The term "celestial navigation" (sometimes called "astronavigation") typically evokes in our minds an image of a sextant and comes along with terms like Universal Time, line of position, nautical almanac, intercept, etc. Indeed, those are among the topics discussed in our spreadsheet project so far:

http://www.navigation-spreadsheets.com/

With these concepts, techniques, and gadgets you can establish your position from astronomical observations.  You can also keep track of your changing position during a trip, for example by bringing the method of dead reckoning into consideration.  Until now, however, an important aspect of navigation (which is relevant whether you use celestial or not) has not been addressed by our suite: trip planning, also known as sailings calculations.

We are pleased to announce that we now provide this additional capability in the latest extension to our suite.  Determining the direction (course) in which to sail (and knowing in advance the length of the journey) is an essential skill that any navigator must have.  For short trips one may directly measure the constant rhumb-line course on a Mercator chart for the path that connects the point of departure with the destination.  However, the bigger the separation between departure and destination, the more extra distance is associated with the rhumb-line path compared to the shorter great-circle path, especially in higher latitudes.  The problem is that attempting to follow the requisite great circle is very difficult since it requires a continuous adjustment of heading.

Thus, on the one hand, the great-circle route (orthodrome) is shortest but it is difficult to steer.  On the other hand, the rhumb-line route (loxodrome) can be well followed along its constant course but it is longer.  Each choice thus has a strength that is a weakness in the other one.  A solution to this dilemma is outlined, for example, in Bowditch which recommends a hybrid path combining the advantages of the two sailing possibilities.  Here the starting point in developing the sailing plan is the shorter great-circle route from departure to destination, but then along that path one identifies waypoints (separated, for example, by 5 degrees of longitude) between which the vessel is to follow easier-to-steer rhumb lines.

We illustrate such a calculation using as an example a trip from San Francisco (USA) to Yokohama (Japan) borrowed from Bowditch.  This classic publication demonstrates the idea graphically using the chart of the North Pacific Ocean shown in two different projections.

First, the great-circle path from San Francisco to Yokohama is found as the straight line connecting the two cities on the gnomonic projection chart.  We mark the waypoints as this path crosses meridians separated by 5 degrees of longitude.  Second, these waypoints are translated onto the Mercator projection chart on which they are connected by straight-line segments representing rhumb lines.  The constant course headings within each successive pair of waypoints can be directly measured on this chart.  Our new spreadsheets perform this exact same function (plus the distance calculations) with even higher accuracy, because they are not affected by the inaccuracies of physical plotting on a chart.

The problem to solve is fully specified by the coordinates of the departure and destination locations.  We have:

San Francisco:
Lat: N 37d 48.0'
Lon: W 122d 33.0'

Yokohama:
Lat: N 34d 42.0'
Lon: E 140d 06.0'

These coordinates enter the spreadsheet sailings.xls in row 2.


The use of this spreadsheet is shown in this YouTube demo video.

The differences between the calculated great-circle and rhumb-line paths are substantial.  The rhumb line is longer by over 200 miles and its (constant) heading is south of west, while the initial great-circle course is north of west sailing into higher latitudes first.  Row 11 displays the coordinates of the vertex, which is the point along the great circle closest to the Pole.  The spreadsheet also shows the (relatively minor) differences arising from the use of a perfectly spherical or slightly flattened ellipsoidal model of the Earth surface in the calculation.

For our purposes the main result of this spreadsheet is the initial great-circle course displayed in the yellow cell C6.  This number, combined with the departure coordinates, completely defines the great circle.  The value is shown with three decimal digits (302.240) in order to cut down numerical round-off errors once it is copied as input into the next spreadsheet: waypoints.xls.


Here, the coordinates of San Francisco and this initial course from sailings.xls enter in row 2.  Then, in column A starting in row 11 we begin entering the longitudes of each waypoint.  All great circles (except those running across both Poles) intersect every meridian exactly once.  The calculated latitude of each waypoint is displayed in columns C, D, E.  The rhumb-line distance and course from the previous waypoint (using the flattened Earth model) is shown in columns F and G.

The initial course is north of west so the latitudes of the subsequent waypoints increase at first.  The courses between them, however, progressively turn away from north and at longitude W 170d (close to the vertex, row 7) it is essentially due west.  The course then heads ever more southward as the path descends back to lower latitudes toward Yokohama.  Figure 2404 in Bowditch shows the E 150d waypoint at latitude N 40d, for which waypoints.xls calculates N 39d 45.5'.  The last waypoint is the destination itself (Lon: E 140d 06.0') with the correctly reproduced latitude (N 34d 42.0').  This hybrid path is still longer (cell F2) than the pure great-circle route, but not by much, and it is easier to steer.

Here is some info pertaining to today's lunar eclipse.  You can see that the Earth is indeed right between the Sun and the Moon, and that the eclipse will be nicely visible (weather permitting!) from western United States.

1) On the Sun-Earth-Moon positional arrangement:
a) the Sun and Moon declinations are very close to being equal in value and opposite in sign/hemisphere, and,
b) their Greenwich Hour Angles (GHA) differ by 180 degrees, which places the two bodies on opposing meridians.

The Moon is of course 15 days "old" and its phase is "full" (100% of the disc illuminated).

2) On the visibility of the eclipse:
The Moon subpoint (a.k.a. GP = geographical position) is:
Latitude (from declination): N 23 degrees 44.9' (very close to Tropic of Cancer)
Longitude (from GHA): 124 degrees west of Greenwich
This location in the Pacific Ocean rather close to Baja California makes this eclipse visible from our area.

The Sun is essentially on the Tropic of Capricorn (declination S 23 degrees 26.2') making this a rare event when an eclipse coincides with a solstice.

The time given is Greenwich time (Universal Time, UT) which is 8 hours ahead of our own Pacific Standard Time.







LATER UPDATE: A picture taken by a friend of this blog:

The data for the following example are from John Karl's "Celestial navigation in the GPS age" (First Edition, 2007), pp. 63-65.  With three spreadsheets we reproduce the computations for the sight reduction of the Moon observation presented in Figure 6.4 on p. 64.  Given the dead-reckoning (DR) position and the recorded time of observation (UT), the Moon sextant altitude (lower-limb) is reduced to intercept distance and azimuth needed to plot the associated line of position (LOP) according to the intercept method of Marcq St. Hilaire.

Input data:
Body: Moon, lower limb
Date: 11 May 2005
UT: 02^h 19^m 14^s
Hs: 25º 21.6'
Sextant index correction: -3.3'
Height of eye: 9 feet
assuming standard atmospheric conditions
DR position: S 34º 13', E 161º 43'

---

moon.xls:

Input:
Date: 11 May 2005
UT: 02^h 19^m 14^s
In cells A2-F2 entered: 2005 5 11 2 19 14

Output:
GP (row 5)
GHA: 182º 25.4' (same as 181º 85.4' in the book)
Dec: N 27º 42.9' (difference of only 0.1')

Semidiameter (SD, cell A8): 15.0'
Horizontal parallax (HP, cell C8): 55.1'

Note that there is no need for increments and corrections (v or d Corrⁿ).




This "almanac" spreadsheet for Moon is available here for a free download.

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alt_corr.xls:

Input:
Height of eye (cells E1, F1): 9 ft
Using standard conditions for temperature and pressure:
Pressure (cell E2): 1010 mb
Temperature (cells E3, F3): 10 degrees Celsius
HP (in arcminutes from moon.xls cell C8): 55.1 entered in cell E6

Sextant altitude (Hs): 25º 21.6', entered as 25 216/600 in cell B1
Index correction: -3.3', entered as - 33/600 in cell B2
Artificial horizon was not used: entered N in cell B4
SD (in degrees from moon.xls cell A8): entered (+)15/60 (positive for lower limb observations) in cell B11

Output:
Observed altitude (Ho) in cells B12-B14: 26º 18.2' (difference of only 0.1')

Note the intermediate result for apparent altitude (Ha) in cells B6-B8: 25º 15.4' (same as in the book)

---

intercept.xls

Input:
Assumed position (AP) is taken to be the dead-reckoning (DR) position:
Latitude: S 34º 13', entered as -34 13/60 in cell A2
Longitude: E 161º 43', entered as (+) 161 43/60 in cell B2

GP of Moon from row 5 of moon.xls:
GHA: 182 254/600 entered in cell C2
Dec: (+) 27 429/600 entered in cell D2

Ho from alt_corr.xls cells B13, B14: 26 182/600 entered in cell E2

Intermediate results:
LHA (cells F2, F3): 344º 08.4' (same as in the book)
Computed altitude (Hc in cells A6-C6): 26º 16.4' (difference of only 0.1')

Main results:
Intercept (cells D6, E6): 1.8 Toward (difference of only 0.2')
Azimuth (cell F6): 15.7 agrees with 16 from the book after rounding

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worksheet.pdf:

The data from this example are summarized in the worksheet below; input data are marked as green, and spreadsheet results (some of which are relayed as input for subsequent calculations) are blue.

The web browsing that I had done during this process did not lead me to any explicit solutions (such as finding someone offering a complete suite of Excel spreadsheets for celestial navigation computations…).  I did find, however, a discussion list of like-minded people that I promptly joined.  This "NavList" has been and continues to be a great source of wisdom.  Recently a very good overview of NavList was published here:

http://columbianewsservice.com/2010/03/they-sail-their-ship-alone/

What is especially valuable is that there are number of members who are accomplished sailors, scientists, and sometimes both, who are a treasure trove of knowledge and experience.  On a number of occasions they presented real-life data they themselves took while sailing, which enabled me to successfully verify the performance of the suite.  In fact, recently one gentleman serving on a merchant ship was in the Indian Ocean taking sights with his sextant and has put my noon spreadsheets to work.

Through NavList I came to learn about John Karl, a physicist, artist, and sailor whose excellent book "Celestial Navigation in the GPS Age" I subsequently obtained.  Finally I had a reference whose style was more in sync with my own thinking; in fact, some of the things I had already done with Excel were even suggested in the book as exercises for the reader!   But the most valuable piece of information in this book for me was a reference to another book titled "Astronomical Algorithms" by the Belgian astronomer Jean Meeus.

The spreadsheets that I had developed up to that point had to do with sight-reduction, that is calculating one's position by solving the relevant geometry on the surface of the Earth.  This however was not a self-contained suite yet, as it required astronomical data as input.  This information, most notably the positions of the observed celestial bodies at the moment of measurement, is precomputed in advance and published in almanacs.  Meeus's book clearly explained how this can be done and even provided analytic formulas that describe our solar system.  The word analytic (as opposed to numeric) is important because this way it is possible to describe the planetary orbits with formulae of very broad validity rather than tables whose size scales with the chosen time period and which would require search and interpolation procedures.  Furthermore, the accuracy of these equations was way better than what is realistically required (and achievable) for celestial navigation purposes.

And thus the idea came whether it would be possible to complete the suite by encoding these formulae into Excel as well.  The formulae were quite large, with lots of terms, and there were a number of steps to keep track of.  Therefore, I decided to first write a program in Fortran, a computer programming language with which I am well familiar.  After I ran the program and convinced myself that the results agree with the official Nautical Almanac, I encoded the same calculations into Excel using the now verified Fortran program as a guide.  This took a fair amount of time, several weeks in fact.  Some of the spreadsheets exceeded 1 MB in size due to the number of terms included.  But it worked and by the end of June 2009 we rolled out a major extension to the suite, now including these new "almanac spreadsheets."  I was pleasantly surprised to see that Excel was able to handle this kind of stuff.  Think about it, a software tool designed for accounting, inventory and whatnot, can be actually used to predict the positions of planets and stars with accuracy better than one minute of arc!  I just think that's awesome, in a freaky-geeky kind of way.

Excel also has the advantage that it (or its sufficiently compatible alternatives and competitors…) are available on pretty much any computing platform.  The spreadsheet app then serves as a kind of "virtual machine" in the spirit of Java.  Thus there is only one version of Navigation Spreadsheets, and it can be run on Windows, Mac, Linux, even on a smartphone, without modification.  On the website there are links to YouTube demonstration videos in which I used an iPod Touch.

So that's the summary of how this project came to be and what it is about.  I want to come back to give occasional updates, maybe give an example or two, and provide additional historical details about the suite from time to time.  Stay tuned.

And so the learning process began. I knew about latitude, longitude and a few other things but then came zenith distance, navigation triangles, assumed position, the intercept... All these were new terms with all kinds of formulae, rules, and procedures attached to them. I read about the noon sun and its special significance. At noon, the sun reaches its maximum altitude for that day and from the associated sextant measurement one can determine one's latitude via simple arithmetic. The slight problem with that is that marking the moment of noon precisely is a bit difficult in practice. Thus the next topic in my little textbook was the concept of the noon curve. Instead of making a single measurement, here one tracks the sun before noon as it rises, then after noon as it descends and extracts the peak of that curve. Having multiple measurements this is automatically more accurate than just trying to mark the moment and taking a single observation when you think you have noon.


At this point I figured that the noon curve method could be easily and rigorously implemented with Excel; after all it's just a quadratic (parabolic) fit, for which Excel is already well equipped. With that in mind I constructed the first spreadsheet, noon_curve.xls. This spreadsheet alone, however, did not quite lead me to "go public" with this project; at this point this was all still limited to my own amusement.


The idea of starting a website and making spreadsheets available to others came after some thinking about the position fix from two altitudes. The "industry standard" method of Marcq St. Hilaire known as the intercept method has proven its merits for over a century. However, what bugged me about it was its reliance on the so-called "assumed position" which was something that seemed semi-arbitrary and not very intuitively clear to me. I was certain that one should be able to calculate the crossings of the two circles of equal altitudes directly without assuming anything. Searching on the Internet revealed nothing on the subject, so I dug into the geometry of the problem, came up with a solution, which I then encoded into another spreadsheet named lops.xls (LOPs stands for two intersecting Lines Of Position marking the position fix.) I was quite sure that I was simply rediscovering something that was already known but I could not find it in any materials that I had at that moment. Now I thought that I had something other people might find useful and as a result I developed the first version of the Navigation Spreadsheets website. It was only fitting that my wife contributed to the website's design, after all this was all her fault! The site was launched on Thursday, February 19, 2009 and not too long after that first customers began to trickle in. (to be continued)

I suppose that the official beginning of this project could be assigned to Wednesday, November 12, 2008, late afternoon Pacific Standard Time. On that day my wife and I were visiting the town of Dana Point on the California coast, where she grew up. She showed me her childhood home, her high school, cool places to hang out. In the evening we drove down to the harbor. As we were walking on the beach, we encountered two fellows, one equipped with a sextant, who were observing the sunset. "Right on the money," one of them said. He was ... << MORE >>

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This blog accompanies the main site:
http://www.navigation-spreadsheets.com/

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