon, along with routines to calculate the OLD and to carry out the iteration procedure described above. Once I enter the date, assumed time, assumed position, and observed OLD, the calculator returns the correct GMT and Longitude in 48 seconds.
I think you might consider putting this kind of description of lunar distance calculations on your web site, as it could encourage more of your readers (and customers) to actually try a lunar longitude determination. If you are interested, I could also send you an explanation you could adapt for your readers on how to calculate the OLD from an assumed time and position. After finding my longitude this way on a trip to Hawaii, I was inspired to include the following comments in a description I wrote of the history of lunars and this procedure:
There are certainly better ways to get a position than the lunar method described here, or for that matter, quicker ways to cross an ocean than to be blown about in a sailboat. Staring at the dials of a GPS tells precisely where you are, but a carefully worked out lunar does more-it connects us with our past, with the navigators of the oceans and the mind who puzzled out the heavens and the earth. The GPS tells position, but a lunar fix reminds you also where you've come from. As Joshua Slocum wrote in 1899, "The work of the Lunarian, although seldom practiced in these days of chronometers, is beautifully edifying and there is nothing in the realm of navigation that lifts one's heart up more in adoration."
Sincerely,
Wendel Brunner
Berkeley, California
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David Burch
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posted April 10, 2004 12:29 PM
Wendel, thanks very much for that detailed presentation. I will check the discrepancy you mention in our example. we look forward to hearing more of your work on lunars. --david
(in passing, i might mention to those who are new to lunars, that the reference to plastic sextants at the beginning of your note is a different topic. As a rule, one cannot due lunars to any usable accuracy with a plastic sextant.)
From: Starpath, Seattle, WA
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David Burch
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posted April 29, 2004 05:12 PM
Great New Article by Bruce Stark
If you do lunars then you likely aleady know about the Navigation Foundation, but if not, I strongly recommend it. Latest edition has a new, in-depth article by Bruce Stark, who has already done so much to revitalize this lost art. Just more fine work from him.
Contact:
The Foundation for the Promotion of the Art of Navigation
PO Box 1126
Rockville, MD 20850
--david
From: Starpath, Seattle, WA
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David Burch
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posted May 24, 2004 07:48 PM
Note on IC for lunars
We just got back from our Inside Passage training trip to Alaska and during the trip did a set of lunars underway. worked out real well, and i will post the data as soon as we get it typed up.
In the meantime, i learned something i want to share... i am surprized it never came up before.
For doing lunars one needs the best possible IC measurment and for this the solar method we describe elsewhere online and in the course materials is the best method. HOWEVER, i just realized an obvious fact — if your sextant has a side error in it you cannot use this method. Generally a side error does not affect sextant accuracy very much, if at all, if you rock properly, and a small side error could even be a bit of an asset when doing conventional IC checks with the horizon or using a star.
But for the solar method (we sometimes call Lewis and clark method) this will not work... and this is really the method that should be used.
hence, first step in preparing for lunars is get rid of all side error in your sextant.... then adjust the indxe error as best you can and then apply the solar method to see what the error really is.
more later on procedures for checking and removing side errors.
--david
From: Starpath, Seattle, WA
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Wendel Brunner
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posted May 30, 2005 10:04 PM
I worked up these moon – Venus distance sights posted by JH above using data from the Nautical Almanac (or actually, the U. S. Naval Observatory web site) to calculate the Hc’s and azimuths for the moon and Venus at 19:08:04 GMT from N 53° 12¢ , W 2° 22¢. Correcting for refraction and parallax to get the apparent, or observed, altitudes, I get 74° 46.8¢ for the lunar distance. This is 0.8¢ less than the observed value reported. I ignored any correction for the oblateness of the earth or for the 0.2¢ semi-diameter of Venus. I find that the hack watch is slow 1 minute and 46 seconds, and the actual position is 26.5¢ to the west. The calculated GMT here is 19:09:50, and the calculated position is N 53° 12¢, W 2° 48.5¢.
Lunar distance measurements and calculations are difficult, and it is hard to sort out errors from observation and from different calculation methods. I sure would like to know what the watch error actually was. I think it would be most helpful when posting lunar distance observations for practice to include the actual GPS position and actual GMT of the sights.
Wendel Brunner
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