August 2008 Archives

Astronomy 101 - Lesson 7 - Mapping the Sky

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Last week we saw how it makes sense to use angles and arcs to identify the position of places on the spherical surface of the Earth. We arrived at a grid of great circles, and the intersections between north-south and east-west circles define two coordinates for each point: a longitude (east or west) and a latitude (north or south). We also saw that two special circles mark the origin of each of the coordinates: the equator marks the 0° line for latitudes, and the prime meridian marks the 0° line for longitudes. The equator is physically defined as being equidistant from both poles, while the prime meridian is arbitrarily selected.

Well, as seen from Earth, the sky looks like a very big sphere with the Earth at its centre. Because of this, it makes sense to use a similar system of coordinates to identify the positions of celestial objects. Note that I'm talking strictly about the apparent position of the objects on the 2D "internal surface" of the sky as seen from Earth — the universe is a three-dimensional place, and you won't be able to use positions represented in this way to, say, chart a navigation course between any of these objects. Of course, if you are in a position in which you may be asked to chart such a course, I expect you'd already know this.

We start by extending some of the terrestrial features onto the sky. For example, the celestial poles are simply "above" the Earth poles: you extend the rotational axis of the Earth upwards until it "touches" the celestial sphere, and that's the celestial pole. And it turns out that these are important points in the sky as well as on the Earth: if you stay out one night and look at the movement of the stars as the night progresses, they all seem to rotate around the celestial poles.

Similarly, the celestial equator is simply a projection of our own familiar line of latitude zero onto the sky; it is a great circle on the sky that is equidistant from both poles. And, having poles and an equator, we can easily get the an equivalent to latitudes; celestial latitudes are called "declinations" and are represented in the same way as latitudes on Earth (as degrees north or south of the equator, from 0° at the equator to 90° at the poles).

Now, the celestial equivalent to longitudes is a bit more complicated. You see, the sky "moves"; that is, as seen from the ground, the sky seems to rotate around the Earth, from the east to the west, at approximately one revolution per day. This means that we can't simply project the terrestrial prime meridian onto the sky to get the celestial prime meridian: the portion of the sky onto which the projection would happen would change by the minute and the coordinates would be useless. We need to do something different.

Let's start talking about the idea of "local sky". From any given point on Earth, at any given moment an observer can see half of the sky (assuming a clear night and an open horizon); which half is something that will change as the Earth rotates. On this view of the sky, we need to define two important imaginary points. One is "straight up", or the point directly overhead (from the point of view of the observer); this we call "zenith". The other is in the opposite direction, "straight down", or the point directly below the observer's feet; this we call "nadir" (if you're standing at the South Pole, the zenith is the south celestial pole and the nadir is the north celestial pole).

Now, the portion of the sky you can see (that is, your local sky) will change continuously, and so will the celestial positions of your zenith and nadir (unless you happen to be at the poles). More exactly, the declination (celestial latitude) of your zenith and nadir will remain static (and will be identical to your latitude on Earth), while the "celestial longitude" will change. But, importantly, this means that what is in your local sky will depend not only on where you are, but also on when you are there - and not just on the time of day, but also on the time of year.

How do we define the celestial longitudes, then? They are, indeed, similar to the terrestrial ones, but we need to define an origin based on celestial features. The origin that was chosen for this coordinate is the line where the sun is at the time of the vernal equinox (spring equinox in the north hemisphere, autumn in the south); this is a fixed location on the sky.

There is one more difference, though: we don't use degrees for this coordinate, we use time. The sky was divided into 24 north-south bands, each one of which takes approximately one hour to move a distance equal to its width across the sky; these bands are then the second basis for the grid system defining celestial coordinates, and the coordinate they define is called "right ascension". Each band is numbered as 1 hour and, predictably, divided into 60 minutes of 60 seconds each; it is similar to degrees, but the notation uses "h", "m" and "s" rather than the symbols for degrees, arcminutes and arcseconds. Also, the hours were numbered from the origin increasing continuously towards the east. There is no east or west right ascension, but a single coordinate that ranges from 0h0m0s to 23h59m59s.

Together, declination and right ascension uniquely define a position in the sky, and can be used to locate any object visible from Earth. They don't tell you directly where to look at, though: they need to be translated according to your local position and to the time of day and year into an actual direction towards which you can look at to find the object being pointed to. But this is something that we'll look into a bit later, as we need more background to work that out.

Next week, we'll start looking into the way the Earth moves and what they mean for our view of the sky.

Astronomy 101 - Lesson 6 - Mapping Our World

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In the last two weeks we saw the process that led us to know that we live in a spherical planet, and also how we managed to discover how large this planet is. Of course, I could have simply told you that in one sentence - "the Earth is a sphere with a circumference of approximately 40,000km" - but actually seeing the methods people used to discover this is as important as knowing this, if not more: it tells us about the process of learning and of dealing with errors that all science needs to go through.

This week, let's take a look at how we can use what we know to help us locate places in the surface of the Earth, and later we'll see how whatever system we use for this can also be used on the sky.

In the ancient world, maps were intrinsically local items: they were centred on an important location -- usually the location where they were made, or alternatively the capital of the empire -- and would tell people the distances and directions to faraway places. This is, of course, only partially useful, since people rarely go from place to place in a straight line. And it grows less useful for larger distances, not only because there's less of a chance of going in a straight line, but also because, for sufficiently large distances, the curvature of the Earth starts to have a noticeable effect.

The need for better maps grew as empires grew larger and as it became harder to approximate the Earth to a flat plane where paths are straight lines. Our planet is, to a first approximation, a sphere, and this gives us a clue that circles, angles and arcs may be more important than planes and lines.

The standard way to measure arcs on a circle, inherited from the ancient Babylon, starts by dividing the circle into 360 equal parts called degrees (or degrees of arc, to avoid confusion with temperatures), denoted by °. One degree is a very small part of a circle: if you extend your arm and hold your thumb up, it will cover approximately one degree of your visual field. However, when you're dealing with a circle the size of the Earth one degree is not small enough; therefore, we need to subdivide it further. One degree is divided into 60 minutes of arc (denoted by '), and each minute of arc is further divided into 60 seconds of arc (denoted by "). An arc of 7 degrees, 25 minutes and 10 seconds is written as 7°25'10".

If these names and numbers (60 minutes, 60 seconds) sound familiar, you're right: that's the same way we divide time. The division of arcs came first, though, and the names of the units of time are derived from the names from the divisions of arcs, not the other way around. You may also ask what is the thing with multiples of 60. This is another thing we inherited from Babylon: 60 is a number that is exactly divisible by a whole lot of other numbers (2, 3, 4, 5, 6, 10, 12, 15, 20, 30) and this reduces the need to use complicated fractions.

Back to minutes and seconds. One second of arc is very, very small. In fact, one minute of arc is approximately the resolution of the human eye: you can tell objects apart if they are at least one minute of arc apart; it's likely that one pixel in the screen of your computer is larger than one minute or arc. One second of arc, or one arcsecond, is the limit of the resolution of optical telescopes, and it's also a unit we'll see much more of later on.

With this in mind, we can start using angles to locate position on the surface of the planet. The way to do this is to establish a series of arcs defining a grid on this surface. We start by doing two primary divisions:

  • first, the equator, which is a great circle dividing north from south located perpendicular to the line connecting the two poles; the position of the two poles is determined by the movement of the planet, as we'll see next week
  • second, a meridian, going through the poles and perpendicular to the Equator, dividing east from west

These divisions define a simple set of two coordinates, one north-south and the other east-west. Now, the equator has a well-defined location due to the fact that the poles are in fixed positions (as we'll see next week), but the meridian we choose as the basis of our coordinate system -- or the prime meridian -- can be basically anywhere. Since the 1880s, by international convention, it is defined as the meridian that goes through the Naval Observatory in Greenwich, just outside London, in England. This position was, not surprisingly, chosen by the English, and adopted internationally simply because the English made very good maps -- it was a de facto standard long before it was set by convention.

Using these two great circles, we can now define a grid that will allow us to define the position of anything on the surface of the Earth with ease: the position of a point is defined by two angles, or arcs:

  • one, known as longitude, is the distance from the prime meridian, to the east or west
  • the other, known as latitude, is the distance from the equator, north or south

This same set of coordinates can be extended to the sky, which also looks to an observer on Earth as a sphere seen from the inside. But we'll look into that next week.

As we saw last week, once it became clear to everyone that we live on a spherical Earth, the next question was "how big is it?". Since one can't well go out and measure it directly, indirect methods have to be used -- which brings us back to geometry. And the first person known to have used geometry to estimate the size of our planet was Erathostenes, who at the time was the librarian at the Library of Alexandria.

Erathostenes knew, thanks to travellers, that in the Egyptian city of Syene (currently Aswan) the Sun would be directly overhead at noon on the day of the summer solstice (which means to us that the city sits at the Tropic of Cancer). He also knew that this was not the case for Alexandria, where he lived, and this meant that he could use this difference to calculate the size of the globe.

What he did was to measure the angle of the rays of the Sun in Alexandria at noon on the summer solstice, and using his knowledge of geometry he knew that this was the same angle formed at the centre of the Earth by lines going from there to each of the two cities (Syene and Alexandria). This is a lot of words to explain what can be seen in the (greatly exagerated) figure; the red lines represent the rays of the Sun, each bold dark line represents one of the two cities, and both angles labelled as α are the same. He measured this angle to be 1/50 of a full circle (or 7°12'), and he knew the distance between the two cities to be 5,000 stadia. This, of course, gives us a size of 250,000 stadia for the circumference of the Earth.

And now we have a problem: how much is that, exactly? The answer is, we're not sure; no one knows how much was a stadion for Erathostenes. Archaeological evidence points to a length of 185m to a stadion in Athens, which would result in a circumference of 46,250km, or about 15% larger than reality -- 40,075.16 km. He might have been much closer, though: different places had different stadia, and some evidence points to smaller ones being used in Alexandria. Also, the numbers in his calculations (1/50 and 5,000) are suspiciously round; since we only know of Erathostenes' work through third-party accounts, it is likely that the numbers were rounded at some point and that his calculations were much more precise.

This calculation reinforced the idea that the Earth was fixed, as it was definitely too big to be moved; and, being fixed while everything else moved, it was also clear that it was the centre of the universe.

A few centuries later, in the 1st century AD, another eminent figure tried his hand at calculating the size of the Earth: Ptolemy. He worked using the positions of the stars and their apparent change in position as a person travels, and he reached a circumference of 33,300km. That is very wrong, but since his methods make sense and the results were reproducible, this value was accepted for a long time and was used by everyone until the Middle Ages.

Around the 3rd century AD, the idea of a flat Earth was revived, mostly by Christians who were intent on throwing away all of the "pagan" knowledge of the time. Among the general population, the idea of flat vs. spherical Earth had its ebbs and flows from them on, but among educated people there was never any doubt about the shape of the planet. In the 1300s, with the Renaissance, the spherical Earth stages a come back and this becomes common knowledge once more.

It is at this time that Ptolemy's underestimated numbers had an important effect in history: if we assume that the Earth is only as large as he thought it was, then the western tip of Europe is not that far away from the eastern tip of Asia, and this is one of the factors that inspired the westward navigation of the Portuguese and Spanish in the late 1400s, which resulted in the colonisation of the Americas. As you see, astronomy can indeed change the course of history, and we'll see more examples of this in the future.

That's it for this week. For more detail on this subject, I suggest reading the Wikipedia article on the History of Geodesy. Next week, we'll start to add detail to our map of the Earth and of the sky.

Partial lunar eclipse this weekend

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A partial lunar eclipse will grace our skies during this weekend and, while timing (and temperature) will certainly be less than ideal for Australian observers, it should provide a good spectacle for those brave ones who choose to go out.

In Melbourne, the eclipse will be cut short by the Moon setting (and the Sun rising) before it ends, but (weather permitting) one should be able to see it almost at its maximum. The umbral eclipse will start at 5:35am on Sunday (19:35 UTC on Saturday) and the maximum eclipse happens at 7:10am; infuriatingly, the Sun will rise at 7:02am and the Moon will set at 7:04am... but, if you live further west, you get to see more of it, and observers in Perth get to see the full umbral eclipse.

Still, if the weather is good and you can get a good, unobstructed view of the western horizon and you can get yourself out of bed at 5:30am on a Sunday, it will be a good show. If the weather is not that good, you can check a live webcast of the eclipse (as seen from Birmingham, UK) at the Lunar Observers website — assuming the weather there is good, of course.

Astronomy 101 - Lesson 4 - Spherical Earth

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Nowadays, everyone knows that the planet we live in is, to a first approximation, shaped like a sphere. However, from the earliest known writings, we see that most cultures saw Earth as being a flat and finite area. That is relatively easy to understand: in ancient times, very few people ever travelled more than a few kilometres from their place of birth; the distance any given person might travel was limited by how much he or she could walk, and that is not very much — especially when you don't really have a reason to move that much. And, from the point of view of a person moving very little, the world is indeed flat; the curvature of the Earth has no visible effect in a small scale.

Therefore, we have the representations of the world made by the early Mediterranean peoples showing that sea at the centre, the known European, African and Asian lands around it, and a very large ocean all around that. The South American Incas had a similar view, with themselves at the centre and land extending a certain distance in all directions, with a more or less rectangular shape and four corners. Egyptians also saw Earth as flat and surrounded by mountains on which the large tent that is the sky is grounded, and even the Bible offers a similar view. Other cultures had more metaphorical views, such as that of the disc of the Earth lying on the back of four elephants, who are on top of a giant turtle swimming on an infinite sea; it's unlikely that these representation were ever intended to be taken as literally true, however (unless you happen to live in Discworld).

The path that leads to our current understanding of the Earth starts, as so much else, with the ancient Greeks. Having pioneered the study of geometry, the Greeks had a love for symmetry and perfection and tended to try to impose this love on the natural world. Pythagoras, in the 6th century BC, was one of the earliest proponents of the idea that the Earth must be a sphere, as that is the most perfect shape for an object, and the Earth, being the place where man and the gods live, must adhere to this ideal of perfection. Of course, he had no evidence for that; this is purely an aesthetic argument. Plato, a century later, defended the same ideas for the same reason, but he was taken much more seriously than Pythagoras, and his ideas had a large influence on people of the time.

A few decades later, Aristotle defended once more the idea of an spherical Earth, but differently from his predecessors he had evidences supporting his idea. In his book "On the Heavens", he came up with three basic arguments:

  • one has a different view of the stars from places at the south than from places further north; some stars appear higher in the sky the further south you go, as if from places in the north your view of the sky is obscured by the curvature of the Earth

  • the shadow of the Earth projected on the Moon during a lunar eclipse is always round; if the Earth were anything but a sphere, you'd see different shapes for the shadow depending on the angle of the Sun at the time of the eclipse

  • and, since objects tend to fall towards the centre of the Earth, the force of attraction towards the centre would tend to cause the Earth to assume a spherical shape; this argument is incredibly advanced for its time, as it basically stands on the idea of gravity overcoming the tensile strength of the material forming the Earth to shape it like a sphere

By the 3rd century BC, then, everyone in the western world knew that the Earth was an sphere, fixed in the centre of the Universe, with several other spheres (planets, Moon, stars) rotating around it. This allowed people to ask the next question: how big is this sphere? Aristotle guessed that it was very large (because the effects of travel on the position of stars are very small), and this reinforced the idea the Earth was fixed; but how large, exactly, is it? We'll see about that next week.

Carnival of Space #66

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Another week, another carnival. The 66th edition of the Carnival of Space is now up at the Mars Odyssey blog. It has enough articles to keep you busy for a week!

But, in case you're a fast reader, this may be a good time to note that I forgot to mention issue #65, at 21st Century Waves, and issue #64 at Music of the Spheres. There, now go read and be happy!

Portal to the Universe

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2009 will be the International Year of Astronomy; many projects are under way to mark the occasion, and one of them is the Portal to the Universe, a web site that will be launched later this year and that intends to aggregate everything that is being said on the Internet about astronomy. One of the people involved in this project is Dr Pamela Gay, who blogs at Star Stryder and can be heard, with Fraser Cain, in the Astronomy Cast podcast.

And this is where you come in: if you publish any type of web content related to astronomy, Dr Gay wants to hear about you. If you are involved with any type of astronomy club, centre, observatory etc.), she also wants to hear about you. Read about it in her blog post, get in contact with her and make yourself heard. Also, help propagate this message by posting about it in your content, whatever it happens to be.

By the way, Dr Gay and Fraser Cain have just posted (a few days ago) the 100th edition of their podcast. If you haven't listened to it yet, now is a good time to start. Congratulations to the Astronomy Cast team!

Astronomy 101 - Lesson 3 - The Night Sky

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The sky at nightAfter the basic introductory material of the first two weeks, it's time to start looking at the main subject of astronomy: the sky. If you go outside at night in a location with reasonably dark skies and you stay up all night, you will be able to see about 6,000 stars. That may look like a lot, but it's not: our constellation, the Milky Way, contains approximately 200 billion stars, and it is only one of billions and billions of galaxies in the visible universe. 6,000 is really not that much.

What we do see are only the nearest stars to our own, plus a few very bright distant ones. We also see other types of objects: for example, planets, easily recognised because they will change locations from day to day against the background of stars. In dark skies you can also see other galaxies: Andromeda, our nearest large neighbour, can be seen easily with the naked eye as a fuzzy blob, and the Large and Small Magellanic Clouds, two dwarf galaxies which were once thought to be orbiting the Milky Way, can be seen very clearly from southern hemisphere locations. And, if you look to the skies for a while, you will almost certainly see artificial satellites moving very quickly as they circle the Earth, and very likely the odd meteor will show up from time to time.

[caption id="attachment_140" align="aligncenter" width="418" caption="Tridimensional visualisation of the stars in the constellation Orion"]Tridimensional visualisation of the stars in the constellation Orion[/caption]

One interesting aspect of the human brain is its ability — or need — to identify patterns in the world around it. That is the reason why we aggregate the stars into constellations: they are nothing more than patterns formed by connecting the "dots" in the sky. It is very important to know that the stars in any given constellation are not physically together; the pattern they form is an artefact of "projecting" a 3D universe into an apparent 2D sky. If you were to move a large distance from the Earth, the patterns in the sky would be significantly different from the ones we know.

In general, the constellations are figures formed by the brightest stars we can see; some do look like what they're supposed to represent (for example, Scorpius (pictured, right) or Orion), but some require a significant degree of imagination (such as Pisces or Lynx). Every culture on Earth invented its own set of constellations; and, as the sky we see today is not significantly different from that of hundreds or thousands of years ago, many of those are similar, but many more are entirely different. Some cultures (notably the Incas and the Australian aborigines) even recognised "constellations" that were in fact dark areas of the sky.

Our current set of 88 official constellations owes much to Ptolemy, who defined an original set of 48 constellations covering the North sky (which was all he could see), many of which survived to this day. When European explorers started moving south of the Equator, they started inventing new constellations with the new stars they saw. This brought a new set of themes to the sky; mythical themes still abounded, but some biblical terms also made their way onto the sky during these period, as did some pieces of modern scientific equipment (Horologium, the clock, is one example). At some point, over 150 constellations were known and widely used, with some overlap happening between them.

In the late 1800s, the organisation that was to become the International Astronomical Union defined the current set of 88 constellations, including their well-defined boundaries. This allows us to refer to objects that are not part of the figure of a constellation as still belonging to it if they fall within its boundary. Also, the whole sky is covered by these 88 constellations.

As a result of this, many constellations were abandoned and forgotten; my personal favorite is Felis, the cat. Others, such as Argo Navis, the ship of the Argonauts, were split into several (in this case, Carina, the keel, Puppis, the poop, and Vela, the sails).

It's worth noting that most cultures also had individual names for the brightest stars in the sky, and that most of those stars are still referred to by name. Most of the surviving names are Arabic (Rigel, Aldebaran, Deneb...), but Greek (Sirius, Arcturus...) and Latin (Polaris, Spica...) are also common.

For unnamed stars, the most commonly used designation is the Bayer designation, which uses a letter of the Greek alphabet followed by the Latin name of the constellation to which the star belongs, in the genitive form; letters are assigned in order of brightness, alpha (α) being the brightest. Most people have heard of the star "Alpha Centauri", and now you know that this refers to the brightest star in the constellation Centaurus, the centaur.

However, the strict order of brightness is not always followed for historical reasons; when the letters were originally assigned (by the German astronomer Johann Bayer, in the early 1600s) there was no way of accurately measuring the brightness, so constellations with similarly bright stars may have them in the wrong order. Orion is a particularly good example of this, with its beta (β) star (Rigel) being the brightest and its delta (δ), Mintaka, being the sixth, rather than fourth, brightest.

This concludes our first look at the universe outside our planet. Starting from the next week, we will spend quite some time closer to home, looking at the Earth, the Moon and the rest of the solar system before venturing out towards the stars.

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