Trivia Testers : Santa

Figure 1

Okay, okay, so Merry Christmas already. Just make it snappy, Pops!
Certain fables in the imagination and fragile minds of a young child are best left untouched, until such time as when they reach a certain age that only a parent can know, when gullibility wanes, worldly wisdom dawns, and the time then comes to guide our wee ones along the bumpy path of a sometimes harsh reality. Most men of the cloth seem to have this innate skill, borne perhaps of uncommon compassion, but there was one stand-in vicar in England who acheived unwelcome fame when he explained with somewhat indiscreet harshness just what might happen to Dear Old Saint Nick and his faithful reindeer, if they truly did have the necessary velocity to deliver their gifts of joy to every home in Christendom during this one blessed night. Let’s just say that he did not break it to them gently. Shocked parents and terror-stricken children listened on as he continued his explanation, thinking the children would really like to know these facts, and he described just how and why Santa, his reindeer, and sleigh would have to be traveling so fast that they would be instantly burned to a crisp at 3000 times the speed of sound. Needless to say, more than a few parents were left with some difficult explanations for their heartbroken children. But we’re grown-ups, right? We can take it! So just how fast would Santa have to go? Within an order of magnitude at least, was the vicar right? (More than one of these may apply.)

  1. We’ll never know. He’s currently ministering to a parish of Yanomamo Indians in the Amazon rainforests of Brazil.
  2. Perhaps he had the best of intentions, but he probably flunked math in high school. Even allowing for a one ten-thousandth of a second stopover in each home would still require travel at a rate in the vicinity of the speed of light (let alone the speed of sound).
  3. The vicar was right, at least within an order of magnitude.
  4. Who cares? Just have a Merry Christmas!

Answer: B and D. No, we’re not going to deal with actual populations, shortest-path algorithms, graph theory, or any of that stuff. We’re only interested in seeing if he was even in the right “power of ten” ballpark (i.e., somewhere between ten times slower and ten times faster than what he’d said). If you assume the shortest path between a few hundred million Christian homes can be represented by a simple-minded rotation of great circle arcs about both poles, separated by, say, 100 feet at 30 degrees North, and making the assumption that if he did equatorward course reversals upon reaching 60 degrees North and 60 degrees South, he wouldn’t miss too many chimneys…and if you even factored in an additional ten degrees northward travel for Alaska, Sweden, Norway, and Finland (sorry, Siberia) that would be, let’s see…

The earth’s equatorial radius is about 3963 statute miles, it’s equatorial circumference is about 24902 miles, it’s circumpolar circumference is about 41 miles less or roughly 24860 miles. If you used 30 degrees of latitude as a reference, 3963 times the cosine of 30 degrees gives a distance from the earth’s axis at that latitude of 3432 statute miles. That, times two pi, gives a circle at 30 degrees latitude of 21564 miles. Multiplying by 5280 feet, then dividing by 100 feet equals 1,138,591 successive longitudinal course lines, between 60N and 60S, around the earth. Let’s take the liberty of being charitable to Santa, and chop that in half, assuming he can skip those bands where there are only vast stretches of ocean (sorry, Tahiti) and round it down to just 569,000. If you then estimate the distance along 120 latitudinal degrees of arc (along each of those course lines) of roughly 8290 miles (using about one third of the circumference), that would give 4,717,010,000 or about four billion, 717 million miles. Then if you added in a bit extra (about half of one percent) for Alaska and northern Europe, it would wind up being about four and three quarter billion miles. Let’s say we allowed Santa & Co. an eight hour shift. (That’s not against union rules, right?) Actually though, being such a consummate globe-trotter, Santa has a big advantage. He can “follow the night”, so instead of only eight hours aloft, as far as the whole world is concerned, Christmas eve for him would actually be 32 hours! (Why? Well, the rotating earth could keep 24 of them at bay until eight more were used up in that lightning-fast feat of gift-giving.) Okay, so covering that distance in 32 hours means a speed of roughly 148,440,000 miles an hour, and dividing that by the speed of sound (at sea level of course) of about 761 mph would give about Mach 195,000. (And that’s without stopping anywhere.) That’s also–ahem–about 22 percent of the speed of light, or for physics weenies, 0.22c. And if you’re wondering about relativity, it would actually work against him, as time would go by on earth some two and a half percent faster (from Santa’s perspective, that is). Yes, from a non-relativistic standpoint, you’d think that he could get more work done. By the time he’d reached 71% of the speed of light, he could make the whole trip in about 10 hours, even though the “time dilation” penalty would only be about 40%. It gets worse from there, though. At 0.98c he might do it four and a half times faster than he could at 0.22c, but thanks to special relativity, time would be going by on earth five times faster! (and at 99% of the speed of light, seven times faster…) So, yup, even if the assumptions behind my calculations are off by a factor of ten…he’s probably still wrong… unless of course, he just meant Jolly Olde England. Then…well…he would be just about right.

By the way, if you’d like to track Santa’s progress (although I have no idea how NORAD can actually do that, given his extreme speed) you can follow along at their “NORAD Tracks Santa” web site.

Too Cold, Too Low
Back in late July, we ran a Trivia Tester about cold air altitude corrections for instrument approaches up where the air is really cold, like Canada, during the winter. However it wouldn’t take arctic air to make you bust minimums. In the case where the surface temperature is just a bit below freezing, say minus 10 degrees C, how much of a cold temperature altitude correction should you add to all published altitudes, on any approach?

  1. 20 feet if below 800 feet AGL, 40 feet between 800 and 1600 feet
  2. three percent of your estimated AGL height below 1000 feet, six percent between 1000 and 2000 feet AGL
  3. For an airport elevation below 2000 feet MSL, it is exactly ten percent of your height above the airport, up to 2000 feet AGL.
  4. The bust would be technical, and legally valid, but negligible: five feet for a 200 foot DH; 15 feet for a 600-foot MDA; and 20 feet for an 800-foot MDA. Since your altimeter can legally be off by 75 feet anyway, this is trivial!

Answer: C. (And from 3000 to 5000 feet AGL, the correction is just 10 feet less than 10 percent.) The reason behind this is that, again, altimeters aren’t self-correcting for temperature. When it’s cold, the air is more dense, and so the same air is “compressed” into a narrower layer. (Talk about density altitude!) The altimeter doesn’t “know” that it’s cold, as it only measures pressures, and it is calibrated for an “ISA” standard atmosphere, which is perfectly dry air at a sea level pressure of 29.92 inches Hg, at a much warmer sea level temperature of 15 degrees C, and a lapse rate of -1.98 C deg per 1000 feet. If you want to see the exact correction formula, it’s this:

So…radar altimeters, anyone?

Improbability Drive, And the Worry Wart
For an instrument rated general aviation pilot, a true nightmare would be to experience a simultaneous double failure of both the electrical and vacuum systems, while in IMC. Just how often does something like that occur, and approximately what are the odds of it happening to you?

  1. infrequently: It occurs less often than once every 1000 flight hours.
  2. remote: Its probability is on the order of greater than one ten-millionth, but less than or equal to one one-hundred-thousandth, per hour of flight, or per event (e.g., takeoff, or landing).
  3. improbable: It has a probability of occurrence greater than one one-billionth, but less than or equal to one one-hundred-thousandth, per hour of flight, or per event.
  4. extremely remote: The probability of occurrence is between the orders of one one-billionth and one ten-millionth, per hour of flight, or per event.
  5. extremely improbable: The probability of occurrence is on the order of one one-billionth or less per hour of flight, or per event.

Answer: E, although possibly D. There can be no exact answer to this question (other than that it is definitely a long shot, well beyond the “one in ten million” range). From the standpoint of the probability or acceptability of operational errors or mechanical malfunctions in designing or operating any aviation device or system, the FAA uses terms to define the odds of something going wrong. One of these is “extremely improbable”. Although not universally applied, this term, as well as the others, are defined in Advisory Circular 120-29.

I asked Bruce Landsberg, Director of the AOPA Air Safety Foundation, if anyone had ever experienced such a failure, and he didn’t think that it had ever actually happened. As he pointed out to me, almost all of the spatial disorientation accidents are due to continued VFR into IMC; on average, there are only three accidents per year because of spatial disorientation that was caused by loss of vacuum or actual instrument failure. (During a recent 10-year period, the actual number of general aviation “disorientation related” accidents was exactly 271, but in fact, only 17 of those were due to a vacuum system or pump failure, just six were because of an instrument failure, and a very unlucky but very small number, just two, had both. The remaining 246, and the vast majority, were simply pilot proficiency or judgment screw-ups.) So don’t get too worried on this one. First of all, most of your flying is likely not to be in IMC. Secondly, most electrical failures don’t happen all at once. (The trick is to keep the ammeter in your scan, or at least to look at it every couple of minutes, while in IMC. If the alternator does take a vacation, you do have a battery, which confers a precious few minutes to try getting to visual conditions.) You may also have a hand-held backup radio. And you might have a standby vacuum (no, not the one for the carpet). Also, you might periodically practice partial panel flying?

So here’s the big picture: Each year, general aviation flies somewhere on the order of 30 million hours. For the last 30 years, it’s been between about 24 and 43 million, with the high in the late 1970s, and a low in the mid-90s. Considering just the last 30 years, that would be, very roughly, a billion hours flown. And it’s probably never happened (or at least been reported). Heck, although it depends on where you live and what you do, in your entire lifetime the odds of the average American being struck by lightning are about a third of a million times greater. (According to the National Weather Service, it’s one in 3000.)