Saturday, September 26, 2009

Where did the Messerschmitt Jet Crash?

Bobby Shaftoe was walking into town, when he saw an experimental Luftwaffe jet aircraft coming across the Gulf of Bothnia (pages 564ff of Cryptonomicon). He estimated that it crossed the shoreline "a couple of miles north of Otto's cabin," which he had just left. Perhaps he based that estimate on knowing the particular clump of trees behind which it disappeared. He did not know either the size of the aircraft, or its speed, which would be the usual information employed in assisting such an estimate.

He counted seven seconds between seeing the fireball of the crash, and hearing the explosion. Using the 'five seconds per mile' algorithm, which he had learned in the Boy Scouts (that is where I learned it), the crash must have been about 1.4 miles from where he then stood. Already, there is a problem; the two estimates don't agree very well, because the airplane had kept going beyond that point on the shoreline.

Bobby walked three more kilometers into Norrsbruck, where he told Günter Bischoff what he had seen (page 583). Bobby specified the distance as "... seven kilometers from where I was standing. So, ten clicks from here."

There are several problems with that statement. The numerical problem may be the most obvious. Just from knowing that a distance of three miles is approximately five kilometers, the metric version of the algorithm must be 'three seconds per kilometer'. The accepted value for the speed of sound (about 330 meters/second in dry air at zero degrees Celsius), means that this metric form of the algorithm is a much better approximation to reality, than is 'five seconds per mile'. Thus, either the time delay was 21 seconds, or the distance was 2.3 kilometers, or perhaps neither number was correct.

The visibility problem arises only if the time delay was indeed 21 seconds, so that the distance was properly 7 kilometers, or about 4.2 miles. I have never been to Sweden, but maps (e.g., ) show several rivers flowing into the Gulf of Bothnia, which implies erosion of valleys. I have lived many years in central Illinois (where Neal Stephenson has also lived) and central Kansas. Both are considered rather flat, and effectively treeless. I would not count on seeing the immediate fireball in either place, at a distance of seven kilometers, because the crash could have happened in a valley. The trees in Sweden would make it even less likely for the fireball to be visible. The ultimate column of smoke would certainly become visible, but it is hard to say just how long that would take.

This military slang usage of "click" for kilometer (better "klick"), was either an anachronism, or a separate creation which died with Bobby and Günter. I served in the U.S. Army during the Korean War, and I never heard that usage. It was widespread during the Vietnam War, and some dictionaries (e.g., ) suggest that it arose during the 1950s. I would guess that it was invented during joint training exercises, involving U.S. forces and other NATO forces. Based on numbers of countries, if not numbers of individual soldiers, the U.S. military was outvoted on the question of yards and miles, versus meters and kilometers. Everyone needed to agree on maps, road marches, and firing tables for artillery. (Even the British, who invented the yard and the mile, abandoned them in favor of metric units.) Perhaps "klick" started as a face-saving joke.


Tuesday, September 1, 2009

Location of Golgotha (Part 2)

Let us now consider how well Lieutenants Goto and Ninomiya might actually have done, in finding the latitude and longitude of the entrance to the Golgotha excavation, where gold was later stored. Stephenson described their method, on pages 789-792 of Cryptonomicon.

I don't know details of a Nipponese sextant of vintage 1940, but it was unlikely to have been better than modern instruments. The best (or at least, most expensive) sextants for amateur navigators, that I have found online, are the Cassens & Plath Horizon Ultra ( ) and the Tamaya Spica ( ). Their specifications are similar. The arc is stated to be accurate within 9 or 10 arc-seconds. The vernier on the drum of the tangent screw reads to 0.2 arc-minute. The aperture of the largest available telescope is 40 mm. Much less expensive sextants (made mostly of plastic) have similar specifications, except that the arc is typically accurate only to 30 arc-seconds.

The angular resolution of such a telescope is about 16.8 micro-radian, or 3.5 arc-second. Thus the precision of angular measurements is limited primarily by the vernier and the arc, both of which might contribute. With good technique, and information about the sun's declination and apparent radius from a nautical almanac, the astronomical latitude could be measured to about the nearest 15 arc seconds, or about 460 meters. This is better than one needs for navigation on the ocean surface. You can see much further than that over open water, and can correct your landfall.

I do happen to know about "a pretty good German watch" of vintage 1940, because I used to own one. Mine was actually made in about 1946, but it undoubtedly used a prewar design. It ticked 5 times per second, and the sweep-second hand jumped at every tick. However, I quickly realized that I couldn't do anything useful with the fifths of seconds. If I was looking at some event, and then looked at the watch, it was almost impossible to determine the time of the event, to any better than the nearest second.

It also wasn't good enough that the watch was "... zeroed against the radio transmission from Manila this morning, ..." If my watch went more than a few months from its last visit to the jeweler's shop, it would be gaining or losing several seconds per day. It was at least as important to determine its current 'rate of going', by checking the watch against a standard for several days in a row.

I have never personally tried 'shooting the sun' with a sextant, but I have read about the technique used at local noon. (I have read every story in the Hornblower series by C. S. Forester, even more often than I have read Stephenson's stories.) One turns slowly, following the sun in azimuth, while rotating the tangent screw to keep the bottom edge of the sun's reflected image in apparent contact with the horizon. The sun's angle of elevation increases as it approaches the celestial meridian, stops increasing exactly as it crosses the meridian, and thereafter decreases. I don't know how easy it is to see these stages, and to say "Mark!" to the timekeeper, exactly when the elevation is maximum.

I would guess that the very best that one can do, is to get the meridian crossing time within one second on the chronometer. [This would apply in the tropics (e.g., the Philippines), because the maximum elevation would be large, and the sun's elevation would change rapidly. At high latitudes, the maximum elevation and the rate of change would be smaller. It might not be possible to identify the time of the maximum, to closer than several seconds.] If the zero and the rate of going of the chronometer are both good enough, then the Greenwich time is also known within one second. The longitude can then be calculated to within 15 arc-seconds, using the information about the sun's right ascension from the almanac.

Unfortunately, Stephenson does not allow Goto and Ninomiya to use this good standard navigational technique, although he does not give an adequate description of what they must have done instead. He specifically states: "They reach it [the highest summit] at about two-thirty in the afternoon, ..." At that time (well after noon), the sun's azimuth and elevation are both changing continuously, and both must be measured, as nearly simultaneously as possible. The best choice would probably be to measure the elevation with the sextant, and the azimuth with the transit, using the compass needle in the transit. This might be dangerous for the observer using the transit, because transits may not have provision to protect the user's eyesight from the sunlight, as most sextants do.

Each observer would continuously track an edge of the sun, but there was no third person to be a timekeeper. One of the observers would say "Mark!" when his own observation was in good alignment, to tell the other observer to stop tracking. The one with the watch would then look away from his telescope, in order to read the time. I doubt that this could be done reliably to better than two seconds in time.

The time of the observation and the almanac identify the latitude and longitude of the sub-sun point, i.e., the point on Earth for which the sun is then at the zenith. The zenith angle, i.e., the complement of the elevation angle, defines the angular separation between the observation point and the sub-sun point. The azimuth angle runs from the observation point to the sub-sun point. One then solves the spherical triangle of those two points and the pole, in order to determine the coordinates of the point of observation.

The near-disaster, which Stephenson imposed upon his characters, was the necessity for using a magnetic azimuth. (If they had stayed on the summit over a night, they could have determined astronomical north by observing circum-polar stars.) The compass needle, between the brackets for the telescope of the transit, might be 12 cm long, or 6 cm from pivot to tip. A protractor of that radius could be divided into degrees, with marks about one mm apart. Even with a decimal vernier on the needle, an azimuth could be measured only to the nearest 6 arc-minutes. That uncertainty in solar azimuth would introduce an uncertainty in the position of the observation point of about 3 arc-minutes, mainly in latitude.

A generous reader could suggest that Stephenson had merely made a typographic error. He may have intended to say, "They reach it at about twelve-thirty, just before local apparent noon, and immediately wish they hadn't because the sun is beating almost straight down on top of them." Notice that this also makes the geometry of the sun's rays much better. Two hours later, the zenith angle of the sun would be about 30 degrees, or even more.

Normally, places in the tropics never bother with daylight saving time. However, the Nipponese armed forces typically maintained the standard time of Tokyo (GMT + 9 hours), and enforced it upon conquered areas ( ). The standard time of Manila is GMT + 8 hours, so that the watch was effectively keeping the equivalent of daylight saving time.

Stephenson carefully avoided mentioning the degrees and minutes for the location of Golgotha, and rarely mentioned dates in Cryptonomicon. However, the pile of gold bars (also on Luzon) was at about 122 degrees east longitude, so that the sun crosses the meridian there about 8 minutes before it crosses the center of time zone +8. Without knowing the date, we cannot guess how fast or slow sun time was, relative to mean time (the equation of time).

At any rate, with an appropriately chosen earlier arrival time, it would have been possible for Lieutenant Ninomiya to shoot the sun in the standard manner. However, the resulting precision of 15 arc-seconds, in both latitude and longitude, is insufficient to allow him to say, "I have the peak exactly -- ..."

There was a corresponding typographic error in another time specification. Stephenson may have intended to say, "At one-o'clock sharp, the enlisted man down in the tree begins to flash his mirror at them, a brilliant spark from a dark rug of jungle that is otherwise featureless." Unfortunately, the very next sentence includes another inadequacy: "Ninomiya centers his transit on the signal and takes down more figures."

What Ninomiya needs to determine at this time, is the displacement (distance and direction) from the summit to the tree near the entrance of Golgotha. A modern transit with a laser rangefinder can do that in a single operation. However, in 1944, there was no such thing as a laser. His transit could only measure a (magnetic) azimuth to the spark. To get a distance, he must do triangulation, using both ends of a baseline of measured length and azimuth. At each end, he must measure the angle between the spark and the marker, which defines the other end of the baseline. The soldier in the tree had to be instructed to keep flashing for a long enough period of time, so that the transit could be carried along the baseline and realigned for the second measurement.

Fortunately, the next sentence offers a possible way out of these difficulties: "In combination with various other data from maps, aerial photos, and the like, this should allow him to make an estimate of the main shaft's latitude and longitude." The mention of "aerial photos" is in the nature of a bad joke. Stephenson has gone to great length to make the point, that in this area of a multiple-canopied tropical forest, it is impossible to see through the foliage from either above or below. Radar imaging from aircraft would be ideal, penetrating the forest to show the ridges and stream valleys of the solid ground. However, that application of radar was a post-war development.

The last opportunity lies in the mention of "maps". We know that a large-scale map of Bundok Site existed, drawn on a linen bed sheet (pages 730-734). Because it was a reasonably accurate representation of the Site, it was probably generated from a well made map at smaller scale.

The history of mapping in the Philippines was intimately associated with the fact that the U.S. controlled the Philippines, from after the Spanish-American War until World War II, and for a few years afterward. (See a brief history by Joseph F. Dracup: .) The surveying and mapping was done with the advice of the U.S. Coast and Geodetic Survey, using the same type of instruments and techniques as had been employed to survey and map the continental U.S. (Not all of the triangulation surveys in the Philippines were done to the same high standard of accuracy as in the U.S.)

The data were presented using Luzon Datum 1911, which incorporated a reference ellipsoid with axes of exactly specified lengths, a single surface point whose latitude and longitude had exactly specified values, and an orientation specified by the azimuth to another point. I have not seen the specifications of Luzon Datum 1911, but it is easy to find the specifications of the analogous North American Datum 1927 ( e.g.: ). The Nipponese captured all of the data from these surveys in 1942, so that they could produce maps that were the equivalent of what the U.S. could produce.

I have not seen U.S. military maps of World War II, but I saw and used maps of Korea, for that war. Almost everything else used by the U.S. in the Korean War was essentially identical to that used in World War II, so I would guess that the maps were equivalent also. The largest-scale Korean maps were at 1:50,000 ( ). Each map quadrangle had its edges labeled by latitude (north and south) or longitude (east and west). A grid of latitude and longitude lines subdivided the interior of the map. The smallest things printed on the map (e.g., contour lines of elevation and the grid lines) were about 0.1 millimeter wide, which represents 5 meters on the ground.

Thus, if one makes the highly optimistic assumption, that every single landscape feature printed on the map is in the correct position within the width of those lines, then its latitude and longitude can be determined to a precision of 0.2 arc-seconds (6 meters), by interpolation between the grid lines. (Personally, I am not nearly that optimistic about the accuracy of any maps of that era.)

This happens to be just good enough, because Stephenson stated that the tenths digits were even, for both coordinates (page 1064). Thus they could have represented fifths of an arc-second, rather than tenths. Ninomiya would have looked for the bends in the Tojo River, as shown on the map, because the entrance was very close to the river. (Note that this means that the observations on the summit of Mount Calvary were totally unnecessary.)

The final consideration concerns styles in GPS receivers. In the present generation (2009), GPS receivers are typically aimed at urban explorers, who might want to find the nearest restaurant or night club, and who need to be told which way to turn at the next intersection. Some previous generations of GPS receivers were aimed at wilderness explorers. They might have wanted to record or relocate mineral deposits, or populations of particular biota, or farm cemeteries with ancestral graves, or sunken submarines, or gold bars.

Some of those receivers came with the capability of changing the datum, to be used to display the positions, at the user's choice. Thus, if Randy's receiver offered Luzon Datum 1911, he could indeed get to the same point, in the map grid, which Lieutenant Ninomiya had selected in 1944.

If everything else had gone well since 1911, Golgotha would be there.