Lunar Eclipse and Overwrought Coincidences

Lunar eclipses are delightful to watch. I've witnessed the more awesome spectacle of a solar eclipse, but lunar eclipses win in the long run because they're more frequent, are visible to half the planet, last a long time and are pretty and safe for viewing.

This one is being aggressively promoted in the media as special because it coincides almost exactly with the solstice.  Since the advance of the lunar nodes (the azimuth of where the Moon crosses the ecliptic) is not in resonance with Earth's orbit around the sun, this particular coincidence is no more or less probable, or interesting, than an eclipse falling on any chosen calendar date. All this coincidence really tells us is that we can finally look forward to increasing hours of daylight and solar insolation (although the weather doesn't start warming up until February).

One coincidence that isn't a coincidence is that this eclipse occurs at a full moon. This "insight" comes from an interview with some new-agey sort of person that I heard on the radio. Well, duh! I suppose we can also add the, um, coincidence that we see the eclipse occuring at night. These facts are about as coincidental as a flipped coin that lands head's-up showing an impression of Her Royal Highness. It is also why this eclipse will not be visible from anywhere in Antarctica -- the explanation of which I'll leave as an easily-solved puzzle for the geometrically inclined.

Apart from the obvious things about lunar eclipses, there are a couple of items that are less widely considered. The first is that when we see a lunar eclipse, if someone is on the moon (anywhere on the hemisphere facing Earth) they will be simultaneously viewing a solar eclipse.

...the view from the moon during the eclipse, with the Earth in front of the sun, would be a spectacular red ring in the black sky.

It should be no surprise that the colour of the scattered sunlight from the Earth's atmosphere (from its visible circumference) is brownish-red (or copper) since that is the tint the moon takes on during the lunar eclipse. In similar fashion, the frequency of solar eclipses for Moon dwellers is the same as lunar eclipses for Earth dwellers. From the Moon, the apparent diameter of the Earth is quite a bit larger than the Moon appears from Earth and so is more likely to cover the sun.

However, there are no "terran" eclipses when the Earth is full (Moon between the Earth and sun) since the Moon's shadow never covers more than a small area on Earth's surface. That occurs when we on Earth see a solar eclipse.

The second interesting thing about lunar eclipses I want to mention refers back to an earlier post of mine that talked about why the full moon always passes high overhead during the winter months at higher latitudes, like here in Ottawa (also at high southern altitudes such as southern Chile); however that, too, is no coincidence. What this means is that the best lunar eclipses are those that occur around the winter solstice, just like the one this week, because the Moon will pass high overhead for optimal viewing.

Unfortunately this is also what I have always hated about lunar eclipses, because the best ones occur when you have to endure cold winter nights if you want to watch them properly. They're never quite so nice when seen through a window, which in any case will be difficult since in many houses the Moon will be so high as to be blocked by the eaves of the roof. In other words, to stay warm while watching this eclipse you'll probably have to deal with window-glass distortion and an uncomfortable viewing angle.

Despite having said all this, my plan (I am actually writing this the night before) is to stay warm and in bed and pass on the joy and the cold. Anticipation for this event by people here in Ottawa may be for naught in any case since it now looks as if clouds are going to spoil the event. Well, there's always the internet so I'm sure there will be lots of photos making the rounds on astronomy blogs Tuesday morning.

Autumnal Fading of the Insolation

The waning of summer has now been replaced by the rapid decline of the autumn as the sun passes the equinox. This is the time of year when the daily rate of change in sunlight -- the interval between sunrise and sunset -- is at its greatest. As we all know, as the days grow shorter and the sun gets lower in the sky the amount of energy we receive from the sun decreases. The why of it is pretty straight-forward although perhaps it is less clear just how great is the difference.

When I was a child and I first learned these facts I can clearly remember standing outside in the frigid winter air and looking at the sun hanging low in the south. It was cold but the sun seemed just as bright (or even brighter due to all the reflecting snow on the ground). I knew that the sun was no further away in the winter -- it's actually closer in the northern hemisphere winter -- and since I was facing it full on the energy I received should also be identical. This is in fact close to the truth, excepting the attenuation due to the longer path sunlight took through the atmosphere to reach my face.

The energy we receive from the sun by facing it, unfortunately, does not hold when you consider the wider expanse of the Earth's surface, both land and sea, which is what counts when it comes to weather and the seasons. At lower elevation angles, the energy is spread over a larger area, which directly lowers the flux. This is called solar insolation, and it is usually measured in watts per square meter, as power crossing a surface. Integrate by time and you get the energy, which matters when you have to take into account length of day.

This is all very qualitative and probably tells no one anything they don't already know. However I like to look into the details since it is intuitive that there are some interesting dynamics at play. For example, during the summer at higher latitudes the days are long, far longer than in the tropics, and the sun is high enough in the sky that the effect of sunlight spreading should be minimal. It should also be evident that at both equinoxes the total daily insolation should only depend on latitude, decreasing as one moves away from the equator, since daylight is 12 hours everywhere on the globe. In winter, the cooling effect due to latitude should be accentuated, and especially so in the arctic and antarctic which for a time has not daylight at all.

It is actually quite easy to do the calculation if you have a little bit of trigonometry at your fingertips. All you need to do is figure out when during the day the sun is in the sky and track its elevation between sunrise and sunset. If, like me, your trigonometry lessons are lost in the dim past you turn to the internet. If you do this you will quickly locate the equations for the sun's declination -- its angular displacement from the celestial equator -- and for the solar elevation angle. One then plugs these equations into a spreadsheet, enters a date and a latitude and spit out the numbers. Except that there are a couple of complications we need to deal with.

First, there is the matter of calculus since a proper calculation requires integrating the solar elevation angle over the time between sunrise and sunset. I took the easy way out and did a numerical integration with half-hour steps, which is a simple operation on a spreadsheet. The second is accounting for all the confounding factors if you want to achieve accuracy: the Earth's elliptical orbit, atmospheric scattering at various incidence angles, local terrain, and so forth. I opted for simplicity by discarding all these items and reduced the significant digits to two. The numerical integration adds its own inaccuracies but this is only a few percent.

With a tip of the hat to the prodigal spherical cow I proceeded to calculate. A summary of some interesting results is in the following table. Since watts and joules and all that stuff is not intuitive to most people I opted to normalize daily solar insolation where a value of 1.0 represents one hour of insolation with the sun standing still at the zenith. That is, directly overhead.

If you peruse the table you should notice some items of interest. One is that the loss of insolation in winter (it's at a minimum at the solstice, give or take a few days) is dramatic once you get to higher latitudes. I chose Iqaluit for my northernmost location since it's just south of the Arctic Circle. Edmonton gets only half the insolation of Ottawa, and Iqaluit gets only one-tenth of that, being only about 1.5% that Quito on the equator received on that same date.

The second interesting item is that at this same date Quito does not get its maximum insolation despite being on the equator. This is because the sun is in the northern sky by just enough to drop the insolation by about 10% from when it is at a maximum, both equinoxes. These are the two dates on which the sun crosses the zenith.

For me, the most striking result is the insolation at the summer solstice. All of these Canadian cities have a higher insolation than anywhere in the tropics. Even Iqaluit gets about the same amount as us southerners. This is of course due to the extra long summer days, when the sun barely gets below the horizon during the brief night, and compensates nicely for the lower solar elevation angle. More interesting is that the daily insolation gets even higher as you go north. This is because the daylight is 24 hours yet the sun is high enough in the sky to have half the intensity at the zenith even at the northern tip of Ellesmere Island, our most northerly point of land. But you have to time your visit to there just right because the warm sun doesn't last for long.

Only about 80 more days until we get our 2.0 zenith-hours of insolation here in Ottawa. I can hardly wait...until the day after.

Statistical Sampling and the Long Form Census

In the ongoing controversy over the federal government's intention to make the census long form optional there is an exceptionally high noise level over issues of governance, political interference in the civil service, tax burden, privacy, value of the long form to government and outside agencies, and much more. I will talk about none of that, and will thus casually sweep from the table all politically-related issues encompassed by the discussion. I do this so that I can focus on the technical aspects of the issue, in particular whether the long form census delivers reliable statistical data when it is mandatory, and when it is optional. The political questions are interesting of course, but these are not my focus in this post.

Stats Can has three broad options on how to go about the long form census:

1. Mandatory
2. Optional
3. Scrap it entirely

From here it is possible to formulate a few simple questions about the entire census exercise:

• Is the mandatory census, as conducted in 2006 and earlier, delivering valuable data? That is, are the results sufficiently reliable to be useful?
• To what extent does making it optional lower the reliability, and will its results meet the test of being sufficiently reliable to be useful?
• Are there good alternatives to the long form which deliver similar results with acceptably high reliability?

If the answer to the first question is negative, the utility of the long form census is highly questionable, raising the possibility of scrapping it entirely. In the case of the second question (assuming that the answer to the first question is positive), it is reasonable to assume that the reliability will be lower (based solely on sampling theory), but we must now determine just how much less reliable it is to determine whether it is worth doing at all. The third question is particularly interesting since some countries do manage to do this pretty well without a long form, however the manner in which it is accomplished is important.

Let's start with the current long form and its reliability. There are two questions to be considered: is the sampling done in a way to meet the target statistical confidence in the conclusions derived therefrom, and; is the data collected accurate? If this question interests you, I suggest you read this article in the National Post since it gives a flavour of the problem of collecting high-quality data before continuing.

As the article shows, whenever you involve humans in any survey there is an issue of quality. It is important to understand that this is entirely independent of the sample size and selection methodology. Humans can lie, be negligent, or simply be wrong no matter what they are being asked. The problem can be managed in part by making the census form easy to use, easy and quick for the responder to get right, and avoiding hot button issues that make people uncomfortable (e.g. tell us about your undiscovered criminal acts and your favourite kinky sex acts). The question about natural gas spending is a good example of a question that will often produce poor data.

The same difficulty is perhaps more obvious in political polls that ask which way you'd vote if an election were held today. These are carefully reported complete with statistical confidence levels, nationally, provincially, and by other responder characteristics (e.g. age and sex). This is all very mathematically rigourous but still totally wrong. The methodology used to calculate the confidence levels (e.g. plus or minus 3.8%, 19 times out of 20) assumes that responders are equivalent to coloured balls in a bag that are randomly drawn; if you have a bag of 100 balls and you draw 10 balls at "random" and you get 5 white and 5 red ones, there is a tried and true mathematical process to say something rigourous about the entire population of 100 balls. Not so with people. Some misunderstand the question, say whatever they believe the questioner wants to hear, lies in an attempt to skew the poll, or would never vote anyway.

Remember, we're still talking about a mandatory long form census. Stats Can spends a lot of effort to make the sample (who is sent the long form) both sufficiently random yet suitable to the types of analysis to which the collected data will be put, and it is not enough. Even the force of law to ensure that the sample integrity is assured isn't good enough. Stats Can knows this and they will apply many supplemental filters to the data in an attempt to tidy it up a bit, if their assumptions about how the data is skewed by human and other factors are reasonably close to the mark. However, they can never be certain.

If we now degrade the data by making the long form census optional, it becomes even more difficult to filter the data. As Minister Clement has stated, an expected reduction in the response rate will be compensated by increasing the sample size. That's nice, but it doesn't help Stats Can all that much. The thing is there is no good way to know who is opting out and why. There is an unknowable selection bias. Even if they know the location of each responder (this requirement is apparently also in doubt, in addition to who does or does not respond) the additional unknown selection bias degrades any analysis of the collected data. Are people's reasons political, privacy related, education or comprehension, availability, or other factors? Stats Can can't know. They may also not have the ability to compare the 2011 and 2006 samples to determine what new filters to apply to the data.

At this point there is some question as to whether the long form census should be done at all if the data and its analysis are degraded by some unknowable amount, especially in advance of the census being conducted. If the quality is deemed likely to be too low for its intended purposes, both by government and other organizations at large, it may be appropriate to scrap it entirely. The cost of an optional long form may even turn out to be higher than a mandatory form: more forms sent out and potentially processed, plus more filter analysis to recover some of the lost data quality, countered by lower costs of enforcing compliance.

If the results of the long form census are still deemed to be important even though it is untenable due the loss of quality, there are alternatives, but perhaps not ones that are doable. Some countries maintain databases that integrate all the data that all government departments know about the country's citizens. This super-database can be good enough to make a long form census superfluous. Except, and (in my opinion) fortunately, this isn't done in Canada. We have set some pretty strong barriers between government departments and agencies which make it very difficult to share and integrate data across those barriers. As an example, reporting income to CRA that was obtained in, shall we say, peculiar means, does not end up in the hands of the RCMP or other police forces. CRA can't even give your identity to Elections Canada without your consent. These barriers are actual respected and so can generally be trusted, which encourages more honestly when dealing with branches of the government. It is highly unlikely that Stats Can's needs would supersede the necessity and enforcement of those barriers. In other words, super-databases are not in the cards, and for which I am most thankful.

I don't know how this will eventually shake out since it has become a political hot potato that has already resulted in the resignation of the head of Stats Can (presumably regarding a perception of political interference) and is generating a surprisingly heated debate across the country. In closing I will note that our Mayor, Larry O'Brien, says that the City of Ottawa will if necessary replicate what the long form census produces by some means. It will be amusing to see how he proposes to do this since, as a minimum, the City does not have the funding to do a local "census" and does not have the federal government's power to force compliance in responding to any questionnaire the city sends to a sample of residents.

What a weird issue this is that has captured so much of the country's attention. Who would have thought it.

Mercury and Venus

Back in January I mentioned that Venus was about to become prominent in the western sky after sunset. Ten weeks later it is out there, shining brightly. While that's a pretty sight, it's even better for the next week or more since it has been joined by Mercury, it's smaller but still quite bright cousin.

With the skies so clear now, and probably Friday and Saturday as well, not to mention the unseasonably warm evening air, if you have never seen Mercury before, this is an ideal opportunity. Venus is the bright beacon you simply can't miss, and Mercury is easily found by looking a few degrees to the right of Venus and down just a bit. If you extend your arm with your palm facing the sky, the width of your hand is approximately the distance between the two planets. Mercury isn't as bright as Venus, but unmistakable once you know where to look.

Here in Ottawa I would say the best time to see them is between 8:00 and 8:30, when the sky is dark enough to make both easily visible and still far enough above the horizon that low obstructions won't be in the way.

I took a picture of them with a camera phone Thursday evening a little after 8 PM. Venus comes through ok, but since Mercury was a barely brightened pixel even with an image-manipulation program, I circled its position so you can get a good idea where it is relative to Venus. Keep in mind that their position and separation will change each night. Mercury will reach its highest point in the night sky next weekend (April 10), and will then quickly swing in between us and sun to become invisible once more.

Evening Star

For casual sky watchers, today is noteworthy since Venus now sets after sunset and so becomes an evening star. It will be in this favourable viewing position until October. Because Venus is the 3rd-brightest celestial object in the sky (the Moon and the Sun are brighter), it is very noticable no matter where you live, even in brightly lit urban centres.

Venus was actually in superior conjunction this past Monday (January 11), when it was opposite the Sun as seen from the Earth. Since it passed slightly south of the Sun, it continued to set before the Sun at the latitude of OttawaNepean for a few more days (see diagram). By the end of January it will have moved far enough eastward to be apparent to anyone looking to the southwest in the early evening. A good time to see Venus for the rest of the winter will be while you're commuting westward after work.

Since the ecliptic is tilted steeply upward in the west during early winter evening, Venus will climb in altitude above the horizon very quickly as the days go by. This increases its visibility so that it is very easy to for everyone to appreciate without going to any special effort. Enjoy the view.

Unfairly Tossing a Fair Coin

I had a good laugh while reading this CBC article about teaching students how to cheat at coin tossing. The article does not relate the ways by which this can be done, however the techniques have been well-understood and used or abused for a very long time. If you are unfamiliar with the "tricks" involved, it may be difficult to picture how it could be done; that is, how the tosser can cheat without rousing the suspicion of observers.

In a traditional coin toss the coin is launched from a flat position with the spin imparted by an off-axis force from (usually) the thumb. A coin flipped in this way has two superposed motions: a gravitationally-determined parabolic arc and a rotation about an axis that contains a diameter of the coin. The result is a coin spinning rapidly end-over-end and travelling through the air in an arc. Since the spin about the coin's diameter is many times the rate of the (roughly-speaking) half-revolution about an invisible centre -- usually near the parabola's focus -- and both rates are not precisely controlled, the outcome is sufficiently uncertain that the outcome of the toss -- heads or tails -- can be said to be random. It isn't entirely random: it is possible that a tosser with a sufficiently fast eye and hand could interrupt the coin's flight at just the right moment to reach a desired outcome. However it is good enough to be an uncontroversial way to decide the kickoff in a football game.

That's was a simple toss, but there are other ways to spin a coin. Further, these various motions can be combined to create some complex, seemingly-random spins that are often easier to manipulate. If you want to find out all the ways to spin a coin (and cheat) I recommend you read the description provided by the physicist and statistician E. T. Jaynes. He went so far as to train himself in the various techniques and then test the results. From the raw data of many, many hundred of trial runs, he became very accomplished at achieving almost certain outcomes with tossed coins. The book where his description can be found is called Probability Theory: The Logic of Science, which you may want to avoid purchasing since it is heavy, large, expensive and highly technical. Happily, early drafts of the text are available online. One place is here (browse down to chapter 10) provided you have a Postscript viewer handy. If not, do a web search and you will find versions in other formats, though you may have to hunt a bit in each one since there is no consistency in the work's structure and content across its many versions. You'll also need a good imagination since Jaynes provides no diagrams, just precise descriptions.

If you don't want to do all that work, let's review a few of the spin types that a fair coin can exhibit. The most trivial is where the spin axis goes through the coin's centre and is orthogonal to the coin's surfaces; in other words, the spin axis is at a right angle to that in a traditional coin toss. No matter how much the coin spins, the result of the "toss" is trivially identical to the face that is showing at the start.

Next, we can launch a coin spinning on its orthogonal axis along the same parabolic arc as in the traditional coin toss. However, if no additional spin component is employed, this form of travel will not affect the outcome: it will still show the same face as if the coin were not launched. You can try this by placing a coin in your palm and accelerating it upward without changing the attitude of your palm. You could also do it with your thumb but it is more difficult to centre the force so that the coin does not tumble. In either case, this is difficult to do while also imparting spin around the orthogonal axis.

A slight variation is to launch the coin so that the same face of the coin points inward (along the coin's orthogonal axis, and approximately toward the parabola's focus). This is much like the tidally-locked orbit of the Moon around the Earth where the same face (but not quite due to libration) always faces the Earth. This is nearly as easy as a flat toss, but this time the outcome of the coin toss will be opposite to the face shown at the start.

Now it gets more complicated. Consider a spin axis that is intermediate between the two preceding cases of the orthogonal and diameter axes. The resulting motion is a wobble that is much like you get by spinning a plate on a table if you launch it an angle to the surface (where the plate touches the table at only one point). You do this with a coin by centering the force imparted by your thumb off to one side of the coin while simultaneously pulling your thumb towards yourself. You may have to brace the coin with another finger so it doesn't slide off your thumb, or you can direct your thumb's motion at an angle that is off the vertical. It can be learned with only a little practice.

The wobble amplitude (the angular difference between the highest and lowest excursions of the rim at any fixed point at the coin's edge) is important to creating the impression of a fair coin toss; to an untrained eye a large wobble is hard to distinguish from a spin about the diameter axis, if the spin rate is high enough. However if you think about it for a moment I think you'll see that the outcome of the toss is exactly the same as the original, trivial case where the spin axis is orthogonal (no wobble, and the top face doesn't change). Combine this with a flat or tidally-locked arc and you can get any outcome you desire. According to Jaynes, depending on the resiliency of the landing surface you can often let the coin land on the floor without greatly reducing the predictability of the outcome. This tactic also has the advantage of allaying the doubts of observers about the toss's fairness. It is of course better when the surface is soft enough that the coin doesn't bounce, such as a natural-turf football field.

There is more to it if you want to get into even more arcane combinations of spin axes that give highly deterministic outcomes while appearing to have random, unpredictable flights. Jaynes pretty much goes through all the possibilities in his book so I'll leave off here and let the curious read what he has to say on the topic.

A House Stands On Its Own

Imagine that you commission a house to be built for you and your family. All seems to go well during its construction. Every inspector has finally given the nod that all is well, and that the routine problems that crop up in any custom job have been addressed. The contractor is about to hand you the keys and collect your cheque for the final invoice when you learn something disturbing. Many of the trades people who built your house were not model citizens: some were murderers, swindlers of the elderly, child molesters and, yes, even washed-up politicians. What are you going to do?

If the inspectors have done their jobs, the house will stand. It will not suddenly disintegrate or have a lower value because the character of one or more of the workers is called into question. Regardless of character, even if one or a few workers were less than capable at their tasks -- say, attempting to hammer in a nail using a stapler -- presumably this was caught by the contractor or an inspector and the incompetent person was forthwith escorted from the job site. The work was inspected and the house will stand. Believing otherwise will only cause you to suffer monetary or other stress-related difficulties down the road. The house is a house regardless of what you choose to believe.

Now let's move on to a similar situation that is getting a fair amount of media attention: the hacking of the email and files of the CRU (Climate Research Unit) at the University of East Anglia in the UK. Scientists and technicians associated with this group played a role in the IPCC's report on climate change. The leaked material shows that scientists are as human as you and me, complete with instances of braggadocio, rudeness, conflict (sometimes personal) and sloppy thinking. This is unsurprising. I would hope that no one still believes the stereotype of the scientist grimly in a white lab coat humourlessly pronouncing incomprehensible equations and facts to anyone silly enough to engage them in conversation.

"The e-mail exchanges among several prominent American and British climate-change scientists appear to reveal efforts to keep the work of skeptical scientists out of major journals and the possible hoarding and manipulation of data to overstate the case for human-caused climate change."

The above quote from a blogger at the National Post goes a little far in the interpretation of what the material reveals of real scientists doing real work. To claim fraud and malfeasance is not justified, or at least not yet. Like in the house example, it seems that the prospective owner is calling in more inspectors to inspect the house yet again, and the work of the previous inspectors, all because the poor character of some of the workers has come to light.

It is not surprising that hard-nosed skeptics and outright political opponents of the case for climate change would latch onto any opportunity to re-open the science with the objective of reducing its credibility. However, opposition is not equivalent to proof of anything. For now the house still stands. It may be that the foundations of the house are indeed rotten and every inspector on the job site has missed this obvious flaw, although that is improbable. More likely is that some of the building materials were defective or several studs were incorrectly secured. It would be surprising if it were not so, since perfection is unattainable even with the very purest of intentions and diligence of everyone involved. This is as true for science as it is for house construction; that will not cause the house to fall.

The rhetoric is a distraction. I am content to watch while the inspectors go back in and poke around. Possibly they will suggest some remedial work, though even if it is not done I doubt that the building will fall. This is no house of cards, so a flaw here and there will not bring it down. So be prepared to pay up and move in when the latest batch of inspectors complete their review.

Asymptotic Pricing: Unattainable Cell Phone Rates

"...Recipriversexcluson: a number whose existence can only be defined as being anything other than itself." This fictional entity, which was invented by Douglas Adams, is at the heart of Bistromathics -- also an Adams' invention -- as described in the third book of his five-book trilogy: The Hitchhiker's Guide to the Galaxy. Although entirely fictional, the idea of a recipriversexcluson is not entirely useless to the better understanding of cell phone service pricing, as we will soon see.

Cell phone plans and their pricing can be terribly confusing and opaque. In this post I will not be attempting to cut through all that nonsense; it needs doing, but not here. Instead I will focus on one core portion of the pricing model: voice minute rates. That is, the actual amount you are paying per minute to use your cell phone. So let's drill down into that a bit further. To start, I will consider a simpler model that will exemplify the concept I want to explain: which is -- to coin a term -- asymptotic pricing.

In the past -- and perhaps at the present, but I don't actually know -- Chapter's book stores offered a discount card. How it worked is that for $25 you get 10% off on all your book purchases. The term was one year (if I recollect correctly). To many book lovers that may at first blush seem like a good deal. The first time I was offered this card by the cashier it did momentarily peak my interest. However a few moments thought was enough for me to find the flaw in the offer, so I smiled knowingly and politely declined. Let's look at how it works, with the aid of a graph.

You pay $25 up front, so you are immediately in a losing position. To dig yourself out of this hole you must buy books, lots of books. How much? Well, to break even you must purchase $250 worth of books (before sales taxes) since 10% of $250 is $25. That is, you pay $225 for the books plus $25 for the discount card, which equals the undiscounted total of $250. Of course, beyond that you do begin to see real savings. As an aside we should note that for Chapter's there is a marketing advantage in that you are motivated to buy lots of books, perhaps more than you otherwise would. If you do, they win, and if you don't, they also win. Win-win for them and lose-sorta win for you.

Regardless of how much you spend on books the one total discount you will never see is 10%. You can get arbitrarily close to that number by buying ever more books, but even if you buy all the books in existence, and even all those that existed in the past and that will exist in the future, you will never save 10%. That is because the initial outlay of $25 doomed you as surely as the poor horse whose rider dangles a carrot in front of its nose to make it run faster; the horse will not get the carrot no matter how fast and far it gallops.

In mathematics that 10% figure is an asymptote: a number that can only be reached by a limiting function, in this case with the limit taken to infinity. (There is a pale similarity to the earlier Hitchhiker's reference.) With this simple example in hand we are now ready to tackle cell phone voice minutes pricing.

Forget about data, SMS, vertical services, premium phones and other add-ons to your cell phone plan; it's only voice minutes we care about here and now. Even so, the graph (below) is more complex than the discount card example. In this hypothetical though typical pricing plan we see that the basic plan costs $30 (you can add in other fees and taxes if you wish to make it more realistic) for 150 minutes of air time. Again, we're keeping it simple, so no free weekends and so on; we'll stick with minutes that are chargable.

If you give any thought at all to the price when you are considering signing the contract, it is very likely that the one figure you will calculate is your expected price per minute by dividing the monthly fee by the number of minutes. In the present case that is $30 divided by 150 minutes, which yields $0.20/minute. However, as you can see in the graph, that value is only attained if you use precisely 150 minutes. If your usage is any other amount you will pay more. If you use less, you might pay substantially more per minute. (I've assumed a typical value of $0.35/minute for every minute you use beyond the 150 limit.) This is a bit like coin flipping: flip a fair coin 1,000 times and the likelihood of getting exactly 500 each of heads and tails is much lower than you might guess. At least if you do the flipping experiment many times the average will tend toward 500, but this is not the case with cell phones since you are relying on your (poor) ability to estimate how much you will use your phone:

Three-part tariffs optimally exploit overconfident consumers because overconfident consumers both underestimate the likelihood of very high usage, and the need to pay high overage charges, and underestimate the likelihood of very low usage, and the likelihood of not getting a refund for included “free” minutes.

The first thing to realize is that the first estimate of the per-minute price is an unrealizable optimum. Second, unlike the book card case, the $0.20 rate is not an asymptote since you can (if you're very careful) actually get that rate. However (third point), there is an asymptote in the graph, except it's not where you might have expected it to appear. The asymptote, like in the book card case, is also a limit for minutes taken to infinity. The asymptote is at $0.35, which is the overage rate.

If you think about it, that makes perfect sense. With unbounded use of your phone -- which is still very finite, coming in at 44,640 minutes in a 31-day month -- you will pay as much as $15,601.50, or $0.349495967/minute. That's pretty close to $0.35 though not equal. This limit is approached when minutes of use (MOU) is much, much greater than the 150 minutes in the basic plan.

Well, that's it. You now understand cell phone pricing just a little bit better than before. Just don't bring any of this up at parties or you will shortly find yourself with no conversational partners.

H1N1 Scare, Politics and Vaccine Fears

The progression in the H1N1 pandemic story has fascinated me. We went from panic to complaisance, then back to a more earnest form of panic: huge line-ups to get the vaccine. Much to my amazement, through all of this our various governments, federal and provincial, have stuck to their guns and performed quite well. The Liberals' recently attempted to cast the federal government in a negative light in this matter -- presumably hoping to profit from some understandable frustrations in how the vaccination program is unfolding -- is merely one more failed attempt on their part to latch onto an issue, any issue, that will elevate their and their leader's public standing.

It seemed that while H1N1 was only killing the elderly and the sickly, public sentiment became little more than a collective shrug once the flu-season ended in the spring. In that light, and the typically moderate symptoms suffered by almost all that have been infected, the continuing pressure from the government on the pharmaceutical industry to prepare for a mass vaccination program had the appearance of gross incompetence, and needlessly expense. When the unfortunate deaths of otherwise healthy children occurred recently, the potential seriousness of the threat became very real for most of us.

H1N1 is not killing a disproportionately higher number of people than other strains of the virus, yet there is the danger that it could. Since it more aggressively attacks the respiratory system and is highly contagious, there is an opportunity for a deadly mutation to turn the pandemic into at least a minor catastrophe. Or, it might not.

Either way, the risk of taking the vaccine is lower than the risk of not taking it. Even if you do wish to take the risk with your own health, there is the matter of whether you want to risk the health of others with whom you come in contact if and when you do become infected (rough predictions say 25% to 35% of us will eventually get infected, though not all will get sick).

However, rather than repeat what is said better elsewhere, what I want to say is that the point about supposed vaccine risks is becoming moot. When children started dying, the public mood changed. Now the trend is pro-vaccination. It seems that while we had the luxury of time to sit back and pontificate without expertise, avoiding the vaccine was a valid form of personal expression. Now it isn't. H1N1 kills in modest numbers among the healthy and affluent, kills more among the poor and sick, and could mutate to kill indiscriminately. That knowledge may now be enough to quell the evidence-free tirades against vaccination. I hope this trend will seep into other conversations about the supposed risks of vaccinations for other diseases. It should.

The Struggle For Survival: Britannia Park Edition

Ah, yes, the never-ending struggle between predator and prey goes on all around us the natural world. Who hasn't watched these surreptitiously filmed battles in the wild between majestic beasts: lion vs. gazelle, snake vs. frog, shark vs. dolphin, and the list goes on. It is quite ordinary and unremarkable in most ways, despite the necessary cruelty, pain and death it engenders.

Perhaps less remarkable is when animals -- any animals, not only predators -- compete for the same food. Hyenas do it, as do litters of suckling newborns. It can even be witnessed on the manicured lawns of local parks, it seems.

It was during one of those rare periods this weekend when it wasn't raining that I saw an unusual sight. What looked like a chunk of bread, whether purloined from picnickers or intentionally left for the birds, was being fought over by a small flock of about a dozen gulls. This is a common enough sight if you've ever had occasion to throw food to these flying garburetors. This was not the thing that struck me as unusual. It was what happened next.

Into the mix there appeared a squirrel. It intrepidly jumped in among the squawking quarrelers and grabbed the bread in its mouth. As it was making its getaway, the gulls finally noticed what had happened and the lot of them turned to chase the furry thief. They kept low to the ground, half running and half flying, trying to impede the squirrel's escape.

The squirrel was quickly surrounded and was being flown at from all sides by the enraged flock. The squirrel darted to and fro, avoiding their sharp beaks but not quite able to break from the circle. In the general melee, the squirrel did finally find an opening and darted through it, and broke into a loping run toward a stand of brush a short distance away. The gulls gave chase, dive-bombing the squirrel as it ran.

The bushy-tailed rodent did succeed in making its escape since once among the bushes the gulls were no longer able to fly at it. What was a barrier to the birds was a protective fence for the squirrel. Now free of the pursuit it took its prize up the nearest tree where it presumably ate the bread. Except that we're talking about a squirrel, so it's equally possible that it dropped the bread, stowed it among the branches to be lost forever, or left it to be retrieved and buried a while later by itself or another squirrel.

So ends another majestic episode of struggle among the beasts in the wilds of Ottawa's park system. I have to wonder if it was crows instead of gulls, whether the squirrel would have come out quite so well. A squirrel may be brighter than a gull, but crows should outperform most rodents on an IQ test.

Misinterpreting Trading-Volume Statistics

It seems that every summer there are articles published about how price movements in the stock markets are less than meaningful since many investors, both retail and commercial, are on vacation. The contention is that the resulting low volume has two significant effects:

1. Trends in equities and other securities and commodities are untrustworthy since market action is not fully representative of investor sentiment.
2. Lower volume means higher volatility in individual issues.

Supposedly, so the story goes, that post-Labour Day these effects may be reversed as the totality of the investor community is back on the job. I believe this belief is completely wrong. I will demonstrate this by qualitatively countering the arguments of the purveyors of this wrong-headed idea with some elementary statistics.

Let's start with the first point by quoting from this Reuters article that exemplifies the argument:

Reuters data shows volumes for the Dow Jones industrial average for the full day on Tuesday were only 56 percent of the index's 90-day average daily volume. They were higher for the Nasdaq, but still only 73 percent of the norm.

In Europe, the FTSEurofirst 300 volume was 69 percent of its daily average on Tuesday, while on Wednesday it looked likely to be even thinner -- it was at less than 40 percent with a little more than two hours to go.

As for China, where the Shanghai Composite Index has fallen more than 18 percent this month, volumes have mostly been less than half what they were in July, especially in the past eight trading sessions.

It is harder to gauge volumes for other assets -- foreign exchange trading data, for example, tends to come with a lag, from central banks -- but anecdotally, participants say it has also been thin.
...

Given this, investors might be better off waiting until September to see what is really happening to financial assets.

These percentages can be rigourously treated as a sample over the total population of investors. As anyone will know from various political polls, as the sample size grows in proportion to the population the expected error in the results, the variance and standard deviation decline. In this manner we typically see variances (typically stated with a 5% confidence, or 19 times out of 20) of several percent when the sample size is several thousand out of a voting population of over 20 million citizens, or on the order of 0.02% of the population. Variances are higher when the polling results are broken down further by province, sex or other factors since the sample size is a smaller percentage of the population (or a similar percentage of a smaller population).

Yet for the case of the stock market we have effective population samples of 40% to 75% of peak season trading volumes. The variance in these instances is well under 1%; that is, summertime equities quotes should accurately reflect the thinking of the entire investor community, including those at the beach. The only way around this is to hypothesize that bulls or bears, but not both, are more (or less) active in the markets during the summer. I have seen no evidence to indicate that this is true, and I would be suspicious of any evidence that would be presented since it seems so unlikely.

There is one psychological factor that may strike the market in September that could give a false indication. People who return from vacation tend to be mentally refreshed and therefore somewhat more optimistic. I wonder whether it is possible that these individuals are more likely to press the 'buy' button when they first get back to work. Even if this conjecture is true (and it could actually be the opposite if the news flow is persistently negative over the summer) their belief structure should quickly revert to the norm once they find that their new-found optimism isn't confirmed by their trading returns.

Regarding the second point about volatility, there is validity to the idea. Illiquid issues do show higher volatility (increased short or medium-term variations from longer duration time-lines such as 200-day moving averages), and that volatility does show a modest inverse correlation to trading volume. However, we are talking about volatility and not longer-term trends. This means that if the price of a stock makes a sharp swing up or down, there will be investors that will quickly take advantage of any "mis-pricing". We then see a reversion to the mean. It may not happen the same day, but it will almost always occur.

Canadian investors should be well-versed on volatility since the majority of stocks listed on the TSX have low average daily volumes. This is not surprising since there is a comparable number of public stocks in Canada as compared to the US, yet our economy is less than one-tenth the size. Our investors have to spread themselves thin if they are to avoid piling onto the relatively small number of liquid stocks with a large market cap. The point here is that low-volumes, while resulting in higher volatility, are irrelevant to long holding times. Traders, however, do care since their holding times are typically measured in hours or even minutes.

Expect to see more of the same sort of story in mid-December when the Christmas-New Year's doldrums hit trading volumes. The only difference is the shorter duration: two weeks rather than two months.

Bleeding for the Perseids

Ah, another meteor shower. The Perseids aren't overly spectacular most of the time, although they do conveniently arrive when a parka isn't required. Then again, parkas do have advantages.

I spent ten minutes late Wednesday evening trying to glimpse a few meteors. I had some measure of success. My total count over that brief time was: one airplane, one helicopter and one meteor. At least the meteor was a Perseid, as determined by its apparent source in the constellation Perseus, then rising in the northeast.

The ten minutes was my limit, because I had no parka. That gave the mosquitoes ample time to locate me in the calm air. There is only so much blood I am willing to lose to see a bright (or in this case, dim) streak crossing the light-polluted sky.

The Leonids aren't often much better, although some years they can be spectacular, but at least they come in November when blood loss is a far lesser concern. I am looking forward to wearing a parka.

Avian Citizenship

What makes a bird Canadian? They aren't on the tax rolls and they don't have passports. Yet it seems to be so terribly important to tag the origin of birds that get in the way of large aircraft.

The scientists have a reason to identify the population of the Canada Geese in the US Airways 1549 incident: which is to find ways to prevent or avoid similar occurrences in the future. That's fair enough. Scientists seem to assign them their home by where they nest and procreate. In this case, that would be Labrador.

However we should be wary of the moniker that we humans assign to these birds -- Canada Geese -- since it means nothing of essential value. We could just as easily call the geese Americans, and then wonder why they keep coming up here and pooping all over our cherished Newfoundland-Labrador. Where are all the editorials about that outrage?

Keeping Up With Moore's Law

Throughout my long career the scientists and technologists have been successfully battling to keep Moore's Law on track. It's been so reliable that it has proven perfectly acceptable that, if you need to find a way to increase the speed or capacity of a technology product, you only need to plot the trend to the planned release date. If the trend meets the requirement, you're done. It's been very handy for those of us in the business of planning future products; doing nothing to achieve one's goals is nice work if you can get it.

As the technological barriers are progressively hurdled there are also those higher barriers that are imposed by the fundamental laws of physics. These range from light speed - limiting the speed a signal can travel within and between circuits - to quantum mechanics - the soft boundary between traditional electronics laws, including Ohm's Law, and the weird regime of the quantum. Reading this recent article in the New York Times called to mind this latter limitation: in particular that the quantum world is largely discrete and so you run into difficulty when the number of discrete particles drops to a low integer value.

As you improve the precision and accuracy of micro-circuit construction, the circuit density increases at a faster rate. This is simply because a micro-chip is 2-dimensional, and so density increases with the square of the lineal improvement. Unfortunately as we enter the realm of small quantum units, the number of those units decreases at a similar rate. For example (assuming that all other things remain the same, which they rarely do), if you have 100 electrons within a square circuit area that are employed to perform some function, if you double the lineal density you then have only 25 electrons on hand. While this is grossly simplified it does demonstrate the nature of the challenge: as particle counts drop toward one, the electronic laws that are based on mass quantities of particles become fuzzy in the face of the probabilistic nature of each of those particles. Macro-laws smooth out the individual differences of unpredictable particle behaviour.

Going to 3D has been a dream for as long as I can remember. In the case presented in the referenced article it can provide a way to avoid the quantum scaling problem for a little longer. However, 3D circuitry has its challenges. I remember when, long ago now, there was a move to 3D circuit boards. These allowed more circuit density with the discrete components then in use by allowing easier interconnection among the huge number of pin-outs of the many ICs and other, more basic components. There was a limit to this since components still had only two surfaces to work with for the most part (wires/traces, not components, were on the interior layers). In time the interconnection problem was eased by better CAD software and chips that replaced dozens of discrete components, making 3D circuit boards less of a focal point for further progress.

Within a chip, going to 3D has the same interconnection challenge while also having other penalties. Perhaps the worst are yields and heat. With increasing density there is an initial drop in yield, which is the fraction of components that exit manufacturing without disabling flaws. This can only get worse with multiple layers where the quantity of deposition layers and masks grows linearly with the number of layers in the 3D stack. It could be even worse. The benefits have to be great to justify the battle over refining processes to achieve economically acceptable yields.

Heat is already a problem in 2D microchips. It is more acute with microprocessors than memory chips, but it still exists and would need to be dealt with. Every gate generates heat during active operation (such as reading and writing), and heat generated within a 3D stack may be difficult to conduct to the surface for removal. There are of course many avenues open to attack this problem since it is very well understood. The optical method outlines in the article would most likely have less heat to contend with, but with the cost of other difficulties.

Since I am far outside this micro-electronic world I can only watch and wonder how these various difficulties can be overcome. If economical, reliable and, perhaps above all, scalable 3D solutions are discovered we will be firmly back on the Moore's Law trend line. Within some constraints, for a given technology base circuit density could increase up to the cube of lineal density. The third dimension most likely will not scale as well as the other two, yet even a modest achievement would have impressive results.

Getting Off Fossil Fuels

I have no fantastical insights about just how we're going to get ourselves off fossil fuels, or even any certainty of the costs of not doing so. The matter of Earth Hour this past weekend is one I wrote about, demonstrating a view that, if not the new mainstream opinion, is at least in agreement with the lack of action of others - there was only a 6% dip in electrical power consumption during that hour, even less than what I, a non-participant, expected.

This article in Cosmic Variance certainly stirred some deeply-held feelings among what is arguably a scientifically-astute audience. The referenced New York Times article on Freeman Dyson (warning: it's very long) is also a worthwhile read about an eminent scientist with skeptical views and a eye on a broader cost-benefit analysis of carbon emissions.

As I opened with, it isn't my intention to get into an argument for one side of the other. What I do want to say is that, like it or not, we will have to adopt alternatives regardless of our individual views and knowledge. It is simply because fossil fuels are a finite resource; there will be an end to it.

It is true that there's an enormous amount of coal, centuries worth, but we can't use it for transportation without even more expensive processing than for the tar sands to convert it into usable liquids. As production declines and the costs of new sources rise there will be a secular and permanent upward tilt to energy prices. This will happen regardless of how clean we can burn the fuel; environmental impacts (or lack of them) won't change this outcome.

To me it makes good sense to ramp up exploration and basic research for alternatives. Even if everyone were to turn hard-core environmentalist overnight, there is today no good alternative other than simply doing without. That will wreak havoc with our standard of living. Some would happily accept that while others would not. My point is that, regardless of individual views on global warming, pollution and consumerism, we will have to change, and it would be best if we start working very hard now so that we can get to a position where we can make the change - a non-disruptive change - to alternative energy sources.

Fossil fuels are certainly finite and dirty, yet still very affordable. Yes, even at $150 per barrel. The bigger issue is the opportunity cost of not preparing for the transition. If we wait until the dollar cost forces us, one by one, to make hard decisions in our individual lives, we will have irretrievably lost time we could have put to developing new energy technologies.

Today I do not see a sufficiently broad commitment to doing the necessary work; not among politicians and not among the population. Turning out the lights for one hour, lights that aren't powered by fossil fuels, may make people feel good enough about themselves that they forget about the challenge for another year. That doesn't help.

Unfortunately it looks like it will be awhile until there is a strong enough turn in the public mood, or fear, for us to get serious about steering our economy in a new direction.

Leaving the Lights On

From all indications a lot more households than last year will turn their lights off for one hour this evening - the so-called Earth Hour. This event is being promoted as a way for everyone to show their solidarity with at least the idea of energy reduction and conservation, with a particular emphasis on reduction of carbon emissions as part of the fight against global warming.

I agree with the sentiment while being ambivalent about Earth Hour itself. Of course, as even the promoters admit, the real impact on energy production of this event is negligible; after that one hour of reduced consumption (Hydro One is expecting a dip of over 10%) we all go back to our typical wasteful ways. The objective, however, is one of raising consciousness about the problem even if the immediate impact is nil, and in that I expect Earth Hour will be successful.

My problem is that we are wasting an opportunity to better understand the energy problem. What gets lost in the celebration is that carbon emissions will increase during Earth Hour here in Ontario. The reason is that most of our electrical power generation is nuclear and hydro-electric. Since it is still the heating season, when you turn off all your electrical appliances, including the lights, you lose this carbon-free source of home heating and your fossil-fuel burning furnace will work a little harder. It isn't often understood that pretty much all your electrical power consumption heats your home as a byproduct of running electric appliances and lighting.

This is hardly a novel thought (it's certainly well understood by any expert in the field) but it is not thought about by the general population. This is hardly surprising since few people are conversant with the science, and even those who could understand it do not usually think it through. I believe getting this point across as part of Earth Hour could make the event more useful and educational.

In addition to turning off the lights (especially those outdoors that contribute nothing to home heating), how about parking the car for a day and setting the thermostat to 19 degrees. People need to think a little bit more about all sources of energy. In regions where electrical generation is from fossil fuels (coal, natural gas, oil), there it makes sense to cut the juice. That, too, is a useful lesson.

The lights in my home will stay on Saturday evening. That is how I will be reducing my carbon footprint.

Spectacular Improbabilities

In the past few days there were three stories of improbable events, two catastrophic and one that was potentially catastrophic. They were widely covered so you would almost certainly be familiar with them: plane crash, satellite collision, and submarine collision. While there is much of human interest in two of these stories, there is a common thread running through these separate events - probability - that always catches my interest.

First, there was the human tragedy near Buffalo. In the weeks and months ahead there will be painstaking work to reconstruct in excruciating detail the events leading up to the crash, and, importantly, to learn lessons to be applied in future. This should lead to increased air safety, which is already extraordinarily safe. It's gotten so safe in part through similar learning in past tragedies.

I can still remember a time when news of a plane crash would lead many people to a renewed fear of air travel despite the statistics of just how safe it was in comparison to alternatives. To me it seems novel that there is little of that today. It may be that the general population has finally understood this fact, that it is highly probable that when you get on an aircraft you will arrive at your destination safely and ready to complain about all the petty annoyances of modern air travel. The reaction is so muted that articles like this one in the Ottawa Citizen seem almost superfluous.

So, the likelihood of an air crash is low, especially in this part of the world. That's what makes it safe. However, there are many flights and passengers and so these tragedies happen with some regularity. Keep that thought in mind while I go through the next two events.

The second catastrophe involved no humans except for, perhaps, insurance underwriters, since it occurred in Earth orbit: the collision of an Iridium communications satellite with a decommissioned Russian military satellite. While this is a wildly improbable event, it is very probable that such an event was due. In the words of Douglas Adams, "space is big, really big," yet with the vast quantity of satellites and their detritus in orbit these collisions will occur.

Up until now the likelihood of a satellite collision in any span of time has been lower than for air crashes. This situation will continue for some time since air travel is getting safer, and for now at least traffic loads are down, even while the quantity of junk in orbit is increasing. I can imagine a time within a couple of decades when those lines of probability will cross and satellite (or debris) collisions will have the higher frequency. Luckily there are few humans in space right now, and they're in one location, so the chance of a human tragedy will remain low for some time. Some day this could change.

The third event, the near disaster, was that curious collision of submarines in the Atlantic Ocean. At first glance this has to be the most improbable of this set of events; there are few submarines and lots of water, and they're being steered by humans. This has even been part of the public relations from the navies involved:

“It's an absolute one in a million chance that the two submarines were in the same place at the same time,” said Lee Willett, head of the maritime studies program at the Royal United Services Institute, a London-based military think-tank. “There is no precedent of an incident like this — it's a freak accident,” he said.

Luckily there was no serious damage so it seems almost comical: two vessels with technology to evade detection manage to bump into each other.

Of course this last event is not quite as improbable as stated above since submarines are not randomly distributed over the surface (and depth) of the ocean. Indeed, this is not an isolated event, as Willett claims, and as is detailed in this New York Times article:

Military experts said there were a number of collisions between western and Soviet submarines during the cold war. In 1992, an American nuclear submarine, the Baton Rouge, was struck by a surfacing Russian submarine in the Barents Sea.

Unfortunately I don't have a suitable quote handy, but supposedly the reason why these collisions are not uncommon is that subs tend to closet themselves in the same areas, areas suitable to their covert or hidden activities, and which are far smaller in volume than the ocean.

The same is true of satellites. They are not randomly situated in space, but instead occupy similar orbits that are optimum for their intended applications. This is similar to the situation with the subs, and further explains why satellite collisions can happen. Unlike the situation with subs, and aircraft, the probability of disasters climbs after satellite collisions since the debris clouds occupy a greater volume than their original, intact, form, and even a single bolt at the high velocities involved can ruin someone's day.

We have made improbable events likely by making a multitude of "attempts". Just like with the lottery, it is very improbable that you will win but very likely that someone will win. So it goes with satellites and aircraft, and even it seems with subs.

Adjectivitis and the Weather

Superlatives are superlatively superfluous.  Or something like that. Adjectives are a bit of a disease that you learn to avoid in business (if you dare) since they add nothing and can alienate those the adjectives are meant to impress. Are you intrigued by products that are the best, or new and improved, or indestructable? I hope not. We should all have become just a little bit wary of these verbal attacks on our credulity.

Something like this happens when it comes to Canadians and the weather. Cold enough for you? It isn't enough to say that it's -30 C, we have to add that it is the coldest this winter, this decade or this century. And it isn't just cold, it's very cold or extremely cold. Never simply cold.

We hate the cold, but seem to revel in enjoying how much we hate it. So we harp upon the temperature as if cold weather in January were remarkable. Or perhaps it's just me that's a bit jaded since I've lived in a colder climate than that in Ottawa. I don't remember which comedian said this, but I like the line: Canadians have to be tough since just the weather is enough to kill you.

But the thing that really gets to me is that we just have to add in the wind chill. Now it's no longer -30, now it's -40. To emphasize the point, many will even forget to mention that it's a wind chill number, not actually the temperature. Wind chill is Canadians' way of adjectivizing numbers; saying -40 is apparently a more sophisticated way of saying it's -30 C with a breeze.

The thing is (and you can do this experiment yourself), if you take a thermometer outside where it's -30 C with a howling wind and hold it out there for a while, the reading will quickly drop from +20 C and then slow as it approaches and finally reaches -30 C. The wind chill may be -60 yet the thermometer will steadfastly refuse to go any lower, and it certainly will not go to -40 where the mercury will, the urban legend says, freeze and then explode in your face. None of this will happen. The reason it won't happen is that the temperature is -30 C, not -60 C, despite the attraction of those beautifully-large negative numbers.

If you're adventurous, you can try the above experiment slightly differently, in line with this fascinating snippet from the Wikipedia entry on wind chill:

The method for calculating wind chill has been controversial because experts disagree on whether it should be based on whole body cooling either while naked or while wearing appropriate clothing...

I would love to observe these experts when they're comparing these competing techniques. Who knew scientists were so tough.

So what does it all mean? It means that it's -30 C with a nasty, heat-sucking wind. That's just what it is. When it's windy that thermometer get to -30 C faster; the wind, like a fan pulling air through a car's radiator, causes a more rapid transfer of heat. Adjectives need not apply. It's just that wind chill temperature is more sexy than watts/centimeter/second, which does accurately tell you how cold it feels.

When Environment Canada went to this latter style of reporting in parts of the country years back, people hated it. I think they hated it even more than metric. No one complained that the reports were wrong - they were in fact extraordinarily correct, surprising everyone who knows to never trust a weather forecast - but a statement like "the wind chill is 2,400" just didn't excite people. There were no opportunities for adjectives. Eventually they gave it up and went back to the temperature style reports which, while mostly meaningless, fit better with Canadians' world view, which includes not trusting the weather report.

The only cure for this disease is called spring. Then we can complain about how wet it is and try out some new adjectives to describe that.

It's Getting Cold, So Time to Check Tire Pressure

The fall and spring are the most important times to check the pressure of your vehicle's tires. These are the times when they fall and rise, respectively, the most. This assumes you have no leaks in your tires. To understand why pressure changes we need to look at the ideal gas law:

PV = nRT

• P is the pressure, which I'll measure in psi (pounds per square inch). As an aside, we measure tire pressure in comparison to the ambient air pressure which is about 15 psi. Therefore the absolute pressure for a tire at 30 psi is actually 45 psi. Luckily we don't need to know this to maintain our cars.
• V is the volume. Car tires have stiff casings (carcass) which allows us to assume that volume is constant within a reasonable range centred on the recommended pressure.
• n is the number of air molecules. If our tires aren't leaking the number of molecules has not measurably changed since the last time the pressure was checked.
• R is a constant that is unique to every gas. Since we're all using air to fill our tires rather than exotic alternatives like carbon dioxide, nitrogen or helium, it is truly constant for our purposes.
• T is the temperature of the air. This must be measured from absolute zero (-273 C), so we have to convert the air temperature from Celsius to Kelvin. For example, 20 C is 293 K. You'll have to pretend there's a degree symbol (little 'o' at the upper right of the numbers) since I'm going to be lazy and not figure out how to do that in HTML.

Now we can simplify the above by using our assumptions. What we do is identify and eliminate all the constants since all we care about is the change in pressure due to temperature. This gives us the far simpler formula: P = T. More correctly this should say that P is proportional to T, not equal, though that's a mathematical nicety we can safely ignore.

What this formula states is that the tire pressure changes the same amount as the absolute temperature. This is just what we are looking for to understand why our tire pressure is lower in October than it was when we last checked in August. Let's say the tire pressure was 30 psi when the ambient temperature was 22 C. Here's what we get as the temperature drops:

• 10 C: 28.8 psi (-4%)
  
• 0 C: 27.8 psi (-7%)
  
• -10 C: 26.7 psi (-11%)
  
• -20 C: 25.7 psi (-14%)
  

These are large changes. A change of even 2 psi will degrade performance over bumps and holes and when cornering, and tire wear is accelerated. With lower pressures, efficiency drops, which raises tire temperature at high speeds, which causes increased wear and susceptibility to sudden failure.

The opposite changes occur in the spring as tire pressure increases with the warmer temperature. Of course if you swap between summer and snow tires this effect is avoided somewhat, however you are not home free.

Many people rarely if ever check tire pressure at all, preferring to leave it to the mechanics or oil jockeys during regular oil changes or service visits. If you've ever measured the tire pressure after these visits you will quickly learn you are far better off to take care of it yourself; all you need are an inexpensive barrel pump and a good pressure gauge. I do after every servicing and have found my 30 psi tires filled with anywhere from 25 to 40 psi. Modern tires are forgiving although handling and tire life are impacted.

High Moon

No, the title isn't a typo. If I'd meant to talk about the election again I would have called this post High Noon; it only means that I'm prone to making bad puns. I really do want to talk about today's full moon and how high it is in the sky.

I find that one of the pleasant compensations of the colder temperatures and loss of daylight this time of year is that it is also a time of brilliant full moons. Whether you call today's full moon the Harvest Moon or, more correctly, the Hunter's Moon, its appearance is the same. The full moons surrounding winter solstice in late December are higher, brighter and up in the sky longer than at other times of year. We may lose the sun earlier in the day, but the moon takes its place when its phase is near full.

This assumes, of course, that the sky isn't clouded over. More importantly you need to live at mid to high latitudes to get this effect. Since Ottawa is midway between the equator and the north pole, it qualifies. And the further north you go, the longer the moon is in the sky although it doesn't get quite to high above the horizon. This high full moon phenomenon is no coincidence; it is always true. The reason is that at full moon the moon is almost, but not quite, positioned directly opposite the sun from our vantage point.

I won't draw any fancy charts myself, though you can find those here on Wikipedia and in other referenced articles. The basic mechanics are as follows. The Moon orbits the Earth in nearly the same plane that the Earth orbits the sun; the planes are inclined to each other by 5 degrees. If the orbits were in the same plane there would be both a lunar eclipse and a solar eclipse every month. Instead the moon usually misses the Earth's shadow at full moon.

Now because of the opposite positions of sun and moon there are a couple of interesting geometrical facts. One is that at full moon, moonrise and sunset occur at the same time (again, not exactly but close enough), as do moonset and sunrise. This means that when the sun isn't in the sky the moon is (and vice versa), so that the full moon is visible throughout the long winter nights. Second, when the sun is low in the sky during the day it dips far below the horizon during the night. Since the moon is opposite to it, the moon must therefore rise high in the sky, reaching it's peak at local midnight (when the moon is due south, and not according to the clock). The same effect occurs in the southern hemisphere, offset with the seasons by 6 months.

The result is that for the next several months of cold weather for the days surrounding full moon each month we will have a brightly lit landscape throughout the night. Since the cold often brings clear skies we are likely to get the full benefit of this effect.

Is it of any use other than being pretty? Perhaps not in the city where there is artificial lighting. It is in the great outdoors where it is very noticable and useful. That's why the Harvest Moon and Hunter's Moon have the names they do; these are activities taking place outside of cities. For urban dwellers it does however have a recreational benefit. Think of midnight skiing and snowmobiling. With only a small amount of care and avoidance of deep shadows among the trees it is possible to frolic all night long, if you remember to dress warmly.

Think about that and look up the next few nights, which will hopefully clear from time to time. Winter does have its perquisites. Make your night-long party plans now.