Longitude: Newsletter Challenge (06/27/06)

With an accurate clock set to UT (GMT), you time the Sun passing it's highest point (zenith). Then use a calendar and the "equation of time" tables to correct for the 6 monthly cycle of variations in the time of the zenith (about + or - 14 minutes). Assuming a 24h clock, your longitude is:
15 x (12 - corrected decimal time in hours) degrees east.
Accuracy depends only on the clock and your error in timing the zenith, of course. (Interesting question: how is that done?)

A few observations -

The sun reaches its greatest elevation at local noon, and unless you are between the Tropics of Cancer & Capricorn, it can not be at your zenith (the point directly overhead). Reaching its greatest elevation is refered to in celestial navigation as a transit, where a body crosses your meridian of longitude.

The Equation of Time describes the differences between Mean Time and Solar Time, and is a 12 month cycle which is not symmetrical about any 6 month period. You can see a graphic representation of the EoT by Googling "analemma".

Timing a transit depends on taking a number of sights on the selected body and using the point where the body has its greatest elevation. The difficulty here is that the elevation changes very slowly at transit, so it helps to take sights before and after the transit and take the time midway between two sights of equal elevation, or plot them on graph paper.

These are some of the techniques used in celestial navigation, a subject much broader than the (relatively) simple timing of a transit.

I teach the United States Power Squadrons (www.usps.org) Navigator and Senior Navigator courses. Taking sights and reducing them to a positional fix is not particularly difficult, but be warned that it is a different technology and so has its peculiar jargon and underlying concepts. One site that is pretty good is: http://www.celestialnavigation.net/practice.html .

More celestial navigation sites can be found through Google.

Thanks for an insightful reply and valuable links! I believe the EoT is also not quite a 12-month cycle, but changes slightly from year to year. If so, what is the actual repetition period?

I think this is related to the precession cycles of the earth's rotation and orbit, but I don't know offhand what they are - just that they are long compared to my lifetime and calculated into the tables in the Nautical Almanac. That's good enough to locate my position based on how well I can take sights, not on data errors.

The Aztecs had this one taped....the precessional rate is 2160 years to transit each sign of the zodiac, so 12 x 2160 = 25,920 years for a full rotation.

wikipedia has a good explanation.

Yeah, I know about the 25Kyr primary precessional cycle, but I was just too lazy to look up its value. IF I ever go sailing on the oceans, I still won't worry about precession as much as trying to take a good run of sights from a rolling deck. I'm a stinkpotter and a river rat, and happy to know that there are only 3 directions on a river - upstream, downstream, and crossstream. 8^)

Exercises for the student: 1) What is the cyclical magnitude of the primary precessional cycle? 2) How much error does that contribute annualy? 3) What are the other major sources of cyclical changes in precession?

Short answer: If one sits in the same spot night after night and notices that stars and sun set or rise over a uniform surface, like the ocean, at slightly varying times from day to day (or at a particular elevation above the horizontal, which requires intimate knowledge of the shape and size of the earth), and an astronomer friend has been good enough to point out that at other latitudes the times of such events vary at different rates, and occur at slightly different times each day, which he and his astronomer friends have previously recorded and put in tables for convenient reference, and the person who has gotten all this figured out decides to sail around the world, for example, he would be able to tell, after determining his latitude by using his sextant pointed at the north star on a known calendar day of the year, be able to compare the time (use of a clock) of the setting or rising or passage through a fixed elevation of a relatively fixed (dosen't move much in a human lifetime) reference point (star) to determine where on earth one has traveled after a little tedious ciphering.

Long answer requires a book.

The time piece, when set to UCMT (zero degrees longitude) can be used at any other longitude to determine that longitude. Simply sight the sun at high noon, then read the time on the time piece. By using the difference, knowing that the circumference of the globe is segmented by 360 degrees and 24 hours, and taking that ratio, you can precisely determine the longitude you are at. Latitude has always been easy as you just sight the sun at high noon, at its azimuth, and correct for the tilt of the earth at that time of year. The invention of a suitable timepiece, able to withstand the gyrations of a sailing ship, and the harsh environmental conditions was a significant engineering marvel. Up to that time, there was little understanding on thermal coefficient of expansion of metals, which had to be studied and then dealt with. It resulted in the first bimetal parts so that thermal changes would be canceled out to maintain accuracy. Then there was the attempted stealing of credit for the invention that occurred too. There was a great PBS program on this a few years ago.

The book was better than the PBS show. Read LONGITUDE for this great story. The prize was offered after a military expedition was lost at sea returning from overseas because they didn't know their correct location - the same problem caused the Spanish armada to founder on the British shore. It only took Parliment about six years after the disaster to get around to offering the prize. The committee was run by the astronomers because it was thought that the prize would be won through that means. The clock solution was unexpected and brilliant. Harrison was an interesting person. His first clock was accurate enough to win the prize - but he refused the award at that time and went on to build three more clocks - each smaller and better than the one before. The final clock was a radical design departure from the previous designs and was the first accurate pocket watch!

One hour marks one twenty-fourth of a spin, or fifteen degrees. And so each hour's time difference between the ship and the starting point marks a progress of fifteen degrees of longitude to the east or west. Every day at sea, when the navigator resets his ship's clock to local noon when the sun reaches its highest point in the sky, and then consults the home-port clock, every hour's discrepancy between them translates into another fifteen degrees of longitude.

Those same fifteen degrees of longitude also correspond to a distance traveled. At the Equator, where the girth of the Earth is greatest, fifteen degrees stretch fully one thousand miles. North or south of that line, however, the mileage value of each degree decreases. One degree of longitude equals four minutes of time the world over, but in terms of distance, one degree shrinks from sixty-eight miles at the Equator to virtually nothing at the poles.

You don't have to use the sun's view of noon to determine longitude. You can use a sextant to measure the angle at which you are viewing the sun at the clock's idea of noon (which is a lot easier to be precise about). That and a compass to let you know which way you're facing gives you a both longitude and latitude.

Actually, if you're good at the math, which I don't care to be right now, you can do it any time of day with the sun or night with the appropriate stars. IF you have the accurate clock.

I'm afraid this won't work with the Sun only. You can be at any point on a semi-circle! Measuring the elevation of many stars by a sextant works if you have accurate star data. You are then where the semi-circles intersect.

This works with any body that you have data for. The Nautical Almanac lists the needed data for the Sun, the Moon, Venus, Mars, Jupiter, Saturn and the 57 "navigational" stars (those brighter than 2nd magnitude). Note that Polaris is NOT a navigational star as its declination is about 89 degrees, 17 minutes. This puts it about 3/4 of a degree from North, making it useful for determining latitude only.

Determining position by taking sights on any of these bodies is basically an exercise in finding the centers and radii of two Circular Lines Of Position (CLOP). The centers are at the Geographical Position (GP) of the bodies - the point on the surface of the earth directly between the earth's center and the body. The radii are determined by the elevation of the body above the true horizon. Time is needed to locate the GPs of the bodies when you took the sights.

Note that there will be two intersections of the CLOPs. One should be wildly wrong, and the other in the vicinity of your dead reckoning position. That's where you are!

Agreed, but not for a single body, as originally suggested by the starter of this sub-threat (#5911). Yes you could use a compass to find the direction of the Sun at any time, which together with it's elevation will give you position - but then you are about as accurate as your dead-reckoning, which depended on your compass! Or am I missing something?

That's basically correct. A single sight gives one Line of Position, which tell you you're somewhere on that line, but not exactly where. That with your DR track can be used to get (at best) an Estimated Position, which is basically a refined higher accuracy best guess.

If you have an accurate timepiece, you don't need a compass, though, as reducing the sight gives two pieces of data - what are called azimuth (direction to the body) and intercept (how far the LOP is from your DR or assumed position). This may be accurate enough for your needs say, if you're in the middle of the ocean and just want to avoid hazards, but even a running fix (two sights taken more than about an hour apart) will be more accurate. Unless the weather is bad, it's not real hard to take sights on two or three bodies within a half hour, and that will get you a fix within 1 to 5 miles (1.5 to 8 klicks) depending on your skill in taking the sights.

Taking sights may seem a bit far afield from the original question of why time is needed for telling longitude, but it's the direct result of having the time availabe, as it allows you to find the one thing sights are based on - the Geographic Position of the body when the sight is taken.

Today, it's all GPS, but if all else fails...
I half-remember a story about an airline pilot that still knew astro-nav, who guided a private pilot lost over some ocean to landfall by simply asking him about the relative positions of the Sun and the Moon.

Electronic navigation is a good thing - I use a GPS on our 30' x 8.5' houseboat on the Illinois River as my usual navigation tool, but I keep track of my position on the Illinois Waterway chart, too. A nearby (1/2 mile or less) lightning strike can damage any electronics, so it's always prudent to have an independent means of telling position.

There are a lot of incidents on ships (sailing and air) where having knowledge of emergency navigation can save your life. For aa basic tutorial, see Bowditch (The American Practical Navigator) Chapter 26. It can be downloaded from the National Geospatial-Intelligence Agency website at http://www.nga.mil/portal/site/maritime/ - select Publications on the left, then select The American Practical Navigator from the dropdown menu. One of the exercises in the USPS Navigator course is determining where an astronaut is at splashdown given what seems to be minimal information.

I am no acquaintant of this jargon, but there is one question bothering me. A clock definitely works for cross-meridians travel, but, practically how for cross-parallels travel?

Hmmmm.... A bit of history may help clarify things. Return with us to the days when Men were Men and Women were not sailors (cue the organ and fade out my voice).

When the European nations were the major economic force (and before accurate timepieces), the quickest (and thus most profitable) way across the Atlantic ocean involved Latitude Sailing. This takes advantage of the prevailing clockwise wind and current flow in the northern Atlantic. Keeping on a course of constant latitude can be done by keeping Polaris at a constant altitude at night, and keeping the altitude of the sun at the noon sight constant, steering due east or west by the compass when weather kept the sky hidden. A trip from London, England to the Virginia coast in the Colonies involved sailing offshore from London, turning south to the latitude of Virginia, and thence sailing west. On the return trip you would sail northeast from Virginia (look at a map to see why) to the latitude of London, then sail east to London. Reasonably quick, does not require a clock, and easy on the Captain and Navigator.

The problem is that Latitude Sailing does not give the shortest (read most profitable) route. That is a Great Circle, directly connecting your origin and destination. Great Circle Sailing, however, requires that you know your position fairly accurately as you are constantly changing your compass course. Not knowing this (particularly at night) is dangerous as England has lots of rocky shores, and if you miss your destination you can easily turn your ship into a shipwreck (decidedly not profitable).

This is where the knowledge of time enters the picture. Time is essential because of the way a sight is reduced to a line of position. You need to know what is called the Geographical Position of the body - this is the point on the surface of the Earth where a line between the body and the center of the Earth would fall. This point moves around the Earth 360 degrees per sideral day for stars (somewhat different for the Sun, Moon & planets). If you know the GP of a body when a sight is taken and its altitude above your horizon, which is a measure of how far away the GP is, you can use spherical geometry to find out exactly how far away the GP is, and in what direction. Look up this method as the Law Of Cosines. Two or more sights will give intersecting Lines Of Position, giving your position. Then you know what course to steer, and can follow the much shorter Great Circle course, making your investors happy.