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Sunday 18 September 2016

DJ, Spin That Shit!

Right, peepholes, time for a bit of fun, methinks.

Today's offering is going to deal with something that many will think is insane - and it is - but there are two things that have been popping up all over the internet with increasing frequency over the last few years which were put to bed in scientific terms centuries or even millennia ago. 

I've always enjoyed reading. In my younger days, I read all sorts of stuff, pretty much anything I could get my hands on. Science fiction, historical fiction, thrillers, you name it.

About twenty-five years ago or thereabouts, I stopped reading fiction. It wasn't a conscious decision, at least at first. Rather, it was a function of discovering some areas of non-fiction that I got hooked on. I ended up with such a long reading list, comprising mostly history and science, that I simply didn't have room on my bookcase any longer under the groaning weight of Simon Schama, Alison Weir, Antonia Fraser, Stephen Hawking, Isaac Asimov, Richard Dawkins, Karl Popper, Thomas Kuhn... There was just too much to get through, and the list has remained fairly constant in size ever since, and even growing on occasion. As I write this, the stack of books waiting to be read runs to about fifty titles, almost all science, with a smattering of history books that have tended, in recent years, to get deferred in favour of whatever new science stuff piques my interest.

There's been one exception throughout this time, and it's become the only fiction I've read in a couple of decades, but I never missed a new publication, and it was always elevated straight to the top of the list. In fact, I'd tend to pause partway through whatever I was reading (although I usually have three or four books on the go at once, so I can change to suit whatever mood I'm in). Sadly, this exception has now dried up, and there's no new fiction to take its place, because the author behind this exception has now, sadly, passed away. I dedicate this post to him, my all-time favourite author: Terry Pratchett.

Pratchett is known to the world as the author of the Discworld series of books. Ostensibly just fantasy stories with a comedic slant, they're really a satire on some of the silly things that humans have believed throughout the millennia. He covered parodies of all the daft religions we've adhered to throughout history, various inventions, such as moving type, news, gunpowder, movies, postage stamps, trains, long-distance communications, computers, etc, as well as satires on real people like Leonardo da Vinci, Capability Brown, Heath Robinson, it's all there.

His world is, as the name suggests, a disc, sitting on the back of four elephants which sit in turn on the back of a giant space turtle swimming through the void. This again is a direct reflection of what was once a popularly-held belief among humans about Earth, although I'm not aware of any of the elephants having to cock a leg to allow the sun to pass by in any of the Earth mythologies...

He also, along with mathematician Ian Stewart and reproductive biologist Jack Cohen, wrote some science books, The Science of the Discworld series, which are genuine science books for a lay audience intertwined with Discworld stories. I can't recommend these highly enough for anybody getting interested in science (even approachable to young adult readers). They do a side-by-side comparison of the science in this world with magic in Discworld (it runs on magic; they can't get science to work there). Brilliant stuff, and extremely funny.

I should also note that there's a fair bit of science weaving through the entire series, but you have to be quick to spot it.

Anyway, to more serious matters. Silly and satirical as Pratchett's books are, it appears that there are genuinely still people who think that Earth is flat or, alternatively, that it sits still at the centre of the universe, while all around us stars violate the speed limit imposed by special relativity in completing an orbit composed of many millions of light-years travel in a single day composed of slightly more or less than 24 hours (yes, this is actually a variable).

It's reasonably certain that, for much of human history, people intuited that the planet was flat. This is hardly surprising. There's an interesting thing about curvature and how it works that means that, on large enough scales, it can appear extremely flat locally. This is fairly easy to intuit, with a little careful thought. Take a square piece of paper 1 cm on a side, and place it on a golf ball. It's easy to see that a piece of paper this size on that size sphere will not sit flat. Now take the same piece of paper and put it on a tennis ball. It will still curve, but it will sit considerably flatter. Do the same with a sphere 1 mile in diameter, and it will sit even flatter. Now take it to the moon, and it will sit functionally flat (assuming the moon was perfectly spherical).* When you get to something the size of Earth, your 1cm piece of paper will be so close to flat that measuring the curvature would be pretty close to impossible in practice.

Our intuitions are incredibly powerful, and have proved immensely useful to us in the evolutionary past of our species. As we've learned in other areas we've explored in previous posts, they can also be incredibly misleading, not least because they have inherent innocence. Intuition tells us that we can't get big hunks of metal to fly, that time flows at a fixed rate, that something can't be in two places at once. All these intuitions are demonstrably wrong. 

Sometimes, we need to gather data before our intuitions can be useful. Others, no amount of data help, and we have to throw intuition out of the window. 

So, how can we be confident that the planet is an oblate spheroid rotating about its axis and orbiting Sol?

First, it's worth pointing out that, in any places with significant maritime histories, it's probable that the popular intuition didn't hold, not least because they'd have had plenty of experience of watching ships disappearing over the horizon. Similarly, any populations living in the high places of the world would have been able to see the curvature.

For most of us, who don't live on the coast or in high places, it's easy to understand why people might have thought the Earth was flat. It certainly looks like it on a naïve appraisal.

The idea that the planet was spherical has been around since sometime around the middle of the first millennium BCE, though it appears in ancient Greek writings with little justification. Priority on the matter is very difficult to pin down, but the biographer Diogenes, writing in the 4th century BCE, attributes priority to Pythagoras, a regular contributor to my own witterings. In any event, it's fairly clear that, among the ancient Greek philosophers, few thought the planet to be anything other than spherical. 

By Graham.beverley - Own work, CC BY-SA 3.0, 
The earliest known account that gives some justification for this comes to us from our resident expert on sexual dimorphism in human dentition, Aristotle. He noted several observational issues that only made sense on a curved surface. Chief among these is the simple fact that there are stars visible to the South from Egypt and Cyprus that are not visible from further North. He also noted that, during a lunar eclipse, the shadow of the Earth on the moon was always curved, as in this image. This can only happen if the planet is spherical.

The first proper measurement on record showing that the planet isn't flat was conducted by Eratosthenes of Cyrene in 240 BCE. It was well-known that, at noon on the Summer Solstice, the sun shone directly down a well in the town of Syene, now known as Aswan, on the banks of the Nile. Eratosthenes took measurements in a well about 515 miles North in Alexandria and measured the difference in angle at seven degrees. Since seven degrees is about one fiftieth of the circumference of a circle, it was a simple matter of multiplication, and he came up with a figure of approximately 26,000 miles. There is some uncertainty on the exact figure, because his distance measures were expressed in stadia, an archaic measure of 600 feet, where the foot was a broadly variable measure, with a range of lengths defined all across Greece. In any event, his figure is somewhere within four to six percent of what we now know to be the correct figure of 24,901 miles.

Eratosthenes was responsible for quite a few other firsts, as well. He measured the axial tilt of the Earth (I'll be coming back to this shortly), created the first world map based on the knowledge available in his day, and invented the discipline of geography. He's also thought to have invented the leap day and to have calculated the distance to the sun. He also invented the sieve of Eratosthenes, a clever way of calculating prime numbers.

In any event, the knowledge that the planet is (approximately) a sphere has been in our species for more than two millennia. This hasn't had much impact on some, quite probably because of a combination of cognitive inertia and the fact that the notion that we live on a flat planet has been made dogma in some spheres, even to the point that suggesting otherwise has been considered sanctionable heresy in societies in which the sanctions included a painful death, if you were lucky.

That's not the end of the story, of course. It should be, but there you go.

In 1687, Newton published his Principia Mathematica, in which was detailed his theory of gravity and his laws of motion. We've seen in previous posts that, when a new, ground-breaking theory is formulated, it isn't always immediately obvious what the full implications of the theory are. Newton's theory was no different, but something that became apparent pretty quickly was something that caused some controversy. It was bad enough, some thought, that heavenly bodies didn't move in perfect circles - the inverse-square law determines that they must travel in ellipses, of which a circle is a special case - Newton's worked implied that the Earth wasn't, as was thought, a perfect sphere. Specifically, because of the opposition of forces, the planet must bulge slightly at the equator and be slightly flattened at the poles. 

There's a wonderfully convoluted tale of how a French team battled for nine years to try to measure this effect by conducting surveys in the Andes, only to be beaten to the result by another team that took measurements in Scandinavia, using essentially the same methods as Eratosthenes had two thousand years before but on a larger scale. This is discussed in Bill Bryson's marvellous A Short History of Nearly Everything

Anyhoo, they confirmed Newton's result, and measured that a degree of arc was indeed longer at higher latitudes, meaning that the planet is oblate.

Image source: NASA
Of course, there's a really straightforward way to show that the planet is a spheroid, but it's one the flat-Earthers love to paint as a conspiracy. Here it is.

This image was taken by the Apollo 11 crew  in 1969. The deniers, of course, will adamantly insist that the entire mission was a hoax, but this doesn't stack up for several reasons.

The first and most obvious reason is the Soviets. Recall that this mission was the culmination of an intensive race between the USA and the Soviet Union, at a time when their rivalry was reshaping the world (figuratively, of course; it was still an oblate spheroid). Remember that this race began immediately in the post-McCarthy era. The Soviets, who had every reason to undermine NASA's claim to having landed on the moon, tracked the entire mission themselves. Indeed, the Soviets had their own unmanned mission, Luna 15, running at the same time, with the Soviets and the Americans sharing information to ensure that they didn't collide. Luna 15 ultimately failed its mission, crashing into the lunar surface. However, the Soviets acknowledged the Americans' achievement, and the then Soviet President, Nikolay Podgorny, sent a telegram to President Nixon offering 'our congratulations and best wishes to the space pilots', acknowledging a success that, had it not actually happened, the Soviets would have had every reason to deny, even if only for propaganda purposes.

More importantly, both missions were independently tracked from the local observatory here, by the Lovell radio telescope, just a few miles from where I'm sitting, at Jodrell Bank.

Moreover, the Apollo astronauts left something behind that's still used to this day, and which put in a cameo appearance in season 3 of The Big Bang Theory. It was the Lunar Laser Ranging Retroreflector Array, which is the only experiment from the moon landings still running. It's a specially-designed panel of clever mirrors on a panel 2ft across.
Image source: NASA
These mirrors are designed to reflect light directly back to source with minimal scattering. We see examples of these all over the modern world; the reflective surfaces on road signs, the reflectors on bicycles, cat's eyes, etc. Indeed, the eyes of cats themselves are a natural retroreflector (in fact, all camera eyes are, and this is why, if your flash is too close to the lens on your camera, the result is that eyes come out glowing red like a demon). In the case of the LLRRA, they're used to precisely track the moon's orbit, and provide information about the moon's gradual recession from Earth, a function of transfer of angular momentum from the tides.

In short, there can be no doubt whatsoever that we live on an oblate spheroid. There are many other lines of evidence, but none of the above is explainable if we live on a disc, and that's even before we get into why the sky is red at sunset but blue when the sun is overhead, and other easily observable effects (this is a function of Rayleigh scattering, named for British physicist Lord Rayleigh, in which, because the photons are travelling through significantly more atmosphere, the shorter wavelength blue photons are more attenuated, leaving the longer wavelength red photons; see Give Us A Wave).

As we observed, another thing that Eratosthenes was responsible for was calculating the planet's axial tilt which, contrary to what one might hear on Faux News, is the real reason for the season. Indeed, it's the real reason for all the seasons.

Earth is tilted at approximately 23.5° from the vertical. If this were not the case, every day would have pretty much the same length regardless of latitude or time of year. Indeed, if the axis weren't tilted, it would have been considerably more difficult to even define a year. We'd still have been able to do it, of course, by noting the positions of stars, but even this would have been significantly more involved. The reasons for this are fairly simple. First, as the planet orbits the sun the axis always points in the same direction (this is why it always points roughly at Polaris, the North Star, regardless of the time of year), which means that, at different times of the year, the Northern and Southern Hemispheres alternate in terms of which is more directly pointed toward the sun. As the year progresses, the point on the planet's surface closest to the sun oscillates between the tropics. In the Northern Hemisphere, the Winter solstice occurs when the sun is directly over the Tropic of Capricorn. The summer solstice occurs when the sun is directly over the Tropic of Cancer. The spring and autumn equinoxes respectively occur when the equator is directly beneath the sun. This progression is what drives the seasons.

The other consequence of the axial tilt is that we see different constellations rising above the horizon at different times of the year and, of course, this is the basis for the Zodiac.

Moving on, what about the axial rotation?

Again, naïve intuition can certainly make it seem that we're sitting still while the universe whirls around above our heads, but there are massive problems with this. The most critical is that, in order for this situation to hold, one of our most fundamental laws would have to be breached. 

To put this into perspective, let's look at our closest galactic neighbour, the Andromeda galaxy. Andromeda sits a little over 2.5 million light-years away. Assuming, for the sake of simplicity, that Andromeda orbited Earth directly, in a circular orbit, we can apply a simple calculation to see the distance it would have to travel in a single orbit. We're all familiar, I hope, with the formula for the circumference of a circle, 2πr. A quick puff of chalk dust tells us that Andromeda would have to travel approximately 16 million light-years in a day. Now, light can travel about 16 billion miles in a day, give or take, which sounds a lot, but this is only 1/365th of a single light-year, and constitutes the maximum distance that anything can travel in a single 24-hour period (see The Idiot's Guide to Relativity). For anything with mass, of course, even that speed is impossible, according to special relativity.

We've already met one consequence of living on a rotating planet, in the form of the moon's recession due to conservation of angular momentum, but it's quite a subtle one. Let's look at a consequence that's more pronounced and apparent.

Image source: Wikipedia
There's a simple experiment we can carry out on Earth. This experiment, first conducted by Lèon Foucault, who we met in The Certainty of Uncertainty as the scientist who first gave an accurate measure of the speed of light, is among the most elegant experiments ever devised. It consists of a simple pendulum, now known as Foucault's Pendulum. The idea is straightforward enough to intuit with a little care. You set the pendulum swinging and, because of the rotation of the planet, the plane of swing of the pendulum precesses. In other words, the angle of swing changes throughout the course of the day. 
Image source: Wikipedia

Here's a simple diagram showing this in action. As you can clearly see, the bob is precessing as the planet moves around. This is actually a disc, so Foucault's Pendulum in and of itself doesn't demonstrate that the Earth is a sphere, but we've already done that. More importantly, a simple translation in the experiment means that the it can show that we live on a sphere. This might seem odd, especially for those who've been paying attention since the inception of this blog project, because we've been insisting that this shouldn't happen, because of Noether's Theorem. However, what's actually happening is that this translation has a direct impact on the experiment because it constitutes a change in the experimental setup that isn't merely a translation.

The amount of precession is a function of latitude. At the equator, the plane of swing remains fixed in relation to the Earth. As you move North from the equator, the precession increases, with the angular momentum proportional to the sine of the latitude. It reaches maximum at the pole, at which it rotates 360 ° in 24 hours. If we pick a point partway between, Paris, for example, where Foucault first conducted this experiment, the plane precesses at 264° every 24 hours. If you move South of the equator, the same thing occurs, except that the precession is in the opposite direction.

This experiment has been conducted at both poles, and there are many examples all over the planet at different latitudes and it behaves precisely as Foucault predicted. Although the deniers insist that the South Pole is a fiction, there's actually an example of this experiment as a permanent fixture there.

This is a manifestation of the Coriolis force. First formalised by Gaspard-Gustave de Coriolis in 1835, the Coriolis force is an inertial force (or pseudo-force, like centripetal and centrifugal force) arising from motion relative to a rotating frame. There are many manifestations of it, not least that airline pilots must correct for it when traversing latitude. Indeed, even snipers have to take it into account for shots over distance. 

Another beautiful - if deadly - manifestation of the Coriolis force is hurricanes. Because of this effect, hurricanes in the Northern Hemisphere always rotate anti-clockwise, while in the Southern Hemisphere, they always rotate clockwise (the effect on tornadoes tends to be negligible, the critical balance there being between pressure and centrifugal force).

I'm conscious that this is already a long read, so I'm just going to address a few quick arguments raised by flat-Earthers for completeness.

The first is one that was presented to me on Twitter a few weeks ago. The FE proponent in question showed a video of a rocket lifting off, with a camera mounted on the rocket looking down toward the Earth, and objected that, if the Earth were a rotating sphere, we should have been able to see the planet rotating sideways as it ascended. The reason we don't, while far from being simple, exactly, is at least trivial once you grasp the consequences of relativity. Specifically, the reason we don't see the ground moving sideways is the same reason your hat doesn't blow off when you're riding in a train, namely that the motion of the planet is already imparted to the rocket prior to lift-off. In simple terms, the rocket and the surface of the planet are both moving at precisely the same velocity. We know from Newton's Law of Inertia that a body will remain in its state of motion until a force acts upon it. When the rocket lifts off, it continues moving with the surface, as it was before, but now with a force applied upwards away from the surface. For the rocket not to remain above the point of lift-off would require the imparting of a sideways force to overcome inertia in that direction (they are affected by the Coriolis force, as are all fast-moving objects in rotating frames).

Another objection deals with space shuttles on re-entry, and suggests that they should experience cross-winds. Orbital altitude is a function of velocity, which means that for an object to remain in orbit at a certain altitude, it needs to keep moving at a certain speed. When the shuttle is re-entering the atmosphere, the first thing it does is execute a de-orbit burn, which slows it down until it meets the atmosphere. Once inside the atmosphere, the shuttle is within the rotating frame of the Earth, and we simply reverse the principle above for lift-off.

One final example of an objection is the idea that rockets can't thrust in space. Once again, we need only refer to Newton here, and his third law of motion, namely 'every action has an equal and opposite reaction'. A naïve view might suppose that a rocket needs something to push against, but this overlooks this law. There have been a few instances in science fiction films in which somebody becomes untethered from their craft, and the mission leader always tells them to throw away whatever they have in their hand, a spanner (wrench, for our transatlantic cousins) or some other tool is thrown in a direction away from the craft, which exerts a force equal and opposite on the thrower, propelling them back toward the craft. When rocket fuel is expelled from a rocket, the fuel exerts a force equal and opposite to the force of the expulsion. That's why rockets work in space.

De Chelonian Mobile - valé, Terry. You're missed.

Hope this was enjoyable. I'd wanted to deal with expanding Earth nonsense in this post, but I think this is long enough, so I'll address that in a future post.

Thanks for reading. As always, crits and nits welcome.

Edited to add:

I got confronted with a picture and a question, and thought I might as well tag it on here. Here's the picture.

The implication should be fairly obvious. The FE proponent wants to know why this doesn't happen with the planet's oceans if we're living on a spinning ball.

Of course, the answer is fairly straightforward to anybody with a tiny understanding of gravity and its relationship to mass. We haven't had an equation for a while, so here's a nice one: \[ v=\sqrt \frac {2Gm}r \] Where v is the velocity, G is the gravitational constant, m is the mass of the sphere and r is the radius of the sphere.

Taking a vaguely standard mass of 58.5 grams and radius of 3.17cm for the tennis ball, a quick calculation tells us that the escape velocity for the tennis ball is a massive 5.65cm/h.

The mass of the Earth, on the other hand, is 5.976x1024 kg, with a radius at the equator of 6378 km. Plugging those numbers in to the equation, we get an escape velocity of 11.2 km/s. The actual rate of rotation of the planet at the equator is approximately 465 m/s, which is less than 5% of escape velocity. 

That's why the oceans don't fly off into space.

A further edit: I was confronted with the following on Twitter and again thought I might as well tag it on the end here with my response:

*There's a famous physicist, Fritz Zwicky, who was fond of growling that people he didn't like were 'spherical bastards', because they were bastards no matter which direction you looked at them from.

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