In a surprising result the internet is responsible for some amazing discoveries!

Distance from Dublin to San Francisco

Distance is 8194 kilometers or 5091 miles or 4424 nautical miles

The distance is the theoretical air distance (great circle distance). Flying between the two locations can be longer or shorter, depending on airport location and actual route chosen.

http://www.timeanddate.com/worldclock/distanceresult.html?p1=78&p2=224

Distance from Dublin to Johannesburg

Distance is 9412 kilometers or 5848 miles or 5082 nautical miles

http://www.timeanddate.com/worldclock/distanceresult.html?p1=78&p2=111

The Earth: Shape, Size, and State of Rotation

The Shape of the Earth

It is common mythology that until the time of Columbus "everyone thought that the world was flat." The falsity of this is shown by a figure I presented in class, a representation of a real globe which represented the state of European knowledge before Columbus sailed. Although Columbus -- and everyone else! -- was right about the shape, however, he was quite lucky in a different respect. He thought that he would have to cross a western ocean of only moderate size before reaching China. Of course, if the Americas had not been in the way, he would have faced an impossibly long journey of more than 15,000 miles! So he was wrong about the size, but not about the shape. (By the way, I have read speculations to the effect that Columbus purposely underestimated the likely length of the crossing in order to persuade Ferdinand and Isabella of Spain to fund his daring expedition.)

You may be surprised to learn that a correct understanding of the Earth's shape was current even two millennia before Columbus. Moreover, as we will see, even the size had been pretty well estimated, although that bit of information did not get reliably handed down. But how did such things come to be known?

As usual, we begin by asking how you yourself would demonstrate that the surface of the Earth is curved. If the world were much smaller, of course, it would be easy. Suppose for instance you were standing on top of a gigantic beach ball, looking at the surface on which you stand. You would certainly see a conspicuous "dropoff" on all sides. Unfortunately, the Earth is much too big to show that effect.

Still, there are alternatives. Indeed, by combining arguments of the following sort, Aristotle was able to conclude quite persuasively that the Earth is indeed spherical. You could:

Travel long distances north-south and watch the stars. Suppose you see a star low on the southern horizon, and then walk or sail many hundred miles south. The star will now appear to be higher in the sky, well above the horizon, as Aristotle knew from the accounts of sailors. If the Earth were flat, this would not happen; a star which was low on the horizon would remain so as you travelled.

(You have to observe the star at the same time of night, of course, rather than at some earlier hour when the star has not yet risen into the sky. The diurnal effects of rotation have to be correctly compensated for.)
To take an extreme example, a person on the equator sees the North Star low on the northern horizon. As she travels north, the Pole Star is seen at higher altitude until, when she is right at the North Pole, it is overhead. For the same reasons, people living in Australia see constellations invisible from here in Canada (and vice versa: they cannot see the Big Dipper).
Watch a boat sail away from shore. Of course, it gets smaller in appearance by virtue of its increasing distance, but it also seems to disappear over the horizon as though it is going down the far side of a hill. The hull disappears first, and soon only the tops of the masts and flags are visible.

This would not happen on a flat Earth; the ancients recognized and correctly explained the phenomenon as due to the curvature of the Earth.

Rely on analogy. There is a semi-philosophical argument based on the fact that the sun and moon are seen to be round. It might seem plausible, therefore, to suggest that the Earth must be spherical as well, on the grounds that this seems to be a favoured shape in the heavens. (This is actually a weak argument, of course: the sun and moon are `up there' in the heavens, while the Earth is `down here' and may be something quite different.)

Watch eclipses: One can notice, as did Aristotle, that whenever there is a lunar eclipse, the edge of the Earth's shadow (which we see projected onto the lunar surface) always looks like the arc of a perfect circle. (By the way, the Earth's shadow is not responsible for the changing phases of the moon, as we will learn in the next section. Only during eclipses, which are very rare, do we notice the effects of the Earth's shadow at all - or at least that part of it which is projected onto the Moon's surface.)

Aristotle concluded, rightly, that the inevitable circularity of the shadow is because the Earth itself is a sphere, reasoning that a sphere is the only shape that always casts a circular shadow. It is certainly true that a flat object (like a coin) can cast a circular shadow, if correctly oriented, but you would expect at least sometimes to see a shadow which looks flattened or ellipsoidal.

For all four of these reasons, Aristotle (probably quoting other thinkers as well) concluded, rightly, that the Earth is a large nearly-spherical ball. This knowledge was not lost in the succeeding millennia, despite the mythology surrounding the expeditions of Christopher Columbus. You should not necessarily believe everything you read!-- except for what I am writing here, of course.

The Size of the Earth

How would you measure the size of the Earth? The obvious way is to travel all the way around it, keeping track of how many miles you have covered. But why go that far? Why not travel, say, a quarter of the way around it, and multiply your answer by four? (If you need to put a fence around a square field, you need to pace out only one side of the field to find out what length of fencing you need in total.)

The problem is to work out when you have covered exactly a quarter of the Earth's circumference, but there is in fact a very simple way! All you have to do is start at the Earth's equator (the place from which the sun shines from directly overhead on March 21), and then travel North until the Pole Star is directly overhead, proving that you are right 'on top of the world.' When you first set off on your journey, the Pole Star is right down on your northern horizon; your journey of 10,000 kilometers changes that and proves to you that the Earth is about 40,000 km in circumference.

In other words, you can tell from the different altitude of various stars how far North or South you are on the globe. Although the principle is quite clear-cut, the notion of travelling thousands of kilometers is a bit daunting, but there are a couple of practical simplifications one can imagine adopting:

don't travel quite so far; and
use a particularly conspicuous star as a reference object to make the observations easy.

Indeed, to make the measurements really straightforward, let's observe the sun itself! - it's by far the brightest star in the sky. In a modern analogy, we are relying on the obvious fact that if we travel south (say, by flying down to Miami in December) we will see the sun at higher elevation (and lie out on the beach to sun-bathe). By quantifying that effect, we will determine the size of the Earth.

The procedure is quite straightforward (and some of you may even wish to try this experiment if you travel south during the Winter Break):

at one location on the globe, perhaps here in Kingston, work out how high the sun gets in the sky at the middle of the day, using the shadow of a vertical stick in the ground to measure the angles; and
from some other location due South of the first, make the equivalent determination on the same day (or as soon thereafter as can possibly be managed). Note that you must make the two observations as close together in time as possible because the sun's altitude changes significantly as the seasons progress, even at a fixed location, thanks to our orbit around the sun. (For this reason, using a prominent bright star is better, since its altitude will not vary with the seasons and it won't matter if you get held up a few days or weeks until you make the second set of observations. The trouble is that no star is bright enough to cast shadows. Still, there are other ways of measuring angles.) Then:
later, at your leisure, measure the distance between the two locations. Comparing the angles measured at the two sites tells you how far around the globe they are from one another, as a fraction of the full circle, and combining this with the distance between them in kilometers gives you the size of the Earth itself.

This is the logic which went into the important calculations made by Eratosthenes, as described on page 88 of your text. He knew that on June 22 the light of the Sun shone straight down a well at mid-day in the city of Syene, implying that the Sun was directly overhead then. Meanwhile, in Alexandria, which is about 500 miles north of Syene, a vertical stick cast a shadow at mid-day, which he accounted for by realising that the Earth was curved. (The stick is made vertical by comparing it to a plumb bob or hanging weight, so that it points straight down to the centre of the Earth.)

Eratosthenes hired people to pace out the distance, and in this way he was able to get a very good value for the size of the Earth -- long before Columbus! As it happens, he made a few small mistakes which more or less cancelled out. For instance, the distance which was paced out was not very precise, and Syene is not due south of Alexandria. But the cleverness of the idea is undeniable.

Degrees of Correctness

At this stage, I would like to emphasise a thought which will recur in other contexts later. Eratosthenes was roughly correct about the size of the Earth -- at least, that's what modern historians of science have concluded, although there is apparently some lingering doubt about the exact size of the units of measurement which he was using. But to me it would not matter a great deal if he had gotten an answer which was only half as large as the true circumference, or three times as big. The critical point is that Eratosthenes recognised the nature of the problem, found a method, and was able to derive an answer which was correct in spirit in the sense that he correctly deduced that the Earth was an immense body which was very much larger in extent than the then-known lands of the Mediterranean basin, the home of Greek civilisation at the time. The sense of the discovery is the wonderful thing, not the mere accident that the numerical value was also correct.

The Rotation of the Earth

The ancients assumed, naturally enough, that the Earth is at rest, with the whole cosmos rotating about us once a day, just as it appears. Part of the motivation was no doubt religious. They must have felt that we are in a divinely ordained place, at the centre of the universe, with everything else dancing around us in attendance. But there is also the practical consideration that you might expect to feel the motion if it is true that this large solid body is spinning like a top. Shouldn't it rumble and bounce like a fast-moving chariot, for example? And would we not feel a wind blowing in our faces if we were truly being carried towards the East in some unceasing diurnal motion? (Of course, we now recognize that the atmosphere is simply being carried along with the rotational motion.)

You may be surprised to learn that there are direct ways of proving that the Earth rotates - that it is we who are spinning, rather than the cosmos around us - and that it would still be possible for you to measure the Earth's rotation even if we were completely shrouded in clouds so that you could never see out to the stars and sun! (These are the objects which indicate to us that something is moving.) Even the ancients could, in principle, have reached the correct conclusion, although it requires a fairly deep physics understanding and, in one case, the ability to make some fairly difficult measurements.

How is this possible? There are three methods:

Study the "figure" of the Earth:

The Earth is not a perfect sphere: it is about 25 miles (40 km) thicker across the middle than it is from North pole to South pole. This arises from the rotation of the Earth (just as a thrown piece of spinning pizza dough flattens out). Despite its obvious rigidity, at least here at the surface, the Earth's materials are elastic enough [remember our discussion in earlier lectures] to deform under the influence of the spin. If you somehow stopped the spin, the Earth would relax into a perfectly spherical shape under its own enormous self-gravity. The material would not have enough structural rigidity to maintain the body of the Earth in its present flattened shape.
By the way, Jupiter is gaseous rather than rocky, and consequently much less rigid in the outer parts. It also rotates very fast, once every 10 hours. As a consequence, it is noticeably flattened, even to the eye when seen through a telescope. By contrast, the Earth looks perfectly spherical, whether seen from space or as a scale model under casual inspection. The flattening is simply not very pronounced, but careful surveying of the Earth does show it.
This would have been a hard test for the ancients to employ. They would have needed to carry out elaborate and precise surveying over very large distances - not easy. Moreover, they would have needed a deep understanding of the strengths of materials and so on, so as to interpret the discovery of the Earth's flattened figure. Little wonder that they did not make the discovery!
Study a Foucault Pendulum:
Imagine constructing a pendulum at the North Pole, hanging from a free-turning "universal joint". With a good push, set the pendulum swinging in a particular direction - say, towards the star Sirius. As the day progresses after that initial push, the Earth turns "underneath" the pendulum, which continues to swing back and forth in the same direction in space since there is no sideways force to make it turn. In effect, the "platform" on which you are standing (the Earth itself) turns beneath the pendulum. From your point of view, however, the pendulum's swing appears to be shifting in direction, one full rotation in every twenty-four hours.
The behaviour of a Foucault pendulum is slightly more complex at lower latitudes, such as here at Kingston, but the principle is the same. Indeed we have one in the middle of Stirling Hall! As an experiment, take a look at it before a lecture one day, taking note of the direction in which the pendulum is swinging; then look again before the next lecture a couple of days later. You will see that the direction has indeed changed from our point of view.
Note that the pendulum has to be suspended on a special universal joint so that it is free to turn (or rather not to turn: the hypothetical pendulum at the North Pole keeps the same orientation in space). If I put my child into a swing and give her a big push to start, so that she is swinging (say) in the direction of Lake Ontario, I will not find her swinging parallel to the shoreline a few hours later! Her swing is mounted at a couple of points on a rigid framework which turns with the Earth, which therefore carries the swing with it.
Study `Coriolis Forces.'
Visualise yourself standing on a moving "slidewalk," such as you find in an airport terminal. Now imagine tossing something (a set of keys, say) to a friend standing beside the slidewalk. The sideways motion imparted to you by the slidewalk, coupled with your outward toss, carries the keys at an angle so that you will miss your friend unless you `lead' him with the toss.
In like fashion, objects on the Earth close to the equator are being carried to the East at about 1000 miles per hour (the Earth is about 25,000 miles in circumference, and spins once around every 24 hours: see the figure on page 48). Things farther north are moving less rapidly, and something at the North Pole doesn't move through any distance at all - it merely turns around, on the spot, once a day.
Now, suppose someone fires a rocket from the equator towards a target due north of them (say, the people of Ecuador decide to attack Ottawa). The rocket would seem to veer off to the East, just as the keys missed our friend. This is a real effect which has to be considered by rocketry experts and even by naval gunners shooting at targets tens of miles away. This effect is a manifestation of the so-called `Coriolis force.' (See pages 286-289 of your text.)
These considerations explain the familiar spiral of hurricanes. If a big low-pressure region forms in the Earth's atmosphere somewhere just north of the equator, you might expect the air around it simply to move directly into the low-pressure center. But the air moving up from the equator has an excess component of sideways motion compared to the air which was originally located on the north side of the new-formed hurricane. This makes the air spiral or swirl in, in a counter-clockwise direction. In the southern hemisphere, hurricanes spiral the other way!
By the way, this effect is much too feeble to affect day-to-day small-scale phenomena like water flowing out of sinks and tubs, despite the fact that it is commonly believed that "sinks drain the opposite direction in the Southern hemisphere." In fact, the way sinks drain out (or bathtubs empty, or toilets flush) depends on their shape, how you pull the plug, etc. It is true that in the Northern hemisphere the water on the south side of the tub is moving eastward faster than the water on the north side, but the difference is extremely small. Very carefully controlled experiments, with specially designed basins, can show the effect, but in real life the 'Coriolis forces' arising from the Earth's rotation are not important - unless you live in hurricane country, or expect a naval bombardment!

rebelcounty

Tuesday, August 08, 2006