Monday, July 12, 2010

Final Posting

Altogether, the previous contributions have about exhausted those topics in Stephenson's tetralogy, which had attracted my attention, while still satisfying the criteria given in my introduction. That is, they should be at an introductory level in sciences, mathematics, engineering, or technology. There are other topics which don't fit the level or the area, or for which I just wanted to add something, to go along with Stephenson's correct coverage.

As a first example, I wonder about the economics of the ship Minerva, which was inspired by the classic 'Dutch East Indiaman', as a heavily armed merchant ship. In particular, I cannot believe that Minerva's crew was just the right size at 105. That was the total number of people on board when Minerva arrived at Manila (page 700 of the Confusion), after her trip to Japan. I can think of at least one economic reason why the crew should be smaller than that. I can also think of other economic reasons, probably more cogent, why the crew should be larger than that. I doubt that I will ever know the true economics of long-distance oceanic trade, at the beginning of the eighteenth century.

A second example involves Minerva's visit to Japan, particularly the resonance between the oscillations of mercury in partially filled flasks, and the "characteristic waves" at the harbor entrance (pages 686-689 of the Confusion). Proper consideration of this phenomenon demands knowledge of hydrodynamics, which is certainly not a topic of introductory physics. We can start to see where problems may lie, by a more elementary analysis.

Enoch Root had first noted a resonance between the waves and the oil "lanthorn", hanging from the ceiling of his cabin. At a guess, the lantern may have been equivalent to a simple pendulum about 0.4 meter long, so that the natural frequency of it (and of the waves) was about 0.8 hertz. Enoch 'detuned' it by shortening its chain "a few inches".

A flask is described as "an egg of fired clay: .., stoppered at one end by a wooden bung." I have a mental image of it as resembling an American football. If the inside of each flask was indeed the same size as a football, it would hold about 60 kilograms of mercury, when full. The filling fraction was not specified, but 1/2 or 2/3 might be reasonable. Jack Shaftoe was able to hold a partially filled flask "at arm's length", "though it took the strength of both arms." It is difficult to know the physical capabilities of a former galley slave, then about a decade out of training. That is an awkward position, especially since he "was too tall to stand upright in the cabin." My guess that it would be effectively impossible to perform that feat, if the actual flask was as much as twenty percent larger in each dimension, than a football.

Thus the first hydrodynamics problem is: "Can a partially filled flask, of reasonable size and shape, have a fundamental mode at 0.8 hertz?" The second hydrodynamics problem is: "What must be the boundary conditions of the harbor shoreline and seabed, and of the wind across the water surface, so that a characteristic wave at 0.8 hertz is strongly excited?" Note that this is a higher frequency than the roll of the ship itself, which could also possibly be in resonance with a different characteristic wave.

The solution proposed by Enoch would indeed solve the problem of a dangerous resonance. When each flask was made "brim-full", it was effectively not an oscillating system at all. A much quicker and easier solution was also available. If each flask, as received, was 'stood on end' and wedged into that position with straw, it would still be an oscillatory system, but its natural frequency would be quite different from 0.8 hertz. Only if that frequency also matched a different characteristic wave at the harbor entrance, could a resonance occur.

My final comment is about the analemma, which is a plot of the 'equation of time' (difference between mean solar time and apparent solar time), versus declination of the sun, over the course of a year. Stephenson made it a sufficiently important topic in Anathem, that a multiple-exposure version of it appears on the cover or dust jacket of the book. In its own way, Anathem is as much about the history of science, albeit on a different planet, as are the books of his earlier tetralogy. I am such a technology freak, that I had always considered that the analemma depended on knowing mean time, as measured by a sufficiently good clock. This is because I was never exposed to a proper course in History of Astronomy, with the mathematics. I now know that Ptolemy understood the difference between mean solar time and apparent solar time in the second century AD, based on earlier observations. I have still not personally read his Almagest, wherein he apparently did not consider mean solar time to be very important.

Thus it is quite reasonable, that on the parallel planet Arbre of Anathem, an analemma (large enough to be visible from space) was laid out at the temple of Orithena, in about the year - 2850 (all dates from pages xiv-xvii). It, along with the temple, was buried by the volcanic eruption of -2621. The laws of dynamics, which allowed the construction of good clocks, were discovered sometime after - 500. (Note that the passage of 2350 years on Arbre, between the discovery of the analemma and the invention of clocks, is comparable to the passage of about 1550 years between the same events on Earth.) The volcanic ash which covered the analemma on Arbre, had been dug away fairly recently before the events of Anathem (+ 3689). Thus, the ancient analemma was about 6500 years old, when Fraa Erasmas saw its image (page 405), and the analemma itself (page 526).

The diagram of the analemma on page 405 is notable, because of its exact bilateral symmetry. This implies that the perihelion and aphelion exactly coincide with the solstices. (I use the terms appropriate for Earth, although we are considering the orbit of Arbre.) The analemma for Earth is only approximately symmetric, because those orbital points are about 12 degrees apart, rather than exactly coincident. The analemma on the cover/dust jacket is also bilaterally symmetric, but its creation, as multiple images, requires technology at the level of 20th-century Earth.

To me, the most interesting thing about the analemma, is that it changes its shape with time. This change is driven mainly by the 'precession of the equinoxes', which is the manifestation of the precession of Earth's axis of rotation, in response to tidal torques exerted on Earth by Sun and Moon. There is also a slower change of Earth's elliptical orbit, relative to the fixed stars, caused by the varying gravitational forces exerted on Earth by the other planets. I don't know of any 'fossil' analemma on Earth, as much as 500 years old. That is not a long enough time for the change of shape of the analemma to be grossly obvious.

However, the situation on Arbre is quite different, because of the 6500-year age of the original analemma. In fact, that age is almost exactly 1/4 of the length of the precession cycle on Earth. Earth and Arbre are such close homologues, according to Stephenson, that it should also be 1/4 of the precession cycle on Arbre. In that case, at the time of the story, the perihelion and aphelion should exactly coincide with the equinoxes. The analemma would have a very different symmetry. The two loops of the 'figure eight' would be identical, so that the analemma is invariant under a rotation by 180 degrees around its crossing point. However, the analemma would seem to 'lean over', rather than standing erect along the meridian.

This is undoubtedly more that most readers would want to know. Even I would have been quite content with a mere half sentence: "... , different than the analemma that Fraa Erasmas knew."

Well, that's it. I will consider other topics from Neal Stephenson's work to appear here, only if someone asks me to.