The Whabblog

Fuzzy Patches in the Sky: The Proof of the Pudding!

2010-12-27T17:23:00.000-08:00

“And God said, “Let there be lights in the vault of the sky to separate the day from the night, and let them serve as signs to mark sacred times, and days and years, ¹⁵ and let them be lights in the vault of the sky to give light on the earth.” And it was so. ¹⁶ God made two great lights—the greater light to govern the day and the lesser light to govern the night. He also made the stars. ¹⁷ God set them in the vault of the sky to give light on the earth, ¹⁸ to govern the day and the night, and to separate light from darkness. And God saw that it was good. ¹⁹ And there was evening, and there was morning—the fourth day”

- - Genesis, Chapter 14.

In previous blogs in this series, trying to reconcile the creation story of Genesis with the limited speed of light left us scratching our heads. According to Genesis, God created the stars to light up the night on the fourth day of the seven days that He took to create the universe. The purpose of creating the stars, as you just read in Chapter 14, was to provide light during the night. In a single day, though, the light generated by creating even the nearest star, Alpha Centauri, would cover only about 1/1500 of the distance to the Earth, so even Alpha Centauri wouldn’t have appeared in the sky for four years after the fourth day. All the other stars in the sky would have taken longer – some over a thousand years longer – to add their luster. If God’s purpose in creating the stars was to light up the night on the Fourth Day, it is difficult to understand why He placed them much too far away to achieve His goal.

But we also discovered that the stars aren’t the only sources of light in the night sky. Take the fuzzy patch of light in the Southern Hemisphere known as the Small Magellanic Cloud (SMC). At the beginning of the last century, astronomers were able to use Henrietta Leavitt’s discovery of Cepheid Variables in the SMC to determine, for the first time, that the SMC lies at least 30,000 light years away (close to 200,000 light years distant by modern calculations). In a single day, therefore, light from the SMC covers only about one seventy-three millionth of the distance to the Earth. If the SMC was created, along with the rest of the Universe, some 6,000 years ago, there is no way the cloud would have been visible to Ferdinand Magellan and his band of explorers in the early 1500’s; no way it (and its Cepheids) would have showed up on Henrietta Leavitt’s photographic plates in the first decade of the 20^th century, and no way it would be visible today. We would be completely ignorant of SMC’s presence until about 194,000 A.D., when its light would finally arrive!

The fact that the SMC (and its neighbor, the Large Magellanic Cloud) has been visible throughout recorded history is proof that the universe is at least 200,000 years old, a far cry from 6,000. But the story of Cepheid Variables, and the headaches they create for Genesis, doesn’t end with the SMC; not by a long shot. Almost 600 years before Ferdinand Magellan set out on his voyage of discovery and saw the Magellanic clouds that would come to bear his name, a Persian astronomer named Abd al-Rahman al-Sufi left a diagram suggesting that he had noticed the existence of another mysterious patch of light, this one in the constellation Andromeda, the maiden. Shown in the left figure below, the diagram includes a maiden, Andromeda, a fish, and a mysterious collection of dots just to the right of the fish’s nose. This is the first record of anybody actually noticing the mysterious smudge of light that is now known as the Andromeda Galaxy. If you have clear skies tonight, and live in an area with minimal light pollution, this smudge will be visible to you, too. As shown in the figure below and to the right, find the Great Square of Pegasus in the northwest about an hour after sundown. Count two stars over and then look up a little to the location indicated by the red arrow. The fuzzy patch you see is the focus of today’s blog.

By the beginning of the 20^th century, centuries of observations with previous generations of telescopes had revealed many other wispy patches just like the Andromeda Galaxy. With no way to determine their distances, though, these “nebula” had become the focus of a battle royal in astronomical circles: Were they some sort of mysterious agglomerations of matter inside our own galaxy, solar systems in the making perhaps, or were they other galaxies, far further away than our Milky Way, but otherwise resembling our own?

Solving this thorny question would fall to no other than Edwin Hubble, the famous American astronomer that the Hubble Space Telescope is named after (we’ll see presently one of the primary reasons why that naming was so appropriate). In the 1920’s, Hubble was fortunate enough to be one of the first astronomers to utilize a marvelous new telescope on Mount Wilson in Southern California, the first in the world to have a primary mirror more than 100 inches (8 feet!) in diameter. It was Hubble who realized that this telescope would be powerful enough to actually resolve the brightest stars in the Andromeda galaxy. Luckily, Cepheid Variables are supergiant stars that have moved off the “main sequence” – a concept I’ll be glad to clarify in the comment section, if anyone’s interested – and have swollen to a gigantic size as a result. Even the dimmest Cepheids outshine the Sun by a considerable margin, and the very brightest, those with the longest periods, are 100,000 times as bright. Hubble knew that if he could find Cepheids in the Andromeda Nebula, he could measure the distance to it, and put an end to the raging debate once and for all.

The figure below is an example of one of the photographs of the Andromeda “nebula” taken by Hubble with his new gigantic telescope. And lo and behold, as you can see from the label in the center, Hubble was successful in his quest; some of the individual stars in these photos did indeed demonstrate the periodic brightening and dimming cycles that marked them as Cepheids! Carrying out the usual computations based on comparing the brightness of these Cepheids as measured in the photographs against their true intrinsic brightness as determined by their location along the period/luminosity function, Hubble announced his determination to the world: The Andromeda “Nebula” was actually 600,000 light years away! At that enormous distance, it couldn’t be a part of our galaxy, but instead must be a galaxy in it’s own right, a magnificent “island universe” like the Milky Way containing billions and billions of stars.

The mystery of the nebulas was solved, and in one magnificent step, Edwin Hubble was able to provide humanity with a true perspective on the size of the cosmos. The Earth wasn’t the center of the universe; it was located in a spur of a spiral arm of a galaxy named the Milky Way, an insignificant spec of the universe surrounded by untold numbers of other galaxies in every direction, all at unimaginable distances from our own. The modern value for the distance to the Andromeda galaxy is 2.5 million light years.

The fact that you can go out tonight and actually SEE the Andromeda galaxy, when its light takes a full two and a half million years to reach us, starts to reveal just how completely nonsensical the timeline for the creation of the universe in Genesis really is. And the fun doesn’t stop there. Hubble was able to resolve Cepheids in many other galaxies, some as far as 10 million light years distant. As he analyzed the light from the unimaginably distant stars in these galaxies, he noticed another quality of their light that would profoundly change our understanding of the history of the universe and how it all began.

But that’s a topic for another blog!

The Cloud in God's Eye

2010-06-29T21:57:00.000-07:00

Henrietta Leavitt published her finding that the intrinsic (the “real”) brightness of a Cepheid variable was strongly related to its period (the duration of its fluctuation cycle) in 1912. It didn’t take long for astronomers to realize the important implications of this relation. If they could just figure out the actual distance to even one Cepheid (obviously, that would have to be a cepheid closer to our neck of the woods than the collection Henrietta discovered in the Small Magellanic Cloud), they could calculate the distance to the SMC using the inverse square law!

How? Remember a couple of blogs ago, when I explained the inverse square law using a pair of identical candles? The two conditions needed to made the law work were, one, you had to know the actual distance to one member of the pair (in my example, it was the distance to the nearer candle), and two, both candles had to be identical, so they would shine equally brightly if placed at the same distance from you. That way, after your friend put the second candle some unknown distance away, you could be confident that the only source of their (now) differing brightness was their unequal distance.

With those two factors in mind, consider our old friend Delta Cep. Long before Henrietta Leavitt turned her attention to the Cepheids in the SMC, Delta Cep’s period had been determined to be 5.4 days, and its apparent magnitude (brightness) had been pegged at 4.3 at the dimmest point in its cycle (by the way, that’s about mid-way between the brightest and dimmest stars visible to the naked eye. You can find Delta Cep yourself in the circumpolar constellation Cepheus if you consult a star chart and live in an area with reasonably dark skies). Today, we know from parallax measurements with the Hubble Space Telescope that Delta Cep is about 891 light years away. Courtesy of Leavitt’s discovery, any Cepheid whose period is equal to Delta Cep’s has the same intrinsic brightness; a pair of such stars would be like identical candles. If an astronomer from 1912 had known Delta Cep’s distance, then using the Period-Luminosity functions from Leavitt’s SMC sample, he or she could have easily calculated how bright a Cepheid with Delta Cep’s period would be if it was located in the SMC (or, if you like, the brightness of Delta Cep itself if it were in the SMC).

The figure below illustrates how this would have been done. First, you would have located the point along the X-axis that corresponds to the log of Delta Cep’s 5.4 day period (about the 0.5 point). Then, you’d have drawn a straight line up from the 0.5 point to the lower of the two period-luminosity plots inside the figure (that’s the one that plots the Cepheids in the SMC at the dimmest point in their cycles). Next, you’d have drawn a horizontal line from the plot across to the Y-axis (the horizontal red line in the figure), and simply read the magnitude where the line intersects that axis. When I do that, just by eye I get an apparent magnitude of approximately 15.8.

Finally, all the conditions would be met to employ the inverse square law! The difference between an object of magnitude 15.8 and an object of magnitude 4.3 corresponds to a difference in brightness of 11.5 units along the magnitude scale. In turn, that difference translates into the fact that Delta Cep’s putative twin in the SMC would be roughly 33,750 times dimmer than the real Delta Cep (the magnitude scale astronomers use to measure brightness is another log-based scale, so it’s highly non-linear). The inverse square law tells us that brightness falls off with the square of the distance, so Delta Cep’s twin would be about 184 times as far away as Delta Cep (squaring 184 yields approximately 33,750). Finally, since Delta Cep is itself 891 light years away, its twin (and therefore the SMC itself) must be 184 multiplied by 891, or 164,000 light years distant - a figure within respectable shouting distance of the modern estimate, which is around 210,000 light years.

Of course, I cheated. Astronomers in 1912 didn’t know the distance to Delta Cep, so they couldn’t compute that actual distance to the SMC. They would have had to be content with the knowledge that, however far away Delta Cep was, the SMC was some 184 times further away than that. A pretty strong clue, you’d have to admit, that some of the objects in the night sky were very, very far away, indeed.

But astronomers of the time wanted more: they wanted the actual distance to the SMC! Energized by the prospect, a contemporary of Henrietta Leavitt’s, Earnest Hertzsprung, set out to measure the distances to known Cepheids within our galaxy using several distance-measuring techniques that I won’t get into here. Although none of these techniques was as accurate as the trigonometric parallax method, Hertzsprung still managed to derive crude distance measurements to not one, but 13 Cepheids.

Those distances gave Hertzsprung the tools he needed to make a preliminary determination of the distance to the SMC using the method described above. The value he obtained, 30,000 light years, was far short of the modern value of over 200,000 light years. But even Hertzsprung’s gross underestimation was large enough to deal another devastating blow to the Biblical story of Creation. According to literalist biblical scholars, the universe is only about 6000 years old. Furthermore, according to Genesis, all the sources of light in the night sky were created in a single 24-hour period – the fourth day after God initiated the Creation process (and there are records of peoples in the Southern Hemisphere having seen the SMC as long as thousands of years ago, so it was definitely present in the sky during biblical times). But if the SMC was created only 6000 years ago, the light from the cloud has not had time to cross more than a small fraction of the distance between it and the Earth, even with Hertzsprung’s gross underestimate. Given our current understanding of the true distance to the SMC, the cloud could not appear in our skies until after 200,000 AD! That’s so far in the future I can’t even imagine it.

Is there an alternative way to save the biblical story? In one of the earlier blogs in this series, I considered a rather unlikely account for how the literal account of Creation could be reconciled with the fact that even the nearest stars are so far away that their light takes years to reach the Earth. On the fourth day of creation, God would have had to place those stars within the confines of our own solar system (so that their light had a chance to make it to the Earth before the fourth night descended), and then He would have had to whisk them out to their current positions. While God would have had to hurry some of these stars along pretty fast – at 26 light years away, a star like Vega would take longer than 6000 years to reach its present position even if it was moving as fast as our fastest jetliners – our spacecraft travel much faster than that, so I don’t think God would have found it a problem.

But even this rather far-fetched account would fail miserably for the Small Magellanic Cloud. 6000 years is not nearly enough time to transport the SMC even 30,000 light years away unless God was moving all the stars in the cloud far faster than the speed of light! Not only would that violate all known laws of physics, but it would once again have rendered the Small Magellanic Cloud invisible; its light could not reach us if it was moving away at more than the speed that light travels!

Unfortunately for those who believe in the literal interpretation of Genesis, we’re not finished with the biblical devastation wrought by Cepheid variables. In the next installment, we’ll tackle some Cepheid-fueled discoveries of an astronomer named Edwin Hubble. You may have heard of him, or at least of the telescope named after him.

You'll love it at Leavitt's!

2010-06-24T20:14:00.000-07:00

By the beginning of the 20^th century, progress in determining the distances to the stars had slowed to a crawl. The trigonometric parallax method had pretty much pooped out. Evidence was growing that the stars vary greatly in absolute (intrinsic) brightness, precluding any direct application of the inverse square law.

Fortunately, it is always darkest just before the dawn. The combination of two advances, one technological and one scientific, was about to provide astronomers with a new and very powerful distance measurement tool. The technological development was the newly acquired capability to take photographs of telescopic images. Among other benefits, “astrophotography” would take much of the guesswork out of determining the precise position of a star relative to other stars in its immediate vicinity, breathing new life the trigonometric parallax method and revealing the distances to many additional stars. Meanwhile, the scientific discovery in question built on a much earlier discovery, made soon after Galileo first turned his telescope to the heavens, that some stars fluctuate in brightness on a fixed, predictable schedule, like Old Faithful.

Here’s the story. In 1794, a young English astronomer named John Goodricke identified a distinct subclass of these “variable” stars that would eventually be called “Cepheid” variables. Although Goodricke had no way of knowing it at the time, Cepheids are giant yellow stars, much brighter and more massive than the Sun. In addition to being large and luminous, Cepheids display a very distinct fluctuation pattern. The figure below this paragraph shows the pattern for “Delta Cephei”, (or “Delta Cep” for short), one of the few Cepheids actually visible to the naked eye (the North Star, Polaris, is another). Like all Cepheids, Delta Cep brightens quickly, dims more slowly, and then brightens quickly again, yielding a distinctive “shark fin” shape to the plot of the star’s change in brightness over time (if anyone is interested in why Cepheids behave this way, I’ll be happy to explain it in the comment section).

Delta Cep fluctuates on a very regular schedule, completing one full cycle every 5.4 days; this is known as Delta Cep’s period. In common with Delta Cep, all Cepheids fluctuate on very regular schedules. However, different Cepheids have very different fluctuation periods that range from about one day (for the shortest period Cepheids) to over four months (for the longest). In the figure below, I show the fluctuation periods for three actual Cepheids that range from a single day to over thirty days. As you can plainly see, the characteristic “shark fin” shape is present even for the longest period Cepheid, just greatly stretched out in time.

Regardless of their fluctuation period, Cepheids are quite uncommon. By 1900, only about 30 were known, none close enough to measure a parallax angle and determine its distance. Back in 1891, however, Harvard College had deployed a 24-inch telescope on a mountaintop in Peru – the first large aperture telescope to be deployed in the Southern Hemisphere. From 1893 to 1906, this instrument was used to take hundreds of photographs of the Small Magellanic Cloud (hereafter, the “SMC”), a prominent southern sky object named after Ferdinand Magellan, whose crew brought it to the attention of Europeans following their around-the-world voyage in 1519. As shown in the left-hand photograph below, the SMC looks like a small fuzzy patch to the naked eye. The right-hand image, taken through a telescope, reveals the “cloud” for what it really is: a “dwarf galaxy” containing many millions of stars!

The telescope in Peru was big enough to resolve the SMC into individual stars. Exploiting this fact, in 1905 an extremely diligent Harvard astronomer named Henrietta Leavitt began a long-term project to scrutinize the SMC photographs in order to identify variable stars. After several years of painstaking effort, Leavitt found about a thousand variables, 25 of which pulsed with the distinctive shark-tooth shape that marked them as Cepheids. The shortest-period member of the 25 had a fluctuation period of about one day; the longest-period member, about 128 days.

Next, Henrietta proceeded to measure the apparent (observed) brightness of each SMC Cepheid right at the top and the bottom of their fluctuation cycles (i.e., at the brightest and dimmest points of their respective cycles). The range turned out to be quite substantial, with the brightest member of the 25 being thousands of times more luminous than the dimmest. Henrietta surmised, correctly as it turned out, that the distance to the SMC was so great that any further differences in distance between her Cepheids and the Earth must be quite inconsequential; to all intents and purposes, they were equally far away. In turn, this meant that the large differences in apparent brightness among the Cepheids in her sample had to reflect actual differences in their “real” or intrinsic brightness – as opposed to an artifact of their being at different distances from Earth.

Henrietta’s finding that Cepheids differ greatly in intrinsic brightness gave tremendous importance to her next discovery, which came about when she plotted the fluctuation period of each of the 25 Cepheids (X axis) against their apparent brightness at both the top and bottom of their fluctuation cycle (Y axis). I’ve taken the liberty of reproducing the actual plot from Henrietta’s scientific paper on this topic below.

For both the top and bottom plots, the key thing to notice is the very strong relation between each Cepheid’s period and its intrinsic brightness: the longer the Cepheid’s period, the brighter it is. This “period-luminosity” relation showed up even more clearly in the next graph, which shows what happened when Henrietta re-plotted the relation after converting each Cepheid’s period to log-linear coordinates (this just means that she took each Cepheid’s period in days and determined to what exponent the number 10 would have to be raised to equal that number. For example, a Cepheid with a period of exactly 10 days would have a value of “1” in her log coordinate system; a Cepheid with a period of 100 days would have a value of “2”, and so on). Now, the “period-luminosity” relation was revealed to be so strong that if you fit a straight-line function to the dots, as Leavitt did in the figure, all 25 Cepheids fell very close to the line at both the minimum and maximum points of their brightness cycles.

I’m going to end this blog with an explanation for another aspect of these graphs that might be puzzling. Although I’ve explained the log scale along the X-axis, you might still be confused on the other (Y) axis by the fact that the numbers (the brightness scale) go from higher at the origin (the bottom) to lower at the top. This means that the brighter, long-period Cepheids on the right of the period-luminosity function are associated with a lower number on the brightness scale than the dimmer, lower-period Cepheids down and to the left.

Why is the number lower, when I just finished noting that the longer-period Cepheids on the right side of the graph were brighter than the shorter-period Cepheids on the left? The blame belongs to the ancient Greeks, who classified all the stars they could see in the night sky according to their perceived importance. They declared the very brightest stars, like Sirius, of “first” (“most important”) magnitude. Stars of the second magnitude were dimmer, and therefore less important, and so on, all the way down to stars of the sixth magnitude, so dim that they were barely visible (and therefore least important). Later astronomers adopted this magnitude scale, assigning bright stars (like long-period Cepheids) to lower magnitudes than dimmer stars (like short-period Cepheids). And while we’re on the topic of magnitude, it’s worth noting that the ancient Greeks included only stars in their magnitude scale, not brighter objects like Venus, the Moon, and the Sun. When later astronomers began to add these objects to the scale, they had to go to smaller (negative) numbers to convey their apparent brightness (that is, the brightness as measured from here on Earth). For example, when at its most dazzling, Venus shines with an apparent magnitude of -4.5; The Sun shines with an apparent magnitude of about -27. On the other end of the scale, stars that are way too dim to be seen in anything but telescopes have apparent magnitudes that extend all the way “up” to +30. As you can see from Henrietta Leavitt’s plots, the Cepheids in the SMC had apparent magnitudes all the way up in the mid teens, though they were far too faint to be seen with the naked eye. So the rule of thumb is: The lower the magnitude, the brighter the star.

In the next blog, we’ll continue the discussion of Leavitt’s period-luminosity function and what astronomers could do with it.

Breaking the Inverse Square Law

2010-06-14T23:51:00.000-07:00

As we discussed in the last blog (tragically, a full month ago now), it took over a century of dedicated effort on the part of several generations of astronomers before Friedrich Bessel finally succeeded in measuring the distance to a nearby star, 61 Cygni. Right on his heels, contemporaries of Bessel’s reported similar success with measuring the distance to two additional stars, Alpha Centauri (the closest star of all) and Sirius, the brightest star. As we discovered, the enormous magnitude of these distances, none less than 25 trillion miles, combined with the finite speed of light, spelled serious trouble for the Genesis account of creation.

In fact, though, much, much greater trouble for Genesis lay ahead. I’m going to begin to tell the story of that trouble in this blog. Before proceeding, though, I want apologize for the long delay since the last installment. In the course of writing this series, I came to realize that, by providing a little more historical background and detail than first envisioned, I could kill two birds with one stone, and close another gap in most people’s knowledge of astronomy. We’ve all become used to the stunning photographs provided by the big telescopes of our day, most notably the Hubble Space Telescope. But have you ever wondered how astronomers actually exploit the crispness of these images to do science; that is, how they use telescopes to answer questions and roll back the boundaries of our ignorance? The story I’m about to tell provides a very concrete example of one way that astronomers use the power of their telescopes to unlock the secrets of the cosmos.

So let’s get to it. Encouraged by the initial success of Bessel and his colleagues in the late 1830’s, astronomers spent the better part of the next 60 years trying to measure parallax angles for a vast number of other stars, including some of the brightest in the sky. Believe it or not, in all that time they managed to obtain reliable angles for only a handful of additional stars, none more than 26 light years away. The problem was the Earth’s atmosphere, which causes the images of the stars to jump around constantly. Small thought these jumps are, they prevented astronomers of the time from making precise enough positional determinations to also measure the apparent movement of the stars as the earth travelled from one side of the Sun to the other. We can only imagine scientists’ frustration with this state of affairs, since their telescopes had long since revealed that the heavens were absolutely teeming with millions upon millions of stars, most too faint to be seen with the naked eye. Would there never be a way to measure how far away they are?

Clearly, an alternative was needed to the trigonometric parallax method. When we look up at the night sky, one of the most obvious features of the stars is how much they differ in brightness. A few, like Sirius, blaze like little beacons. Others are just barely visible. On the straightforward assumption that bright stars are closer to us than dim stars, couldn’t astronomers use a star’s brightness to determine its distance?

Mathematically, the answer is a straightforward “yes”! The relation between the brightness of an object and how far away it is follows a really simple relation known as the inverse square law. To get a feeling for this law, suppose you took two identical candles, placed one 10 feet away from you and the other 20 feet away from you, and lit them both. How much brighter would the nearer candle be than the farther candle?

Intuition might suggest that the more distant candle, being twice as far away, would be half as bright. Not so, however. The geometry of the situation dictates that exactly four times as much light would reach your eye from the nearer candle than the farther candle, so the farther candle would be only 1/4 as bright. Note that the denominator of that fraction, 4, is 2 times itself, or 2 squared. Next, suppose you measure the brightness of the more distant candle after moving it an additional 10 feet away, so it is now 30 feet away from you, or three times as far away as the nearer candle. Your measurement would reveal that it is only 1/9 as bright (3 squared). At risk of belaboring the point, if you moved the more distant candle out another 10 feet, making it 40 feet away (4 times the distance of the nearer candle), it would appear only 1/16 as bright (4 squared). And so on.

Now suppose you had a friend position the second candle instead of yourself, so that you didn’t actually know how far away it was. Could you use the inverse square law to compute its distance? Easily! For example, suppose you measured that candle’s brightness to be only 1/25 the brightness of the nearby candle. Since 5 squared equals 25, you would know that the more distant candle is five times as far away as the nearer candle, or 50 feet. It’s a classic example of the power of a simple mathematical relation to solve for an important unknown!

The inverse square law applies equally well to stars as to candles. If two stars shine with the same “intrinsic” (astronomers call it “absolute”) brightness, and one star is four times as far away as the other, the more distant star will be 1/16 as bright. As long as you know the distance to the nearer star, calculating the distance to the further one is trivial.

Now, back to the beginning of the 20^th century. Fired up by the inverse square law, an astronomer of that period might have measured the brightness of a target star that had not yielded a measurable parallax angle (and whose distance was therefore undetermined), and compare that brightness to the brightness of a star of known distance, such as Sirius, which is about 8 light years distant. Suppose he found that his target star was 1/16 as bright as Sirius. The inverse square law would tell him that his star was four times as far away, or 32 light years from Earth.

Unfortunately, applying the inverse square law is only that straightforward if the two stars whose brightness is being compared have the same intrinsic brightness, like two identical candles. In our stellar example, the target star would have to have the same absolute brightness as Sirius. The problem is, stars do not all shine with equal brightness; they range from thousands of times dimmer than the Sun to a million times brighter. Is our target star 1/16 as bright as Sirius because it is actually much brighter than Sirius, but also much further away than 32 light years? Or is it 1/16 as bright because it is the same distance as Sirius (or even closer) but intrinsically much dimmer? Without additional information about the star, you just can’t tell.

Might there be a way to get around this problem and still utilize the inverse square law? In the next blog, we’ll find out the answer.

Of Parallax and Paradox

2010-05-13T22:34:00.000-07:00

And the evening and the morning were the third day.

14 And God said, Let there be lights in the firmament of the heaven to divide the day from the night; and let them be for signs, and for seasons, and for days, and years;

15 And let them be for lights in the firmament of the heaven to give light upon the earth; it was so.

16 And God made two great lights; the greater light to rule the day and the lesser light to rule the night; he made the stars also.

17 And God set them in the firmament of the heaven to give light upon the earth,

18 And to rule over the day and over the night, and to divide the light from the darkness; and God saw that it was good.

19 And the evening and the morning were the fourth day.

Genesis, Chapter 1, verses 13-19

Last week, in part two of the current series, I told the story of how astronomer Ole Roemer first discovered that light does not move from one place to another instantaneously, and how he made his initial calculation of the actual speed that light travels. In retrospect, I should perhaps have given more emphasis to the many additional experiments that have been conducted to measure the speed of light in the centuries since Roemer did it. These measurements, involving ever-more accurate devices and improved experimental techniques, are the reason that we have such an exact value (299,792.458 kilometers per second) today. Some discussion of these experiments would have established, beyond a shadow of a doubt, that 299,792.458 kilometers per second is not a “hypothesis” or a “theory” about the behavior of light that is subject to debate, or could be overturned in some future experiment. 299,792. 458 kilometers per second is a concrete fact, as real as the hand in front of your face.

The importance of the fact that light’s speed (physicists call it “c”) is a far cry from infinite will become clear a little later on. Right now, I want to begin a different topic with a simple question: In all the times in your life that you’ve been outside at night, and looking up at the planets and the stars, have you ever wondered how, and when, people began to figure out how far away they are? Determining their distances certainly wasn’t easy. For the vast majority of the time that we humans have been around, we lacked the right perspective to even frame the question correctly. That’s because, just like the Sun appears to rotate around the earth each day (an illusion which made it devilishly difficult to understand that it is really the Earth that is rotating), the night sky looks like a two dimensional surface, at a fixed distance from the Earth, with little lights plastered across its surface. Imprisoned, constrained, and completely fooled by this compelling illusion, the ancients proposed all sorts of colorful beliefs about the night sky, all of which were premised on the assumption that the heavenly lights were all the same fixed distance away. One of my personal favorites held that a black curtain surrounds the earth, and the stars are holes in the cloth through which the dazzling brilliance of heaven is shining through.

It was the Greeks who started to part the curtain by employing a straightforward mathematical tool called trigonometric parallax. To get a quick and dirty feeling for the tool without the messy mathematical details, all you need to do is carry out the following simple exercise. Position yourself directly in front of your television at a comfortable distance away. Close your left eye. Raise your index finger of your right hand straight up in the air. Position your index finger directly in front of, and about six inches away from, your nose. Make sure your finger is located directly in front of the middle of your television screen.

Next, without moving your finger, open your left eye and close your right eye simultaneously. You’ll see your finger “jump” quite a ways to the right, maybe even all the way off your TV screen (if your screen is small enough, and you are far enough away). Now, repeat the exercise, but extend your arm all the way so that your right index finger is as far as it can possibly get from your nose. You will still see your finger “jump” when you switch eyes, but the distance covered by the jump will be noticeably reduced, small enough that your finger probably stays well inside the boundary of the TV screen.

The change in the position of your finger with respect to the background TV is simply due to the fact that you are looking at your finger (and the more distant TV screen) from slightly different positions when you switch eyes. The closer your finger is to your eyes, the greater the difference those two positions will make, and the more your finger will appear to jump against the screen. This is where the mathematics comes in. If you know the exact distance between your right eye and your left eye (the two viewing points), and you know exactly how far your finger appeared to move against the TV screen in the background, you can use simple high school trigonometry, invented by the Greeks, to calculate exactly how far your finger is from your eyes.

What’s the take-home message from our little example? As long as you can observe a foreground object (like your finger) from two separate locations (like your left and right eyes), and you have background objects much further away from you than the foreground object, then by measuring the amount that the foreground object appears to shift against the background, when observed from the two locations, you can recover the crucial third dimension, and determine exactly how far away the foreground object is from you.

In the thousands of years since this “trigonometric parallax” method was discovered, it has been used routinely to solve all kinds of useful problems, like determining how far a ship is from the shore. Military types used to need that information in order to figure out how to fire their cannons to hit enemy vessels. Not to be outdone, both the ancient Greeks and the first generation of astronomers to have access to telescopes used trigonometric parallax to calculate of the distance to objects within our own solar system. For example, at about the same time that Ole Roemer was making his first observations of Io, Jupiter’s moon, Italian astronomer Giovanni Cassini was making careful observations of where the planet Mars was located with respect to nearby background stars from his location in Paris. Moreover, Cassini had a research assistant make the same measurements from Equatorial Guinea, several thousand kilometers away (see the figure below). Through trigonometric parallax, Cassini determined a distance to Mars that was only 7% off of the currently accepted value, which we get far more directly by bouncing radar signals off the planet and measuring how long they take to return.

Does trigonometric parallax even work as a tool to measure the distance to the stars? Certainly not by placing observers at a known distance from each other on the Earth’s surface, the way Cassini did for Mars. The shift in the star’s location against a background of more distant stars would be far too tiny to measure. Every six months, however, the Earth itself moves in its orbit from one side of the Sun to the other, covering about 186 million miles in the process. Perhaps that was a displacement large enough for the trigonometric parallax method to work? Starting around the beginning of the 18^th century, astronomers began to pick candidate stars and, with the help of telescopes, note their exact positions, relative to other stars in their vicinity on, say, Dec 21^st, and then again on June 21^st, six months later. The method is illustrated in the figure below. If one of the target stars was observed to move against the other stars in its vicinity, the movement had to be due to the fact that the foreground star was being observed from two different locations, like it was being observed from two eyes, except the two locations were separated by those 186 million miles! If the size of the shift in position could be measured reliably, astronomers would have all the information needed to calculate the distance to the foreground object (the target star).

Simple, right? Well, yeah, except that for the longest time, no matter what star they selected as the target, the poor astronomers couldn’t measure any shift. Luckily for us all (all, that is, except those who cling to the literal truth of Genesis) failure was not an option, and by the early 19^th century, several astronomers were competing to be the first to measure stellar parallaxes and compute stellar distances. Finally, in 1838, an anal German scientist (is there any other sort?) named Friedrich Bessel won the race, and reported a reliable parallax shift for 61 Cygni (a relatively unassuming star in the constellation Cygnus). Bessel had had good reason to suspect that this particular star might be a relatively near neighbor to the Sun (how Bessel made that determination is interesting in its own right, but beyond the scope of this blog), and was therefore a good candidate to reveal a parallax. That parallax was exceedingly tiny, though, so small that when Bessel computed the distance to 61 Cygni, it came out to a jaw-dropping 52 trillion miles.

Suddenly, everything clicked into place. As more stars had their distances determined, it became clear that the trigonometric parallax method had failed for so long simply because the stars are so mind-numbingly distant, which means that the amount they appear to shift was exceedingly tiny. Although other stars were discovered to be closer than 61 Cygni, none were found any closer than Alpha Centauri, at 26 trillion miles away.

Time to connect some dots. At a speed of almost 300,000 kilometers per second, how long does it take light to cross the space between the stars and reach our eyes and our telescopes? In the case of even the closest star, Alpha Centauri, the answer is 4.2 years. In the case of 61 Cygni, the answer is closer to 10 light years. For other familiar stars, like Vega, we’re talking more like 26 years.

According to the Genesis version of creation, these stars all popped into existence in the course of a single day, the fourth after creation, and were lighting up the night sky by the time the fourth day came to an end. But if we take into account the speed of light, and the distance of these stars from the Earth, this is flatly impossible. On the assumption that the stars were approximately the same distance from the Earth 6000 years ago as they were when Bessel and his colleagues first measured their parallaxes (and, to a first order of approximation, that is the case), then their light had just gotten started on its journey to Earth on the day they were created, and would not have reached the Earth for years. At risk of belaboring this point, the night sky would have been completely bereft of stars for the first 4.2 years, and then one single, solidarity point of light, the star we now call Alpha Centauri, would have winked on. It would have taken the entire lifetime of Adam and Eve, for instance, for all the familiar stars in our sky to gradually become visible.

All right, you’re saying. God has a lot of tricks up His sleeve. Maybe he got around the pesky little speed-of-light problem by pulling a fast one on us. Maybe, when He created the stars that would be visible to Adam and Eve in the night sky, He initially placed them all well inside the confines of our own solar system, close enough that they would all be visible on the evening of the fourth day (after all, he had up to 24 hours of wiggle room, so he could put them as far as 16 billion miles out in space, way past the orbit of Pluto, and their light would still have reached the earth before His deadline). Then, maybe God employed some rather adroit stellar dynamics to whisk these stars out of the solar system fast enough that they would reach their present gargantuan distances in time for Bessel and his colleagues to measure them, 6000 years later.

Fair enough. After all, God is omnipotent, so there is really nothing beyond his power to accomplish if He sets his mind to it. But even this incredible kluge cannot save Genesis, because, embedded in the speed of light and the distances to celestial objects lurks a far more profound challenge to the story. That challenge will be the subject of the next blog. Until then, would anybody like to try and trump me and hazard a guess as to what it is?

Science and Religion at the Speed of Light

2010-05-06T23:16:00.001-07:00

In the last blog, I promised to demolish the Genesis account of creation without referencing Darwin, evolution, or anything to do with Earth’s geology. Instead, I’m going to build my case with ironclad facts from the fields of physics and astronomy, facts that have been established over the course of hundreds of years of painstaking inquiry into the nature of physical reality.

Although no one knew it at the time, Genesis began to unravel the very first time somebody looked at the planet Jupiter through a telescope. As most of you know, the person in question was Galileo, and the time was the beginning of the 17^th century. To his great astonishment, Galileo observed four points of light surrounding the giant planet. Observing them over time, he noted that the positions of the lights changed in ways that made it obvious that they were satellites, orbiting Jupiter in the same way that the planets orbit the Sun.

One of these lights, Jupiter’s moon Io, is so close to Jupiter that it completes a full orbit in only a couple of days, going behind Jupiter for a brief period each time (in astronomy-speak, Jupiter “occults” Io). Later on in the 17th century, the prominent Danish astronomer Ole Roemer systematically recorded the timing of these occultations over an extended period stretching from 1671 to 1677. Combining his observations with those of some of his contemporaries, Roemer discovered a remarkably systematic pattern: the time between occultations gets steadily shorter as the Earth’s own orbital motion brings us closer to Jupiter, and then lengthens again as the Earth moves farther away. Reporting his results to the French Academy of Sciences, Roemer hypothesized that

This… appears to be due to light taking some time to reach us from the satellite; light seems to take about ten to eleven minutes [to cross] a distance equal to the half-diameter of the terrestrial orbit

Clever fellow that he was, Roemer combined this time difference with early estimates of the “half-diameter of the terrestrial orbit” (the distance from the Earth to the Sun, from which he could easily calculate the difference in distance between the Earth and Jupiter when they are closest together compared to when they are furthest apart) to compute the speed of light for the first time (prior to that point, many prominent scientists, Newton included, believed the speed of light was infinite).

The value Roemer obtained, about 211, 000 kilometers per second, is considerably lower than the currently accepted value of 299,792.458 kilometers per second. That’s because the radius of the Earth’s orbit wasn’t yet known with any precision in his time. Still, Roemer’s calculation was in the right ballpark, and the realization that the speed of light had a fixed value, instead of being either immeasurably high or infinite, marked a major advance in the history of science and our understanding of the natural world. For example, we’ve already seen how fundamental a role the finite speed of light plays in generating time dilation effects. In subsequent blogs, we’ll discover just how big a nail the speed of light was to drive in the coffin of the biblical story of creation. But first, we have to identify another important nail, in the form of the distance between the stars and us. That is the topic of the next blog.

In the beginning was the Word

2010-04-29T23:52:00.000-07:00

The Holy Bible was written by men divinely inspired and is God's revelation of Himself to man. It is a perfect treasure of divine instruction. It has God for its author, salvation for its end, and truth, without any mixture of error, for its matter. Therefore, all Scripture is totally true and trustworthy. – from the Basic Beliefs of the Southern Baptist Church, currently 16 million members strong

At the core of most major religions is the belief that the universe has not existed forever. Instead, it was created by a higher power at some specific point in the past, usually to provide a backdrop or platform for humanity to inhabit. Interestingly, arguably the most profound scientific discovery of the 20^th century was that the Universe has, in fact, not been around forever. About 13.7 billion years ago, by current best estimates, the whole enchilada started out as an almost infinitely hot and infinitely dense speck, so tiny that even our most powerful microscopes couldn’t have seen it. From there, the speck started to expand and cool, a process that is still going on today. This “big bang” account leaves the question of how (and, if you insist, why) the universe came into being completely open. The Catholic Church, for example, sees in the big bang the undeniable handiwork of God the Creator.

The church may be right. In the absence of any evidence one way or the other, divine intervention has just much right to a seat at the table as any other conjecture. Thus, in concert with those of a religious persuasion, I’m quite open to the possibility that the Universe had a creator. However, there is one particular religious belief concerning creation that I could not disagree more violently with: the specific details of the creation account as described in the Old Testament. Actually, to say I “disagree with” the biblical version of creation doesn’t do my position justice. The Genesis version of the Universe’s creation can’t possibly be correct. It’s just flat-out wrong. It’s untrue. The sooner that all religions and denominations embrace the Catholic notion that Genesis should be taken figuratively, not literally, the better off all of us will be, believers and non-believers alike.

This embrace cannot come fast enough for me. As recently as 2008, a Gallup poll found that fully 44% of US adults agreed with the statement "God created human beings pretty much in their present form at one time within the last 10,000 years or so”. We can assume that a similar percentage would have agreed with the statement “God created the heavens and the Earth sometime in the last 10,000 years” as well, had this question also been asked.

So what does Genesis actually say? Very briefly, Chapter 1 asserts that the heavens, the earth, and all living creatures, including the first humans, were created in the space of six days. Assuming that the subsequent course of human history, as laid out in the Bible, proceeded uninterrupted from that week, various Biblical scholars have computed the time that has elapsed since that monumental week as about 6000 years.

In this latest series of blogs, I’m going to confront this Biblical account of creation head on. But I’m not going to go about it the standard way, which is to champion Darwinian evolutionary theory over the Genesis story of how humans came to be, or to lob geological arguments at the Young Earth Creation (YEC) movement, whose members assert that most of the geological evidence for an old Earth is instead the product of a relatively recent event - Noah’s flood. These conventional ways of framing the issue might well be futile anyway. In the words of the YECs themselves,

“We further deny that scientific hypotheses about Earth history may properly be used to overturn the teaching of Scripture on creation and the flood”.

So what kind of argument am I going to mount instead? I’m afraid you’ll have to check out the next blog to begin to find out!

The Difference a Day Makes

2010-04-12T20:57:00.000-07:00

Last time, I explained how the movement of the Earth around the Sun creates the illusion that the Sun is moving through the zodiacal constellations. There’s just one loose end to tie up. At sunset, which is, as we’ve discovered, a good time to visual local motions, the Earth is moving down and to the right of the Sun; that is, the Earth is moving in the direction of a point below the western horizon. That Earthly motion is what pushes the Sun constantly up and to the left. Below, I’ve included the first figure from the last blog just to reinforce this point. See how the Earth’s movement pushes it continuously to the right of the Sun? And how that motion shifts the Sun continuously leftward?

This “cause and effect” is something to be savored, partly because you can think of it happening at any time of the day or night. For example, remember from the “Beatles” blog that after sunset, the Earth’s rotation “pulls” the Sun along that line below your feet towards the Eastern horizon, on a course that will culminate in sunrise? The Earth doesn’t stop orbiting the Sun just because it’s nighttime, of course, so throughout the night the Earth’s movement is constantly pushing the Sun back and to the left. This has the effect of slowing the Sun’s forward progress to the eastern horizon, slightly delaying sunrise. In other words, the Earth’s movement makes the nights just a little bit longer than if the Earth was stationary. Similarly, the constant forcing of the Sun to the left during the day works to slow down the Sun’s forward motion then too, stretching out the length of the day. The impact is small, adding only about two minutes of length to both day and night. Still, it’s interesting to sit back and consider that each 24-hour day would be about four minutes shorter than we measure it to be if the Earth wasn’t moving around the Sun.

Now, let’s get back to how the Earth’s movement impacts our view of the stars. I’ll ask you to once again imagine you’re out and looking up right at sunset, when the sun is barely above the western horizon. If, every day, the sun is drifting slowly leftward, then every successive day that the sun sets, the stars that, the night before, were just far enough left of the Sun to be visible for a brief period just after sundown, before they rotated out of view, will now be swallowed up in the Sun’s glare, and disappear completely from the evening sky. Meanwhile, in the morning sky, prior to sunrise, just the opposite will happen. Since the Sun is drifting leftward, there’s a little sliver of sky visible just before dawn that was swallowed up in the Sun’s glare the night before. Thus, in the early morning hours, just before sunrise, the Earth’s movement giveth; in the early evening, it taketh away.

On the scale of a single day, these effects are miniscule. Over time, though, the evening losses and the morning gains add up until, over the course of a month, an entirely new zodiacal constellation becomes visible in the pre-dawn sky, while another zodiacal constellation disappears from the sky after sunset. In any six-month period, these effects are enough to completely alter the pattern of stars that are visible overnight.

And that’s really all there is to it. We’re done with our treatment of local motion, and I will wish you a brief adieu while I work on the next topic. Until next time, Whabbloggers!

Your Sign... or Mine?

2010-04-08T20:07:00.000-07:00

All right. In yesterday’s blog, I promised to talk about astrology, so let’s start with the most common form of astrological knowledge: your “sign”. Literally everybody knows what their sign is, right? Mine happens to be Cancer; yours is most likely one of the others, perhaps Leo, Virgo, or Libra; maybe Scorpio, Sagittarius, or Capricorn; possibly Aquarius, Pisces, Aries, Taurus, or Gemini. It is also common knowledge that our signs are dictated by the date of our birth, with each of the 12 signs roughly aligned with one of the 12 months of the calendar year.

Astrological signs are so deeply embedded in popular culture that horoscopes run everyday in major newspapers, and knowledge of one’s sign is often considered an important form of personal information (“Hi! “I’m a Cancer. What’s your sign”)? But what are astrological signs, really? Where do they come from? Why are they called “Sun” signs”? To answer these answers, I’m going to ask you to look at some familiar “landmarks”, and some familiar locations, from a new and different perspective.

If you happen to look up at the stars tomorrow night, one of the first things that will strike your eye is the fact that the stars seem to form recognizable patterns, called constellations. Each astrological sign is named after a particular constellation. People born this month are Aries, because their astrological sign corresponds to the constellation of the same name. So, when you look at the sky in April, are you able to see Aries? The answer is a most emphatic “No”! Right now, Aries is as invisible as astronomical objects ever get!

The diagram below tells the story. It depicts where the Earth is in its orbit at roughly this time of year (the illustration says “The Earth in May” instead of "The Earth in April" because a slight discrepancy has emerged over the centuries between the real position of the sun against the stars, as shown in this figure, and the designation of where the sun is in astrological terms. I'm deliberately glossing over this discrepancy for purposes of this blog). If you draw a straight line between the Earth, and Sun, and the sky beyond, the arrow points directly at Aries. That’s why you can’t see it. Right now, the Sun and Aries occupy the same location in the sky. They rise and set together, so Aries is not visible at night.

Once the sun sets, however, other astrological (or more precisely, zodiacal) constellations ARE visible. Near the Western horizon are the constellations Taurus and Gemini. Cancer is small and not very prominent, but the big spring winner of the zodiacal constellation sweepstakes, Leo, is very prominent; you can identify Leo by the stars that are arranged in the shape of a mirror-image question mark. The other zodiacal constellations visible early tonight, Virgo and Libra, are further to the east, and harder to spot because they are lower in the sky. But, if you wait until later on, and give the Earth a chance to rotate further east, the more familiar summer constellations of Scorpio and Sagittarius will rotate into view.

But wait, you may be saying. I’ve just named the astrological signs for birthdays that come later on this summer and fall. If astrological signs are defined by which constellation the sun is aligned with that month, and the sun is in Aries right now, it must have to be shift position to align with these other constellations over the course of the next few months. That is, the sun must be moving constantly along an imaginary line that connects these constellations.

Well, it is certainly the case that the sun appears to move along such a line. The figure above tells the real story, though. It is really the Earth that is moving along its orbital track. Every day, we shift position to the right along that imaginary line by some 1.6 million miles. This movement forces the line extending from the Earth to the Sun and beyond to pivot leftward. As you can see, the effect is to “push” the Sun through the constellations that happen to be lined up with the Sun’s position in the sky: that’s what defines a zodiacal constellation. By next month, May, the Earth will have moved far enough to co-locate the Sun with the stars that make up the constellation Taurus, the Bull, which is why May babies have Taurus as their Sun sign.

Simple, right? In the figure below, we’ve fast-forwarded exactly half the year, to early November. In that amount of time, the Earth has shifted an enormous distance, enough to push the Sun all the way to the position in the sky that’s occupied by Libra. Then, the Sun “enters” Scorpio, the sign of you mid-November babies (again, the astrological dates for Libras and Scorpios are somewhat earlier than the dates would be if the astrological dates were aligned with the actual present-day position of the Sun in relation to the zodiacal constellations).

That’s pretty much the story, Whabbloggers, at least for astrology. In the third blog of this series, I’ll return to the issue of how the apparent movement of the Sun through the zodiacal constellations changes the stars we see in the night sky. Would you care to steal my thunder by speculating on that in the comments section?

Doing the Local Motion

2010-04-07T22:14:00.000-07:00

Last month, we discussed how the appearance and behavior of the Sun, the Moon, and Venus can be explained through the combined effects of various forms of “local” motion: the Earth and Moon rotating about their axes; the Moon revolving around the Earth; and Venus revolving around the Sun. As you know, our own planet is in constant motion too. Every year, the Earth completes a 600 million-mile journey around the circumference of an enormous circle with the Sun at the center (to get some idea of how truly colossal 600 million miles is, if you were to drive that distance, you would be on the road for almost a thousand years). This translates into a constant speed of 19 miles per second, which is far faster than any supersonic jetliner, or even the shuttle.

With the exception of the Earth’s axial rotation, which is responsible for our day/night cycle, the orbital motion of the Earth is woven more deeply into the fabric of our lives than any other form of local motion. Consider: Virtually every one of our major sporting events, every one of our major holidays, every one of our major religious observances, and even Halloween, occurs at virtually the same time every year. The adherence to that rigid schedule means that these shared cultural events occupy a particular point in space as well as in time. They each have a reserved location on the Earth’s orbital path.

The connection between these all-too-familiar activities and this underlying astronomical truth is not widely appreciated. In part, this is due to the nature of the events themselves. Let’s face it, folks: Tennis tournaments, football games, horse races, and Fourth of July celebrations do little to promote astronomical awareness; to the contrary, they tend to focus our attention firmly on a small piece of terra firma. Even if we turn our attention to the objects in the sky, the impact of our orbital motion remains subtle. There is very little hint of that motion in the behavior of the Sun, the Moon, or Venus, for instance. Indeed, you would be hard-pressed to notice changes of any sort on a day-to-day basis (contrast that with the Moon’s orbital motion, which places it in a noticeably different location in the sky every night). However, if we observe the night sky over a longer period of time, such as the three or so months that make up each of our seasons, the changes wrought by the motion of “Spaceship Earth” finally become apparent; so apparent, in fact, that they blow the observable effects of the other local motions completely out of the water. For, given enough time, the Earth’s journey around the Sun completely alters the thousands of stars that are visible across the night sky.

Tomorrow night, suppose you were to go out about an hour and a half after sunset and assumed the standard viewing position for our visualization exercises, which is facing due south. Your mission, should you decide to accept it, will be to note and memorize a select few of the star patterns that you see when you look up. Then, once a month for the next twelve months, you go out and observe again, constantly comparing the star patterns you see then with the patterns you saw the month before.

With myriads of stars to choose from, which patterns should you to focus on? Good question. On the one hand, I’d like you to bite off enough patterns to appreciate the breathtaking scope of the changes that the Earth’s rotation brings to the night sky. On the other hand, I don’t want to overload your powers of observation and memorization with too many patterns. To help me out of this dilemma, let’s restrict the objects of interest to just those star patterns that figure in that most unscientific of astronomically related disciplines: Astrology!

Astrology? Why would a hard-nosed scientist like me want to have anything to do with astrology? As you know, I love to push back the barriers of ignorance, and going with astrology allows me to kill two “birds of astronomical ignorance” with one stone. For the vast majority of us, astrology has something in common with the highlights of our calendar year: Although astrological concepts are embedded quite deeply into our culture and our consciousness, the connection between astrology and the Earth’s orbital motion is not widely understood. By illustrating the effects of the Earth’s orbital movement with reference to astrological star patterns, I can make explicit the common underlying connections between astrology, the Earth’s motion around the Sun, and the changing view of the stars in the night sky throughout the year.

Am I being too ambitious here? Perhaps, so lets create some suspense, and save the nitty gritty until the next blog. In the meantime, feel free to share your present understanding of these connections in the comments section!

Venus, if you Will!

2010-02-23T21:19:00.000-08:00

It’s mid February, Whabbloggers, and very soon, the evening sky is going to be graced with a rather stunning addition. Yes, our closest planetary neighbor Venus, known since antiquity as the Goddess of Love, is about to assume her position as the dominant “Evening Star”, brightest of all celestial objects save the Moon.

Where has Venus been lately? And why isn’t she the “Evening Star” all the time? To begin to answer these questions, suppose, about a month ago, you headed out to your back yard (or equivalent) at our preferred visualization time, right around sunset. You had taken pains to pick a clear day, of course, so the Sun was clearly visible along the western horizon. The rest of the sky was still bright blue, with no stars visible just yet. Now, where was Venus? Even though invisible, you could have easily visualized her location by imagining a straight line joining you to the Sun and extending out behind it. Venus was almost directly on that line, some 67 million miles past (beyond) the Sun!

Of course, we know from the earlier Sun blog that the Sun is a stationary object, so there’s nothing special about having gone out near sunset as far as imagining the location of Venus was concerned; you could have just as easily imagined the line connecting the Earth, the Sun, and Venus when the Sun was halfway across the sky at noon, or just clearing the eastern horizon at sunrise, or even when the Sun was below your feet at night. To make this point more concretely, take a look at Figure 1 below, which depicts the green/blue Earth, the yellow Sun, and four half-black circles. These circles represent the position of Venus at four points in her orbit around the Sun. Last month, at the time you made your hypothetical trip to your back yard, Venus was in the position corresponding to the circle on the far right of the figure, on the other side of the Sun from the Earth.

In the Figure, you’re looking straight down on the Solar System from directly above the Sun. Another very useful perspective from which to visualize the relative positions of the Sun, Venus, and the Earth is from a vantage point just behind and slightly above Earth, as in Figure 2 below. A month ago, Venus was in the position labeled “Superior Conjunction”. The Sun occupies the middle position, so the side of Venus that was turned toward the Earth was fully illuminated. We couldn’t see it, though, due to a combination of two factors. First, Venus was fully 160 million miles away, so its disk was quite small. Second, and more important, since Venus rose and set at almost exactly the same time as the Sun, it was never in the sky at night, and during the day, it was completely washed out by the Sun’s glare.

It is interesting to compare the source of Venus’ invisibility last month to the invisibility of the New Moon. As we found out in the lunar blogs, the Moon is located between the Sun and the Earth when it is New, so the side facing us is completely dark. Venus is invisible for entirely different reasons.

Like every other body in the solar system, Venus is never stationary. Day by day, our sister planet travels in the standard leftward (counterclockwise) direction all inner solar system objects travel as they orbit the Sun. The closer the object to the Sun, the faster it moves, and Venus actually whips along at over 75,000 miles per hour. You read that right: Venus covers 21 miles every second! You can see from Figure 2 that over the past month, this movement has been taking Venus gradually away from, and to the left of, the sun.

As Venus moves leftward, the distance she is putting between herself and the Sun (as viewed from our perspective here on Earth) is about to pay dividends for sky watchers. Fast forward in your visualization exercise to about three weeks from now, towards the middle of March. Take some time to settle down in your back yard just after sunset, maybe with a cocktail to keep you company. Sipping slowly, you enjoy the gathering dusk while waiting for the first star to appear. Presto: I guarantee that a bright object will pop into view over near the western horizon, and you will be the first on your block to welcome the Evening Star!

Every day thereafter, Venus will slip a little further to the left along its orbital path, taking her ever further away from the Sun. As she climbs higher in the sky, she will hang around for a longer period after sunset before descending down to the western horizon and disappearing from view (this rapid movement of Venus towards the western horizon is entirely due to the Earth’s leftward rotation; it has nothing to do with Venus’s own movement at all).

Let’s continue to follow Venus around in her orbit. As she travels ever further “up, up and away” from the Sun, eventually she reaches the half-way point, equivalent to the Moon’s “half full” position, the point of maximum displacement from the Sun (again, that statement pertains only to our perspective; from her own perspective, Venus stays remarkably close to 67 million miles from the Sun throughout her orbit). The period of maximum displacement will happen over this summer. The black and white ball at the top of Figure 1, and the extreme left of Figure 2, illustrates what position she will take. By then, Venus has swung far enough from the Sun that she is quite a ways from the Western horizon at sunset, and remains visible for several hours after sunset.

Let’s pause for just a moment, consider Venus at this point in her orbit, and visualize her behavior in our sky over a 24-hour period beginning at sunset (i.e., through a full rotation of the Earth). Once it has disappeared, the Sun commences following the standard path below your feet (the path we visualized in the Sun blog) due to the Earth’s eastward rotation. Meanwhile, that same earthly rotation is driving Venus ever rightward, down to and eventually below the western horizon. By the time the Sun pops back into view along the eastern horizon in the morning, where is Venus? Still below your feet, of course, tracing out pretty much the same path as the Sun, just several hours behind. Since Venus doesn’t rise for several hours after the Sun, the event goes completely unnoticed because the planet rises into full daylight.

When the Moon swings into view at the half full point in it’s cycle, our satellite is clearly visible in the daytime sky. Believe it or not, Venus is, too, if you know exactly where to look. The best opportunity comes when the half moon and Venus rise at almost exactly the same time, very close together, and then move in unison as the Earth’s rotation carries them up and across the daytime sky. I’ll be sure to point this opportunity out when it happens this summer. If you can spot Venus, you’ll be one of the few human beings on the planet who’ve ever seen it in broad daylight!

Now, let’s return to the main event, the changes in Venus’ appearance that accompany its orbit around the Sun. You can see from the highly informative Figure 2 that, at the time of greatest elongation, Venus is actually “rounding the curve” and about to start swinging back to the right along our line of sight: A path that will bring it back into alignment with the Sun. Venus is still drawing closer to Earth during this time, so it continues to grow larger. However, as late summer gives way to autumn, each night the distance between our brilliant Evening Star and the Sun will shrink steadily, and the interval between when the Sun sets and Venus sets will shrink as well.

By Halloween, Venus’s orbital motion will have brought it right back into alignment with the Earth and the Sun, but between the two, rather than on the far side of the Sun as it is now. Once again, rising and setting in tandem with the sun, Venus will fade into invisibility in our night sky. This time, her invisibility is caused both by the fact that Venus and the Sun will rise and set together, and also because our point of view is now aligned with the dark side. In that sense, the Halloween “New Venus” and the New Moon of November 6th will have a lot in common!

And what then? Once again, Figure 2 tells the story: Venus’s incessant movement will cause it to move past the sun and off to the Sun’s right. Remember from our Moon blogs what happens to celestial objects that are located to the right of the Sun? When the Sun sets, objects to the right are already below the western horizon, and have a head start when it comes to being whisked around by the Earth’s rotation toward their appointment with the Eastern horizon. So it is that Venus will start to rise before the Sun, and take her appointed place as the blazing Morning Star. Over the course of next winter, the planet will swing further and further to the right, giving her more and more of a head start on the Sun. Consequently, Venus will rise earlier and earlier than the Sun, and stay visible for a longer period of time before dawn arrives and she fades into the morning twilight.

To really imagine this Morning Star behavior, let’s go back to Figure 1 and its “God’s Eye perspective” above the Sun. Pretend you are a specific location along the Earth’s equator (along the very edge of the disk), and the entire green ball representing the Earth is spinning in a counterclockwise direction. Venus is at the bottom of the figure. Now imagine spinning throughout the night. As you pass midnight, which is when you are exactly opposite the Sun on the Earth’s far side, you can see from the geometry in the Figure that Venus is soon going to swing into your view along the Eastern horizon. The Earth will have to complete almost another quarter turn before the Sun does its own appearance act, also to the East.

And that’s pretty much it! When Venus is located to the right of the Sun (from our perspective) she forms the Morning Star. When she is located to the left of the Sun, she forms the Evening Star. Over the course of this year, I encourage you to follow the progress of Venus, starting about three weeks from now along the western horizon. Check out how high she gets in the sky over the summer. In the autumn, track her movement back in the direction of the Sun, before she disappears completely. And then, next winter, if you find yourself with insomnia some night, or happen to get up unusually early, check out the Morning Star!

Figure 2 illustrates one more notable change in Venus brought on by her orbital motion, although you need a telescope or good pair of binoculars to see it. Right now, in the middle of February, the side of Venus that faces Earth is almost fully illuminated. As the planet swings in our direction, the proportion of the lit side that is visible from our vantage point shrinks steadily. When Venus is at her maximum elongation from the Sun, she is exactly “half full”, just like the Moon. Then, as Venus swings back into alignment with the Sun, late next summer and into fall, the proportion of her disk that is illuminated from our vantage point shrinks to a thin crescent (again, these phases are just like the Moon). You will see these phases of Venus clearly if you ever examine the planet in a good pair of binoculars or a small telescope.

Before leaving Venus completely, I’d like to take this opportunity to talk about the place for just a little bit. Virtually a twin of the Earth in terms of size, Venus has always been my favorite planet. I don’t really know why. She certainly isn’t more likely to harbor life than Mars; the surface temperature is about 800 degrees Fahrenheit, 300 degrees hotter than the highest setting on your oven. That temperature is pretty constant across the entire surface, too, despite Earth-like variations in latitude and altitude, and despite the fact that nights on Venus are longer than 100 Earth nights. The long nights are because Venus spins on its axis very slowly, a lot more slowly than the Moon even (you do remember from earlier blogs how slowly the Moon spins, right)? As with many other features, the slow rotation rate of Venus is still a mystery. As for how Venus maintains such a constant temperature, the answer is that Venus’s atmosphere is 90 times as dense as ours. The air pressure at the surface is equivalent to the pressure 3000 feet down in our oceans! Just as there isn’t a lot of variation in water temperatures at that depth, so too is there little temperature variation on Venus.

There is a great deal more that is fascinating about our sister planet, such as the current thinking about why she has such a thick atmosphere, but I will save that discussion for another blog.

Until next time, then, Whabbloggers… when I think I’ll tackle a little astrology!

From the Earth to the Moon

2010-02-10T23:16:00.000-08:00

Several blogs ago, we engaged in a visualization exercise to help understand the apparent movement of the Sun in our sky. In today’s blog, I want to continue the fun by visualizing the Sun’s behavior again, but this time from the point of view of a tourist (you! you!) on the surface of the Moon. The purpose? To illuminate one of the Moon’s best-kept secrets!

Before boarding your spacecraft, I’ll briefly review how the Moon behaves from our vantage point here on Earth. As you know, the Moon is constantly moving in an easterly (leftward) direction, circling the Earth every 27 days or so. The cycle starts with a small crescent that grows a little fatter every night until, about two weeks later, the Moon has the appearance of a big round ball. Then, the pattern reverses itself; each night, darkness steals more and more of the ball from our sight until it disappears completely, and we say that the Moon is “New”.

Through the last two blogs, we’ve discovered that these changes in appearance are completely explained by the change in the relative positions of the Sun, Moon, and Earth due to the movement of the Moon around the Earth. When the Moon is new, it is between us and the Sun; when the Moon is full, we are between it and the Sun. But although the Moon’s constantly shifting position fully explains its phases, a mystery remains. Imagine the Moon in its “New” position, between the Sun and the Earth. The “far side” is fully illuminated by the Sun, and the “near side” is in darkness. If there was a small city on the near side, we’d be able the lights of the city twinkling in the lunar night.

Now, using your powers of visualization, imagine just picking the Moon up and moving it to the position it occupies when full. Where is our hypothetical city now? By rights, it should now be on the “far side”, the side invisible to us. The side that is completely illuminated, and the side we actually see, should be what was the Moon’s far side (the side facing the Sun) when the Moon was new. But that’s not the case. What you see, instead, is the same side that faced you when the Moon was new: The so-called near side, the side containing our hypothetical city. Assuming the city were big enough, you could see the buildings with a telescope.

How can this be the case, when the Moon has traveled all the way around the Earth? You are on your way to the Moon’s surface to answer this question.

Flash forward a couple of days, and you’ve arrived at your destination. Your spacecraft has deliberately landed smack dab on the Moon’s equator (half way between the “top” and the “bottom” of the Moon as seen from Earth), and smack dab along the extreme right edge of the disk, as viewed from here (we want to be able to see you, so you’re just barely inside the edge). In addition, you’ve timed your arrival so that it coincides exactly with new Moon.

Imagine getting out of your spacecraft and lying flat on the ground with your head closer to the North Pole and your feet closer to the South Pole (so that if you look down at your feet, you’re looking south). From our perspective here on Earth, your feet would be toward the “bottom” of the moon, and your head toward the top. Now, stretch your arms out so that your left arm is pointing due East, and your right arm is pointing due West.

Having positioned yourself in exactly the right configuration, all you have to do now is wait, and tell me, back here on Earth, what you experience during the next two weeks. The goal is to connect your experience with the changes I will see in the Moon’s appearance. For a short time after you lie down, you’d be in pitch darkness; your particular location along the right limb is still in darkness. But not for long. Just a short time past new Moon, I see the Moon as the thin crescent, as in the photograph below that you’ve seen before. From my perspective, the area around the right limb (the area around you) has transitioned from being in darkness to being in sunlight. The area off to your west (your right) has not yet undergone this transition, and is still shrouded in darkness.

What causes this transition from darkness to light? Sunrise, of course: The same event that transforms night into day here on Earth. You’ve been right on the dividing line between night and day, and now the Sun has risen right at the location along the eastern horizon that you are pointing to with your outstretched left arm.

24 hours now pass. Viewing the Moon from my vantage point here on Earth, the crescent has grown a little fatter, which means that a larger area of the Moon’s surface beyond the limb (off to your right) has become illuminated. I’ve captured this situation in the photograph of the Moon above. From where you are located, the only way that areas to the west of you can have transitioned into daytime is if the Sun has climbed higher in the sky over to your east, exactly the track it takes across the sky after sunrise here on Earth.

Over the two weeks that the Moon waxes, you stay completely still, and we compare notes every night. At “half full” from my perspective here on Earth, where is the Sun located for you? Straight overhead. It’s high noon on the limb of the Moon, and the temperature is approaching the boiling point of water! Skip ahead anther week, to when I report seeing a full Moon. For you, the Sun has moved all the way across the sky and down to the western horizon, where it is now poised to set. The very next night, I report that the Moon is just past full; the right limb of the Moon has fallen into darkness. That’s entirely consistent with your report, which is that the Sun has now set, and the temperature is plunging. Your long lunar night has begun.

Okay. You’ve been describing how the position of the Sun has changed every day, rising in the East and setting in the West, just the way the Sun behaves for us here on Earth. We know that the Sun’s apparent movement across our sky is an illusion, brought on by the Earth’s counterclockwise rotation. Is the similarity between the Sun’s behavior here and on the surface of the Moon just a lucky coincidence? Hardly. The only way that the Sun can behave in the same way from your vantage point on the Moon, as it does here on Earth, is if the Moon, too, is spinning on its axis in an easterly direction (or to the left, from your perspective on the Moon’s surface).

How can the Moon’s rotation be reconciled with the fact that the same side of the Moon always faces the Earth? At first, this seems rather difficult. If the Moon spins in a counterclockwise direction, like the Earth does, over time, shouldn't new regions of the Moon’s surface become visible to us? Specifically, why doesn't new lunar territory constantly spin into view along the left limb of the disk (as viewed from Earth), and constantly disappear from view along the right limb? Why, in other words, don’t you disappear behind the right limb?

The reason is simple, but subtle. Let's shift perspective for just a moment and pose a different question. If the Moon didn't rotate on its axis the way we’ve established it does, what parts of its surface would we see during the two-week period that it moves from new (barely visible as a crescent) to fully illuminated? Well, since the Moon moves continuously in a leftward direction along its orbital track around the Earth, our viewpoint should be constantly shifting "around" the moon in a rightward direction. We should be seeing new territory appearing constantly on the right side of the Moon's disk (right limb), where you are, while territory constantly disappears from the left limb. This fact is a little easier to visualize with our old friend, the phases of the Moon figure, so I'm including it again below.

Let’s pause and summarize these points. Considering the Moon’s movement around the Earth, the sides of the Moon’s disk where territory should be becoming visible and invisible are exactly opposite the sides where territory should be becoming visible and invisible, given the Moon's own rotation. That’s a big conceptual mouthful to swallow, so it’s worth kind of savoring it, if you have the time and patience.

Are you still with me, patient Whabbloggers? I hope so, because if you are, you have all the conceptual ingredients needed to put the big picture together. A key to the entire issue is the speed at which the Sun moves across the lunar sky from your perspective on the surface. Recall that, approximately one week after your sunrise, the Sun was directly overhead (it was locally noon)? That means that what takes approximately 6 hours to happen on the Earth (the time needed for the Sun to move from it’s position on the eastern horizon at sunrise to fully overhead at noon) has taken a full week on the Moon. A week after that, when the Moon is full, the Moon's rotation has pushed the Sun all the way to "sunset position” off to your right. A day on the Moon is two weeks long, and following sunset, two weeks will pass before the sun once again peeks above your eastern horizon.

It takes exactly one month for the moon to rotate once around on its axis. It also takes exactly the same amount of time for the Moon to complete one revolution around the Earth. Consider: During the time it takes the Moon to revolve from the position it occupies when it is new (right in front of the Sun) to when it is half full (so its position forms a right angle with the Sun and the Earth), it has moved through 90 degrees, exactly one quarter of its orbital circle. If the Moon wasn’t spinning on its axis, exactly one quarter of the far side would have swung into view along the right side of the disk. But, in the week it takes to reach that “half moon” position, the Moon has also rotated, in a counterclockwise direction, exactly one quarter of the way around on its axis, effectively blocking any new territory from appearing. The two motions completely cancel each other out, leaving the same side of the moon permanently turned toward the Earth!

There’s just one loose end to wrap up. For the Earth to complete a day in a scant 24 hours, it has to be rotating at an extremely fast clip, reaching a thousand miles an hour at the equator. What about the moon, where a day lasts a month? It turns out that even at the Equator, the Moon only rotates at about 10 miles an hour! As you move away from the Equator, toward one of the lunar poles, the rotation rate slows down all the way to walking speed and below. Yes: There are places on the moon where you could walk toward the west and keep the Sun permanently fixed at one position in the sky. Someday, I can imagine moon settlers living in specially designed double-wide boxcars on a railroad track that completely circumnavigates the Moon. With the boxcar moving along the track at just walking speed, the residents would live in perpetual daylight, with the Sun permanently frozen in a position low enough in the sky (shortly enough after sunrise) that the ambient temperature would always be a balmy 72 degrees! Yes, you could construct things on the Moon so you lived your life in permanent daylight, and endless summer!

And with that, it’s time for both of us to leave the Moon. Next blog, I will tackle the behavior of another compelling object in our sky, the planet Venus, the Evening and the Morning Star. The question is, how can it be both?

To Everything, Turn, Turn, Turn...

2010-02-10T00:08:00.000-08:00

One of my favorite activities while on a tropical vacation is to sip a cocktail at sunset. Let’s go with that theme for this blog, a continuation of our series to understand the behavior of the Moon. By great good fortune, pretend you are enjoying a month-long holiday at a fabulous resort somewhere along the Earth’s equator. By interesting coincidence, the start of your vacation, two weeks ago, coincided exactly with New Moon. Every day since your arrival, you’ve made it your business to be on your fabulous open patio at sunset, facing due south, enjoying your own favorite cocktail. The drink goes down easily, the tropical breezes soothe your brow, and you haven’t a care in the world. Why would you? You’re only half-way through a long fabulous vacation!

But, you actually do have one small concern. Over the last two weeks, you’ve watched the moon’s position and appearance change as it went from New to Full. Tonight, the Sun is slipping below the western horizon, setting up a gorgeous tropical sunset. Off to your left, the full Moon is coming into view along the eastern horizon. Although the scene is dripping with beauty and romance is definitely in the air, your mind is occupied with trying to puzzle out what’s going to happen next to the Moon, now that it’s full.

To understand this, it is helpful to and try and connect your local view to the zoomed-out “birds eye” view that you would get if you were out in space, many thousands of miles above the Earth’s North Pole. That perspective is exactly what’s captured in the figure above (the same one I used in the last blog to trace the Moon’s behavior while it was waxing). Looking down on the central little ball that is the Earth in the figure, you are in essence seeing the half of the Earth’s surface that corresponds to the Northern Hemisphere. Half of that area is lit up by the Sun (which is off to the right), and is experiencing daytime. The other half, turned away from the location of the Sun, is black, signifying night.

Now, comfortably ensconced in your tropical get-away on the equator at sunset, where does that put you on the surface of the little Earth ball? If you can stop reading and answer that question right now, yourself, then you’re way ahead of this game! If not, well, let me tell you: You’re right at the top of the ball, exactly where the line is separating white from black (day from night). The Sun is setting because the Earth is spinning counterclockwise (eastward), so your position on the Earth’s surface is shifting leftward into the dark side.

As we noted before, the Moon, the Earth, and the Sun are all lined up, with the Earth directly between the Moon and the Sun. That configuration explains why the Moon rises over your eastern (leftward) horizon at this exact point in time. It is only now, once you spin into the dark side, that the Moon becomes visible; a little earlier, while you were still on the lit (daylight) side, the Moon was still hidden below the Earth’s Eastern horizon. In essence, throughout the daylight hours, the Moon was positioned below your feet. However, the eastward-spinning Earth was constantly “pushing” the Moon toward your left until, just at nightfall, it got “pushed” above the eastern horizon and into view.

As the night of the full moon commences, you keep on spinning to the left, which now ”pulls” the Moon ever westward across the sky. Just as you’re about to spin into the lighted side (Sunrise), the Moon passes out of view over in the West.

What then? In the last blog, we discovered that when the Moon was waxing, it was located to the left of the Sun. That meant the Moon was in essence “trailing” the Sun, both setting after the Sun did (always during the night) and rising after the Sun did (always during the day). But, moving at the standard 25,000 miles per hour, the Moon doesn’t stay aligned with the Sun and the Earth for long. From the “top-down” perspective of the “Phases Figure”, you can see the impact of this perpetual counterclockwise motion on when the Moon rises during the next two weeks. Every night, the Moon doesn’t appear until longer and longer after sunset, and doesn’t set in the West until longer and longer after sunrise. A good benchmark for these changes is the “3^rd quarter”, when the Moon is exactly half-way back to the vicinity of the Sun (and halfway back to another New Moon phase). At this point, the Moon forms a right angle triangle with the Earth and the Sun. The Moon doesn’t swing into view in the east until you are halfway through the “dark side of the Earth”, or midnight, and doesn’t again get obscured by the Earth (set in the West) until you have spun around to the point where you are halfway through your “daylight phase”.

Although the moon continues to rise at night throughout the waning period, the gap between moonrise and sunrise shrinks steadily as the Moon churns ever closer to the Sun, and the amount of time that the moon is visible during the day expands steadily. However, that doesn’t necessarily mean that the Moon is getting easier and easier to see during daylight (though it is actually pretty easy to spot if you know where to look). As the Moon approaches closer and closer to the vicinity of the Sun, less and less of the surface is illuminated. You can visualize how this is happening by returning to your vantage point on the patio of your tropical vacation paradise. What does the Moon look like in the sky as it wanes?

The answer is illustrated in the series of phases in the figure below. The top row shows how the Moon looks while waxing, which we covered in the last blog. The bottom row is what you’d see if you kept track of the Moon while waning. Right after full Moon, darkness start to encroach along the right side limb, similar to how brightness appeared while the Moon was waxing. Over successive days, less and less of the near sight is lighted, until the Moon reverts to that familiar crescent shape again. The difference from the waxing phase is that the crescent is on the left side of the Moon’s surface, rather than the right.

This behavior makes perfect sense when you realize that the Moon is moving ever closer to alignment with the Sun from off on the Sun’s right flank. As the Moon moves back toward full alignment, more and more of the sunlit hemisphere slips around to the far side, leaving less and less of the near side illuminated. Eventually, the moon is almost completely aligned with the Sun and the Earth, and only a sliver of near side is visible along the left limb. Of course, the rest of the sunlight region is now behind that limb, heating up the far side of the moon.

A day or so later, the moon slips into full alignment, directly between you and the Sun. Just as it was when you started your vacation, the near side is completely shrouded in cold and darkness. The good news is, we’re back where we started, at the New Moon phase! The bad news is, your vacation is over.

All, right, it’s time to leave fantasy land and get back to reality. Wherever you actually live on the Earth’s surface, I encourage you to start looking for the Moon when you find yourself outdoors, day or night. When you spot it, think about what its appearance tells you about where it is in its cycle and where it will be (and what it will look like) in the coming days. Then, go out and confirm your predictions with actual sightings. In no time, you’ll be an expert on the Phases of the Moon!

In the next blog, I’m going to build on this “introduction to lunar behavior” to uncover and describe something else about the Moon that explains a major part of its appearance, at least as seen from Earth. Hint: Over the course of the Moon’s orbit, regardless of whether the Moon is waxing or waning, only the near side (or a portion thereof) is ever revealed to us.

If the moon is circling all the way around the Earth during this period, why don’t we ever see the far side? If you can answer that question in the comment section, you won’t even have to bother reading the next installment!

Of Wax and Men

2010-02-04T22:16:00.000-08:00

Last time, with a little help from our friends the Beatles, we used our powers of visual imagination to better understand the daily behavior of the Sun in the sky (and below it, at night). As we discovered, the key is to think of the Sun’s motion as the product of the Earth spinning round and round and round on its axis. This time out, we’re going to start to tackle the behavior of our nearest neighbor in space, the Moon. First up will be the remarkable changes that take place in the Moon’s appearance, and position, over the course of each month. What’s in this for you? Suppose you went outside last night and casually noticed the Moon. Could you have answered a companion’s questions about where the Moon would be in the sky tonight? Where it will be tomorrow night? And how its shape is going to alter? Well, if you can get through the next two blogs with me, answering questions like that will become child’s play. You’ll be able to effortlessly analyze the Moon’s appearance at any time of the month and know just where it is in its monthly cycle, why it appears the way it does, and where it is going. Fun, right? But not only that. In the final installment of the Moon series, we’ll build on your newfound understanding to uncover and discuss a fascinating “secret” about our nearest neighbor and its behavior.

As you know already, the Moon’s monthly cycle starts and ends with the “New Moon” phase, when the Moon is completely invisible. What hides it from our sight? The answer rests with two simple facts. First, the Moon is a big round ball. Just like the Earth, exactly half of the Moon’s surface (one hemisphere’s worth) is always illuminated by the Sun, and the other half is always in darkness. The figure below illustrates the second fact: The Moon continuously circles (orbits) the Earth, completing one revolution every 27 days. Notice from the figure that when the Moon is new, that corresponds to the point in the Moon’s orbit where the Earth, Moon, and Sun are perfectly aligned, and the Moon is between the Earth and the Sun. See where the sun’s rays are hitting the Moon’s surface? All the illumination is confined to the “far side”, the hemisphere that’s opposite the hemisphere that faces us (the near side). In other words, when the Moon is new, the far side is experiencing daytime, and “our” side is experiencing night.

As I noted, the Moon takes 27 days to complete one orbit, and another two days before it again lines up with the Sun (the reason for the discrepancy is that the Earth is in constant motion around the Sun. But that’s a topic for a future blog about, of all things, astrology). At an average distance from the Earth of about 239,000 miles, the Moon covers over 750,000 miles per orbit. To go that far in just 27 days, the Moon has to be moving in excess of 1000 miles per hour. That’s rather fast for an object that looks completely stationary when you view it in the night sky, wouldn’t you agree? But I’m getting ahead of myself because, since we’re still talking about the new Moon, you can’t see it yet!

I know I said (and the figure gives the impression) that the new Moon is positioned directly between the Sun and us, but that’s not quite true. Usually, the Moon’s path takes it just slightly above or below the Sun rather than directly in front of it. On the rare occasions where the Moon does pass right in front, some lucky locations here on Earth experience a solar eclipse. However, with that speed of 1000 miles an hour, the Moon doesn’t stay aligned with the Sun very long; very soon, and very quickly, it moves off to the left, and keeps on moving.

One of the primary tools in our “Moon visualization” arsenal is the simple fact that the Moon moves in an eastward direction, which means that it is always moving to the left through the sky. An amateur astronomer took the photograph on the right immediately after sunset, scant hours after the moon was new. That razor-thin crescent is, of course, the Moon, having traveled just a small way to the left (east) of the Sun and, since the orbit is curved, a small way “back” in our direction.

The Moon assumes this form of crescent once every month. You’d have to be really on your toes to see it, though, because it is very close to the Sun, and quickly follows the Sun below the horizon (that is, it quickly sets). Even though the Moon is moving east, you’re viewing it from a vantage point that is itself spinning around in an easterly direction, and this rotation is “pushing” the Moon (and the Sun) down and to the right much faster than the Moon is moving to the left around its orbit. Recall in the Beatle’s blog that I discussed how the Earth’s rotation controls the movement of the Sun after sunset, “pulling” it ever further down and to your left (if you’re looking south) until the Sun swings back into view along the Eastern horizon (dawn)? That reminder should give you a sufficient basis to visualize what happens to the Moon after it sets on the heels of the Sun. At any rate, I encourage you to try. The first commenter who correctly identifies when the Moon “rises” in relation to the Sun gets today’s comment section prize!

The next order of business is to understand why a sliver of the near side is now illuminated. Looking back at the “phases of the Moon” figure, fast-forward about two weeks, to the point when the Moon has traveled exactly half way around its circular orbit. You see from the illustration that the Sun, Moon, and Earth are once again lined up, but this time, with the Moon furthest from the Sun and the Earth in the middle.

See what happens now? The Sun’s rays are falling directly on the near side, lighting it up fully, and it’s the far side that’s shivering in darkness (by the way, this illustration helps you see why lunar eclipses, which occur when the Moon passes through the Earth’s shadow, only happen when the Moon is full).

Now, turn the clock back two weeks, when the Moon was still a thin crescent near the Sun. Every day, the Moon travels about 25,000 miles along it’s orbit, so every day, it draws a little closer to the position that it’s eventually going to occupy when full. Moving ever closer to that position causes us to be able to see a little bit more of the sunlit hemisphere each night; a good way to think about this is to imagine the moon moving, not only out and to the left of the Earth, but also a little more “along side” the Earth, thus revealing a bigger and bigger piece or fraction of the sunlit hemisphere that used to be hidden behind the right limb. After about a week, the Moon is at the top of the figure, and forms a right angle triangle with the Earth and the Sun. At this point, exactly half of the illuminated hemisphere is visible to us, and we say the Moon is “half full”. Where is the other half? On the far side, of course, meaning exactly half of the far side is illuminated too.

But there’s something else that’s important to glean from the figure. Take a look at the little circle in the middle representing the Earth, half of which is lit, and half of which is not. You’re seeing the Earth as it would look if you were directly above the North Pole. Now imagine the little circle rotating in its counterclockwise (Eastward) direction around the pole. Now change your perspective, and pretend to move from your “eagle eye”, way above the North Pole, to your present location on the Earth’s surface, right on the line between the dark side and the light side. That position corresponds to sunrise, of course; the Sun is coming into view on the eastern horizon. As the day wears on, you continue to rotate in a counterclockwise direction. Although the half-full Moon starts out below the horizon, the Earth’s spin pulls it closer and closer to your eastern horizon, until, right at local noon, the Moon swings into view. Yes, when the Moon is half full, it rises right around local noon (and is clearly visible, even thought it’s broad daylight). Your day progresses, the Earth continues to spin, and the Sun slowly sinks into the West. Meanwhile, the Moon continues to climb higher into the sky. At local sunset, you are now positioned right at the very top of the little Earth ball in the figure, and the Moon is now directly overhead. With nightfall, the Moon blazes brightly in the sky. Over the evening hours, though, it follows the sun into the west, until finally setting half-way through the night.

As the Moon proceeds towards full, her thousand mile-per-hour orbital speed opens up more and more distance between her and the Sun. If you visualize the relative locations of the Sun and Moon near sunrise, (i.e., below your feet and off to your left if you are outside facing south), you’ll see that the increasing Sun-Moon distance is creating a longer and longer delay between sunrise and moonrise. The increase also means that the Moon is visible in the night sky for a longer and longer period after sunset.

Eventually, the Sun-Moon gap gets big enough that the Moon is both full, and rises just as the Sun is setting. Again, the figure shows why this is the case: The Moon is only full when it is on the other side of the Earth from the Sun, as far away from the Sun as it can get.

OK. We’ve covered things up to the full moon phase. In the next blog, we’re going to tackle the other half of the cycle, when the Moon starts waning. If you’re like most people, and like me before I started to pay attention to these things, the waning Moon is much less familiar than the waxing Moon. This is because most people stay up after sunset, so they gain plenty of experience with seeing the Moon when it is visible in the early nighttime sky. In the waning phase, though, the constant leftward movement of the Moon in its orbit causes it to move back in the Sun’s direction, on a trajectory that has it approaching the Sun from the right-hand side. As we’ll see next time, this geometry means that the Moon rises later and later each night, after we’ve typically gone to bed, and remains visible longer and longer in the morning. However, since we much more naturally associate the Moon with a nighttime object, we almost never look for (or see) it in the morning, and our nearest neighbor completely exits our consciousness.

But more on that next time.

Meet the Beatles!

2010-01-26T22:18:00.001-08:00

Here’s the opening blog of a quartet. The theme: giving you a better understanding of the behavior of the most familiar objects in our sky: The Sun, the Moon, the brightest “star”, Venus, and the constellations that make up the zodiac. First up, though: The Sun.

As with most of my generation, I grew up listening to Beatles music. “Help!” was the first album my older sister ever bought, and we proceeded to almost wear the vinyl out on her mid-60’s record player. Surprisingly, though, I never paid much attention to Beatles lyrics; I suppose I thought they were largely nonsensical (though frequently clever) plays on words that John Lennon created for his own amusement.

My inattention to Beatles lyrics extended even to McCartney classics like The Fool on the Hill, even though I think the melody is absolutely gorgeous. Sing it with me, won’t you?

Day after Day
Alone on a hill
The man with the foolish grin is keeping perfectly still
And nobody wants to know him
They can see that he's just a fool
And he never gives an answer

But the fool on the hill
Sees the sun going down
but the eyes in his head
see the world spinning round....

Ooh, ooh,
Round and round and round and…

Just typical Beatles nonsense, right? Certainly nothing too profound. At least, that's what I used to think.

But that was before I started what has become almost a weekday ritual in my life. I happen to live right beside Steven's Creek Trail, which parallels the section of a creek that flows out of the Santa Cruz Mountains and into the south end of San Francisco Bay. Most days after work, I combine walking and running along the trail for about two miles out and back. The exercise is great, of course, but it also gives me a daily chance to think and ruminate about topics to post on the Whabblog.

As great good luck would have it, Steven’s Creek (and the trail) is oriented almost exactly north-south, at least along my portion. That chance alignment means that I’m facing virtually due north during the 20 minutes or so that I’m running away from my place, and virtually due south on the way back.

What does this have to do with "The Fool on the Hill"? Well, the chance alignment of the creek with the Earth's axis affords an extended opportunity for me to visualize things. Astronomical things, quite often, like the relationship between the Earth and nearby objects like the Sun, Moon, and even Venus.

The opportunity to visualize comes in handy. Take the behavior of the Sun, for instance. Every day, the Sun rises at some point along the eastern horizon (somewhere off to your left, if you stand outside and face due south, like I do while I’m running back to “McCann Manor”). In a pattern that’s as familiar to us as the back of our hands, the Sun moves steadily across the sky until finally disappearing somewhere off to our right at sunset.

Of course, intellectually, we all know that the movement of the Sun is just an illusion. It doesn’t really “rise” in the East, or “move” across the sky, or “set” in the West. Instead, the Sun’s apparent motion is brought about by the fact that the Earth spins on its axis, completing one full rotation every day.

But it's one thing to SAY that the movement of the sun is an illusion; it's quite another to fully “grok” the reality behind the illusion. What I mean is, it’s actually quite difficult to square sunrise, sunset, and everything in between with the reality that the Sun doesn’t budge one inch. To smash the illusion, imagine yourself outside, facing due south, right at sunrise. Imagine further that the Earth is spinning in an easterly direction, which forces you (and everything around you) to move continuously to your left. Visualize that spin literally forcing the Sun (the stationary object) to move in the opposite direction and track across the sky to the west. Eventually, the continual steady spin pushes the Sun all the way to the western horizon, and then past it.

Now comes the fun part. Just because the Sun has set, don’t stop now. Continue to imagine you rotating in that easterly (leftward) direction. What effect does the rotation have on the position of the Sun after it sets?

We’ve seen that when the Sun was “up”, your local direction of movement was “pushing” it ever further to the west. Once the Sun sets sets, though, that leftward spin starts to “pull” the Sun back toward you; that is, it now “pulls” the Sun in the same direction that you’re spinning. Expressed another way, your rotation is yanking the Sun ever closer to the eastern horizon, with sunrise as the inevitable result.

So you think you have this concept nailed? Here's the ultimate test. Stay up to about midnight, go outside, and face due south. Then, even thought you can’t see it, point exactly to where the sun is. If you get it right, you’ll be pointing straight down into the ground below you, at a position roughly aligned with the North-South meridian. If you stayed up later, you'd still be pointing into the ground, but at a location that's starting to slide off to your left. Are you with me, here? Stay up even later, and eventually you'd be pointing to a position way to your left, just below the eastern horizon. And, then: presto! The constant eastward spin of the Earth forces the Sun back into view along the eastern horizon, and you’re pointing at the sunrise.

This ability, to imagine where the Sun is and how it behaves at night, is the key to abolishing the illusion of a stationary Earth and a moving Sun. Of course, you don't actually have to go outside, or wait until midnight (or later) to visualize it. Try, instead, going out right at sunset, the next time the sky is clear, and face south. Then imagine you are suddenly spinning to the left (east) at much faster rate than the speed that the Earth actually rotates. What happens to the Sun? It’s actually quite easy to imagine it racing around that circle through the ground beneath you, closing in quickly on the point that intersects the location on the eastern horizon where the Sun rises.

And then you'll have done it, patient reader: You'll have matched the observational capabilities of the Fool on the Hill! Day after day, for the rest of your life, the eyes in YOUR head will see, not a moving ball of fire, but a world (our world) continuously spinning round and round and round. And YOU will have a gut-level understanding of a truth about the relationship between the Earth and the Sun that eluded humankind for almost the entire time that we’ve existed on the planet.

The form of visualization exercise I’ve asked you to engage in will pay dividends in the rest of the blogs in this series, too. In the meantime, let’s sit back and appreciate the genius of the Beatles’ lyrics, in addition to their melodies.

The End of the Beginning

2010-01-15T19:13:00.000-08:00

Have we reached the tortured end? Yep, almost (at least for the moment…. hehehehe). But before I leave the topic of time dilation and special relativity, I’d like to address time dilation in the real world, as opposed to the imaginary world of Einstein-style thought experiments featuring vaguely vagina-shaped spacecraft carrying excited male celebrities.

Did you know that, every second of every day, subatomic particles, mostly protons, strike our atmosphere? You may have heard them referred to as cosmic rays. They are most definitely not rays, though; protons are bona fide particles with bona fide masses. And despite the name, cosmic ray, they’re hardly exotic; there are one or more protons inside every atom in your body, and inside every atom in the universe.

As cosmic rays, though, protons travel to us over phenomenal distances often after being spun up by black holes to about 0.99 of the speed of light. When these naked protons (by naked, I mean they’re not part of an atom) strike the gas molecules in our upper atmosphere, about 15 kilometers above the ground, they immediately decay. One of the byproducts of proton decay is a particle called a muon, which continues through the atmosphere in roughly the same path as the originating proton at roughly that same high rate of speed. Now, one thing about muons is that they’re very short-lived particles. They take an average of just two-millionths of a second to decay into other particles. That’s just an average, mind you. Some muons survive for a little more time than two-millionths of a second, while others survive for a little less.

At a velocity of about 0.99 the speed of light, or 297,000 kilometers per second, how far do muons travel through the atmosphere before they decay?

Distance equals speed multiplied by time. In the muon’s frame of reference, 297,000 kilometers multiplied by two-millionths of a second (their average time of existence) yields roughly 0.6 of a kilometer. This means that the average muon doesn’t even come close to covering the distance between the top of the atmosphere, where it was created, and the ground, before it decays. Even the muons that beat the average, and survive for a little longer than two-millionths of a second, don’t make it.

That’s a good thing, because muons are radioactive. Exposure to them can cause cancer. But, is it really the case that muons never travel far enough to cause us any harm? So far, our discussion about how long a muon exists, and how far it travels through the atmosphere, has been from the muon’s frame of reference. In our (the Earth’s) frame of reference, which is at rest compared to the muons, we’ve got to factor in the extra distance that the muon is going to cover due to time dilation. For a muon traveling at 0.99 light speed, time is stretched by a factor of about four. In our frame of reference, therefore, instead of existing (and traveling) for about two-millionths of a second, the average muon exists for about eight-millionths of a second. That is enough time to traverse about two and a half kilometers before decaying.

But that’s just the average muon. For muons that decay more slowly than average, time dilation makes just enough of a difference that they do manage to reach the surface, and maybe even hit you or I. It’s not a lot of muons; statistically, only about four out of every hundred last long enough for the combination of their relative longevity, compared to other muons, and time dilation, to make it all the way down.

From our earlier discussions, you’ll recall that, for the object traveling quickly, time is locally the same as it is for you, and the phenomenon of time dilation is manifest as space compression. In the muon’s frame of reference, it lasts exactly as long as it should (on average, only two-millionths of a second). Thus, the (statistically) most durable muons manage to make it all the way through the atmosphere, not because time dilation extends their lifetime, but because the distance between the top of the atmosphere and the ground is compressed by a factor of four.

You could verify the presence of muons reaching the surface with a simple Geiger counter. Being radioactive little beasties, some of the clicks you would hear when you turned on the counter would be in response to their presence. If you climbed a mountain, and turned the Geiger counter on up there, you would hear more frequent clicks, because more muons make it that far.

The bottom line: Time dilation is real, influencing real events in our real lives. And that’s a wrap on the topic, at least for now. I just have to “close the loop” between time dilation and my late friend Karl, my original motivation to research and write these blogs. Frankly, the connection is kind of a downer. I’ve had four different surgeries in the last couple of years for skin cancer. Would I have escaped the cancer scourge if not for time dilation? I don’t know. Karl died of complications due to surgery necessitated by the strong suspicion that he had colon cancer. I wonder if he would be alive today, too, if the phenomenon didn’t exist.

Ah, well. Time dilation is built into the very fabric of our reality, so there’s no point in wishing. I just hope you’ve enjoyed finding out what it’s all about. I know Karl would have!

A Baseball... and a Terrible Accident

2010-01-14T18:52:00.000-08:00

We’ve arrived at the penultimate time dilation blog, faithful readers! Having explained the nuts and bolts of the phenomenon already, this time I’m going to feast on some of the implications of the effect, such as the solution to the conundrum I posed yesterday. Before we get to that, though, I’d like to take care of an important piece of unfinished business. I want to be sure you understand that time dilation is not confined to the duration of events involving light pulses; it occurs for any and all events on board Shirley.

What is an “event”, when you really think about it? My definition is pretty simple: anything that has a beginning and an end, and takes a measurable amount of time to occur, qualifies as an event. We’ve discovered that when an event occurs in a frame of reference that’s moving relative to you, during the time that separates the beginning of the event from its end, the objects involved travel farther in your frame of reference than in theirs. Since Shirley is moving at the same speed in both frames of reference, the only way that she can possibly cover more space for you than for George is for the event to take more time.

To appreciate the point further, let’s return to yesterday’s version of the thought experiment. Shirley was revved up to very close to the speed of light, and the critical event (the light flash traveling up and down her shaft) took almost 13 minutes (as measured by you). Now, suppose you and George decide to do something completely different. Ignoring the searchlight, George stands up on Shirley’s floor with a baseball in his hand. At just the point where Shirley passes over you (still traveling at the same 299,999 kilometers per second) George lofts the baseball straight up in the air, with just enough speed that it travels upwards for exactly one second, and back down for another second.

Suppose also that a closed-circuit television lets you watch the baseball go up and down from your location on the ground. What would you see? Well, the television signal is a form of light that also travels at 300,000 kilometers per second. Since the television signal is beaming directly from the floor of the spacecraft to your television set, what you see is determined by how far the signal has to travel, which, in turn, is dependent on how far away Shirley is when the images are emitted (these days, about 120 images are emitted every second). We saw last time that in two seconds, extra space is being created at a furious rate by Shirley’s high speed (the distance along Line B is stretching out quickly). This means that each successive television image of the baseball has to travel a longer and longer distance, which means a successively longer delay between when the onboard television camera records the image of the baseball, and when you see that image on your screen. The net result is that you would see the ball moving very slowly upwards, as if you were watching film of the moving baseball in slow motion. As time continued to pass, the movement of the ball would slow down more and more, until it became imperceptible.

The baseball wouldn’t stop moving entirely, though, and if you took a break and went to the bathroom, when you came back you would see that the ball had shifted position. 13 minutes after George released the baseball, you would finally see it return to his hand, and the event would be over.

Actually, in 13 minutes you could take a lot of breaks from watching your television. You could go to the bathroom, have a brief conversation with your neighbor, watch the Kentucky Derby, and read a Whabblog. Meanwhile, George would have no time to do any of these things; he would be fully occupied by throwing the ball up and catching it almost immediately. A text from him to you might say something like: “Really, really rushed! Just barely had time to throw the ball up before I had to catch it again!”

Not so for you, and not so for everyone sharing your frame of reference (which is everybody on the Earth). Imagine all the things that happen across the world in 13 minutes. Thousands of people die; thousands more are born; thousands exchange wedding vows; thousands get notices of a hiring or a firing; and there’s a distinct possibility that a big natural disaster like an earthquake occurs (and the first reports about it came in on CNN). In short, a busy little chunk of everyday life, full of scores of individual events, passes by.

What would happen if you didn’t stop Shirley after the ball-throwing task (or the light pulse measurement task)? Suppose you just let her continue moving at the same speed for, say, two weeks of ship time? You would just go about your normal daily activities during that period, and so would George onboard. What would your texts to each other look like at the end of the period?

When you and George were doing the light-pulse measurement, it took about 775 seconds for you to see the light pulse return to Shirley’s floor, versus a paltry two seconds for him. Thus, for every second of time that was passing for George, approximately 387 seconds pass for you. That same multiplier works for any time scale. Thus, while only two weeks of time are passing for George, 775 weeks of your life unfold.

775 weeks is almost 15 years! At the end of George’s two weeks, if you could exchange photos along with your text messages, he would look identical to before; nobody ages noticeably in only two weeks! But you… you would look noticeably older; depending on how well he knew you, George might even have trouble recognizing you for a moment. And imagine all the pages and pages of news you could put in your text message to him, with 15 years worth of your life to draw from!

If we extended Shirley’s trip a little longer, to say a month, almost 30 years of your life would pass. While George would hardly have time to age at all, there would be a nontrivial possibility that you would have grown old enough to die. And in fact, if you cranked Shirley up even more, to more than 299,999 kilometers per second, the time dilation would grow so extreme that in the month George spent on board Shirley, many thousands of years would pass here on Earth. When he returned, you would be just a distant memory. Shirley would have turned into a very effective time machine.

I told you time dilation was a really freaky phenomenon, didn’t I? But there’s one last aspect to it – the answer to yesterday’s paradox – that may be freakiest of all. Let’s return to our regular thought experiment with the light pulse. From George’s point of view, in the two seconds it took for the light pulse to go up and down Shirley’s cylinder, she covered exactly 599,998 kilometers. That’s not a trivial amount, granted: it’s about 1.5 times the distance from the Earth to the Moon. But 600,000 kilometers or so is utterly insignificant from your point of view, because for you, Shirley has traveled 236 million kilometers, about a third of the way to the planet Jupiter! By virtue of traveling so far, she and George could well have collided with an asteroid, ending both the time measurement experiment and poor George’s life. Meanwhile, from George’s perspective, nothing of the sort would have happened.

One afternoon last spring, while pondering this conundrum during an afternoon run on Stevens’ Creek trail, I had an “aha” experience, and the last piece of the time dilation puzzle finally fell into place. The light pulse moving up and down Shirley’s shaft is the same event for you and George; therefore, it has to have the same history in both frames of reference. So how do we get around the “asteroid collision” paradox? There’s only one way. Shirley, George, and the light pulse have to be at the identical location in the solar system at every point along the pulse’s journey, both in your frame of reference, and in theirs’. That way, if George and Shirley meet with an asteroid along the way, they do so in both frames of reference.

Let’s assume George and Shirley have the good fortune to avoid any asteroids, and the light pulse reaches the floor of Shirley’s shaft quite safely. The speed of light is constant, so according to Shirley’s odometer, she has to have moved exactly 599,998 kilometers to your right when the light flash reaches the floor. The only way to reconcile that fact with Shirley being all the way out in the asteroid belt is if, from George’s point of view, space itself is compressed – literally, scrunched - in the direction of Shirley’s movement. That is the simple, but astonishing truth, folks: Space actually shrinks along Shirley’s direction of motion, by exactly the same factor that George’s time expands for you. In other words, at a speed of 299,999 kilometers per second, everything in Shirley’s path, including her final destination out in the asteroid belt, becomes 387 times closer than it is for you here on Earth. That’s why, from George’s perspective, it only takes two seconds to reach it!

We just saw that if you are part of a frame of reference that moves at a sufficiently high rate of speed relative to the Earth, you become a time traveler, able to take a very short trip and yet return many years in the future. Spatial compression is the amazing flip side to this phenomenon; it means that, within your lifetime, you could travel to very remote destinations in the universe, including other stars and even other galaxies, which to us on Earth are so far away that they remain forever out of reach.

You could argue that it’s totally absurd to have gone to all the trouble of exploring and explaining a phenomenon that never actually happens, because nothing ever goes that fast. Not true, however. There are actually things in our universe that move at close to the speed of light, and experience significant time dilation. I’ll reveal what those things are in tomorrow’s blog.

To the Moon, Alice! And beyond!

2010-01-13T21:31:00.000-08:00

In the third (and final) version of our little thought experiment, we’re going to add a few “dilithium crystals" to Shirley’s engines so that we can crank her up to 299,999 kilometers per second, just one kilometer below the speed of light. By now, you’re familiar with the drill. At the exact instant Shirley passes over you, George flicks the searchlight on and off. Both you and George measure the time it takes for the light pulse to travel up and down Shirley’s shaft. You record your times, and repeat the exercise over and over, until you both get a stable average. Luckily, your stopwatches are smart, so they help by automatically subtracting your brain processing time, and in addition, your stopwatch automatically subtracts the time for the pulse to travel from Shirley’s transparent floor back to your eyes.

When Shirley was going “only” 150,000 kilometers a second, your stopwatch registered about three tenths of a second more for the light pulse to complete its journey than George’s did. Now, with Shirley moving almost twice as fast, a perfectly reasonable conjecture would be that your stopwatch should register a delay that’s about twice as big, or something in the neighborhood of 0.6 seconds. Consequently, when you and George start the first measurement, you go in half expecting to see the light pulse still arriving only a little later than he does.

But I’m afraid that doesn’t happen. You don’t see the flash after 2.6 seconds. You don’t see it after twenty-six seconds. You don’t even see it after two hundred and fifty seconds! As the time keeps mounting, you begin to panic, worried that you’ve somehow missed the flash and spoiled the experiment. Just as you’re ready to call the whole thing quits, lo and behold, the flash finally arrives! Breathlessly, you look down at your stopwatch, only to see that seven hundred and seventy-five seconds have passed. Almost 13 minutes!

13 minutes? For the first time since you started this whole crazy time measurement gig, you scarcely believe the outcome of your own experiment. How can a two second event be taking 13 whole minutes? Your sense of disbelief persists as you and George perform the measurement again and again, only to confirm: two seconds for him; hundreds of seconds for you.

To make sense of this result, we have to go back to – where else – Pythagoras. In the one second it takes the light flash to go from the searchlight to Shirley’s mirrored ceiling, in George’s world, Shirley has been racing away along the line stretching off to your right at 299,999 kilometers per second. Geometrically, Shirley’s movement has constructed a Line B that is 299,999 kilometers long. Eager to find out how much distance 299,999 kilometers adds to the length of the all-important Line A (the line that traces out the path the light pulse takes from your perspective), you frantically crunch through the familiar Pythagorean calculations. Once again, I’ll spare you the numerical details: Line A turns out to be 424,263 kilometers long. That’s fully 124,263 kilometers longer than Line C, which is equivalent to the path that the pulse takes for George. The resulting triangle is shown in the figure right below.

After one second of George’s time, his light pulse has covered 300,000 kilometers, enough distance to reach the ceiling. After one second of your time, the light pulse has also covered 300,000 kilometers (light travels at the same speed for everybody), but since it has to travel up the diagonal of the right-angle triangle, it has 124,263 extra kilometers still to cover.

That’s a lot of extra distance: About 10 times the diameter of the Earth, in fact. Even traveling at 300,000 kilometers per second, 124,263 kilometers is going to take the pulse almost half a second - 0 .414 seconds to be exact - to traverse. In 0.414 seconds, though, Shirley, zipping along at that blinding 299,999 kilometers per second, moves an additional 124,262 kilometers down the line (0.414 times 299,999), extending Line B by the same amount. By the Pythagorean theorem, that, in turn, lengthens Line A by 95,000 kilometers or so.

Just as it was when Shirley was going 150,000 km per second, successive cycles of space/time creation have begun. The extra time needed for your light pulse to cover that additional 95,000 kilometers along Line A is about 0.317 seconds. But in 0.317 seconds, Shirley moves 80,000 kilometers further along. These numbers are pretty big. Moreover, exactly the same-sized cycles will be created by Shirley’s movement during the one second of George’s time that the light pulse is traveling back down Shirley’s shaft, stretching Line A for the mirror-image triangle by identical amounts (I’ve illustrated that in the figure above, too). So, it’s already clear that the time you’re going to measure for the light flash to reach Shirley’s floor will be larger - quite a bit larger – than when Shirley was traveling at “just” half the speed of light.

Still, just as we saw at that slower speed, the numbers in each cycle are shrinking rapidly. You would be forgiven for thinking that the cycles would, again, collapse fairly quickly to insignificance. However, you’d be wrong! As the cycles start to pile on, an important aspect of the geometry of the situation begins to exert an ever-increasing influence. To understand that aspect, I’m afraid we need to revisit the details of the Pythagorean theorem (yeah, I know. Back to math. Sorry about that).

Recall that according to the theorem (shown again above), to get the length of Line A, you first square the lengths of Line B and Line C. Then, you add the squared values together. Finally, you take the square root of the sum. Remember when Shirley was moving only one kilometer per second, and I talked about the relative contributions that Line B and Line C make to the overall length of Line A? Remember how I took pains to emphasize that squaring the lengths of Lines B and C, and then adding them together, has the effect of inflating any initial difference between them when it comes to how much they donate to the length of Line A? Simply stated, when the lengths of Lines B and C are very discrepant, with one relatively long and the other relatively short, squaring the numbers before adding them together ensures that the length of Line A is almost completely controlled by the length of the longer side.

The figure to the right once again illustrates the geometry of the situation back when Shirley was moving only one kilometer per second (slow relative to the speed of light) making Line B vastly shorter than Line C. With Line C so much more in control of the length of Line A than Line B, the math “forced” Line A to be virtually the same length as Line C. Remember, too, that I took some time out to illustrate the case of a triangle where Lines B and C are of equal lengths? In that case, their squared values are equal too, and the fact that you simply sum their squared values means that they start donating equal amounts of their own lengths to the length of Line A.

That’s the situation we’re in right now. With Shirley moving at 299,999 kilometers per second, after just one second Line B and Line C are virtually the same length (Line B is just one kilometer shorter). Consequently, Line C is supplying just a tiny trifle more than 71% of its length to Line A, and Line B just a tiny trifle less.

In the very next cycle of time/space creation, though, Line B increases by another 124,262 kilometers, making it considerably longer than Line C. As the space/time creation cycles pile on, Line B grows ever longer, while Line C (of course) stays the same. Consequently, it’s Line B that begins to control a larger and larger proportion of the overall length of Line A. Critically, this means that Line B also starts to donate a larger and larger proportion of the portion of its length that was just added in the latest space/time creation cycle.

The top triangle of the three triangles in the figure below illustrates how the situation has evolved by the 20^th cycle of space/time creation, close to the number of cycles that completely “closed things out” when Shirley traveled at only 150,000 kilometers per second. Shirley has moved almost 1.5 million kilometers down the line, making Line B almost five times as long as Line C. In the very next (21^st) cycle, illustrated in the middle triangle of the figure, Shirley moves an additional 30,941 kilometers down the line. Crucially, with the big (and growing) imbalance between the lengths of Line B and Line C, the vast majority of that additional length is donated directly to Line A. Specifically, Line A lengthens by about 30,303 kilometers, only a fraction less than the increase in Line B itself.

It takes just slightly more than a tenth of a second for the light flash to cover that extra 30,000 or so kilometers. In that amount of time, Shirley moves another 29,999 or so kilometers away, and again, virtually all of that additional length is donated to Line A. As the cycles of space/time creation continue to mount, and Line B grows ever longer compared to Line C, the proportion of the additional length of Line B that’s donated to A grows every larger too, until they’re virtually identical. The result is illustrated in the bottom triangle of the figure above by the relative spacing between the large number of light pulses along the diagonals (those spacings are obviously not completely accurate; they are meant to just illustrate the general point). Eventually, things almost settle into perfect equilibrium, where on each cycle, almost as much new distance is added to Line A as was added on the immediately previous cycle. As a result, on each cycle almost as much new space is created for the light pulse to have to cover in the future as it just covered in the past. Even though it’s traveling at 300,000 kilometers per second, the light pulse makes very little headway in its journey toward the top of the triangle (Shirley’s mirrored ceiling), or (after one second of George’s time) in its journey back down to the floor.

Luckily for your patience, though, the amount of Line B that gets donated to Line A never - quite - equalizes. On every successive cycle, the temporal window of opportunity for Shirley to move further down the line grows just a little bit smaller, and the old girl covers just slightly less additional distance compared to the cycle before.

Eventually, the cycle IS choked off, and the light pulse DOES reach the ceiling (floor). As shown by the pile-up of light pulses toward the end, though, it takes many, many, many extra cycles for that to happen. The final Line B distance says it all: by the time the pulse reaches the ceiling, Line B has stretched to over 116 million kilometers, and by the time the pulse reaches the floor, Shirley has covered twice that distance, for a total of 232 million kilometers (in your frame of reference), an impressively large fraction of the distance to Jupiter!

Compared to that colossal distance, the length of line C, which remains fixed at 300,000 kilometers, pales into insignificance. This is why the shape of the triangle is so squashed, and why Line A is now almost identical in length to Line B (of course, the true triangle is a great deal more squashed than this figure can do justice).

Let’s pause here to take stock. We have exactly the opposite geometry of the situation we found when the spacecraft was traveling at only one kilometer per second, and it was Line B whose length was insignificant compared to Line C. See, now, why time dilation is magnified by such a colossal extent when the spacecraft gets very close to the speed of light? Line B has an opportunity to get so long compared to C that virtually all of the length of Line B is donated to A; thus, as Line B grows, so grows Line A, creating ever more distance for your version of the light flash to have to cover.

And, of course, light always takes more time to cover more distance.

With this description, we’ve virtually finished the quest to understand the “time” in time dilation! The key to the entire phenomenon lies in how quickly the spacecraft is traveling, and how quickly the light pulse can reach the floor of the spacecraft. If the pulse can do so in a relatively small number of extra cycles of space/time creation, Line C will always win the competition with Line B for who donates the bigger proportion of their length to Line A, and things won’t get out of hand. If, however, Shirley is moving fast enough, Line B wins the “who donates the most to Line A” sweepstakes, and the time/space creation cycles acquire a life of their own.

Believe it or not, there are only a few loose ends to tie up, now. One of those ends is really nifty, however, so I’m going to leave you today with a teaser to it. We’ve seen that, in your frame of reference, when Shirley travels at very close to light speed, she covers over 230 million kilometers before the light pulse returns to her floor. That distance puts her way out in outer space: in the context of our solar system, 230 million kilometers away marks a location well inside the asteroid belt that lies between Mars and Jupiter.

Meanwhile, what distance would the odometer onboard Shirley herself read when the pulse reaches the floor? 599,998 kilometers, of course: Her speed, 299,999 kilometers per second, multiplied by the two seconds of elapsed time that it takes for George to see the flash. But that is only about one and a half times the distance to the Moon, and is many millions of kilometers short of Mars, let alone the asteroid belt. So. At the exact point in time when the light pulse returns to the spacecraft floor, time dilation appears to have created an enormous discrepancy between where Shirley is located in George’s world, compared to where she is in your world.

This discrepancy raises a major conundrum. Since, from your perspective, Shirley travels all the way into the asteroid belt, it is entirely possible (though unlikely) that an asteroid lies somewhere along her path, and Shirley actually hits the asteroid in a high-speed collision that immediately pulverizes her, and poor George, into dust. If that happened, you’d obviously never see the light pulse, because there’d be no glass floor around to reflect it back to you. You wouldn’t care much about that, because you’d be mourning the loss of poor dedicated George. Meanwhile, from George’s perspective, Shirley doesn’t travel nearly far enough to even enter the asteroid belt. The high-speed collision never takes place, the light pulse reaches the floor with no problem, and George lives to record the time of that event.

Can exactly the same event have two such different histories? How can George both live and die? How can Shirley get pulverized and not get pulverized? I invite you to speculate on how to resolve this paradox in today’s comment section! Or, if you like, just sit tight and wait for me to provide the (in my humble opinion) quite mind-boggling answer!

The Mystery of the Missing Time

2010-01-12T19:01:00.001-08:00

All right, faithful readers! Time’s a wasting, and we have some very important business to take care of today. You and George have been independently measuring the time for the light flash to travel up and down Shirley’s shaft while Shirley is traveling 150,000 kilometers per second (half the speed of light). I captured the geometry in the illustration in the last blog, and I show it again in the left side of the figure below. From your stationary point of view on the ground, in one second Shirley’s movement creates a Line B that’s 150,000 kilometers long. Her motion thus forces your version of the light pulse to have to climb up and down the hypotenuse of the two adjoining right-angle triangles. From Pythagoras, when Line B is 150,000 kilometers long, it adds almost 36,000 extra kilometers to Line A. Adding the same amount to Line A on the way back down (the hypotenuse of the mirror-image triangle), a total of just less than 72,000 kilometers of extra distance is created.

At the speed that light travels, that extra distance should have added about 0.236 extra seconds to your base measurement of two seconds. Instead, though, your average measurement was around 2.301 seconds, or about seven tenths of a second longer than it should be.

Not a big deal, you say? Don’t count on it! To understand the significance of this seemingly trivial discrepancy, I’ll repeat the straightforward question I posed at the end of the previous blog. During the extra 0.236 seconds that the light pulse takes to cover the almost 72,000 extra kilometers along the two diagonals, what is Shirley doing? Answer: She’s continuing to move away from you at the same 150,000 kilometers per second!

But wouldn’t this constant movement make Line B even longer?

Indeed it would! In the 0.118 seconds it takes the light pulse to cover the extra 36,000 or so kilometers up the first hypotenuse, Shirley moves about 17,000 kilometers further away, stretching Line B from 150,000 kilometers, the length it is in the triangles on the left side of the figure above, to 167,000 kilometers, the length in the triangles on the right side of the figure (of course, Shirley is moving at the same speed while the light flash is going up to the ceiling as she is when the flash is going down, so her motion stretches Line B by the same amount while the light pulse is going in both directions).

From Pythagoras’ theorem, any time you add length to Line B, you’re adding length to Line A. Doing the math (I’ll spare you the details), lengthening Line B by 17,000 kilometers adds roughly 8000 kilometers to Line A; this is the source of the number 8 at top of Line A on the “uphill” triangle. But hold on a minute! Isn’t your version of the light pulse going to have to take even more time to cover that additional 8000 kilometers? Sure. Since light travels so fast, its not much time, about 0.028 seconds to be precise (the time above the “8” in the figure). Still, even that little sliver of additional time is enough for Shirley to slide an extra 4,000 kilometers to your right, which is also added directly to the length of Line B. I haven’t shown this in the figure, but 4000 extra kilometers along Line B lengthens that all-important Line A by a little under 2000 kilometers, which takes the light pulse an additional 0.0066 seconds to cover, which allows Shirley to move even a little further away… and so on.

See the pattern? Each additional increment in distance along Line A increases the travel time for your light pulse. That increase in travel time creates an additional temporal “window of opportunity” for Shirley to move even further away, and further lengthen Line B. The bottom line: your spacecraft is now traveling fast enough to set up repeated cycles of space and time creation!

At the same time, though, these cycles are shrinking very rapidly. Within only about 18 cycles in total (each one smaller than the last), the amount of additional distance being added to Line A is driven to virtually zero, choking off any opportunity for Shirley to move further away. The whole space/time creation thing comes to a screeching halt and, even from your perspective, the light pulse reaches the ceiling. Then, the whole routine of space/time creation is repeated while the light pulse comes back down the shaft. Eventually, even in your frame of reference, the pulse reaches the floor, gets reflected back to your eyes, and you hit your stopwatch.

Adding up all the additions to the length of Line B donated by Shirley’s movement from the point of departure (which corresponds to the time when she passed directly over you, and you and George both started your stopwatches) to the return of the pulse to her floor yields a final length of about 173,000 kilometers for Line B, which adds a total of about 46,000 kilometers to Line A. Multiplying 46,000 kilometers by two gives the total increase in the distance your version of the light pulse has to travel, compared to George’s version, of 92,000 kilometers.

How much extra time does it take light to cover that much extra distance? You guessed it… approximately 0.301 extra seconds (actually, it’s .306666 seconds; I’m rounding to keep things straightforward). In other words, exactly the amount of additional time you measured on your stopwatch for the pulse to complete its journey, over and above the flat two seconds measured by George!

We’re really starting to get somewhere, now, readers! The notion that, from your frame of reference, time and space mutually construct each other over successive cycles is kind of amazing. And make no mistake: these cycles are perfectly real. Your stopwatch doesn’t lie about the extra time that the event has taken. As for space, suppose there was an odometer onboard Shirley that recorded the distance she covered from the point she passed over top of you to the point when the light pulse returned to the floor, two seconds later (as measured by George onboard). Since Shirley is traveling 150,000 kilometers per second, the onboard odometer would read exactly 300,000 kilometers. In your frame of reference, though, by the time the light pulse hits the floor, Shirley has traveled almost 50,000 kilometers further. That’s a lot of extra distance out into space!

When you really think about this, it raises all sorts of puzzles, which I’ll explore in future blogs. To avoid straining your brain right now, you might be forgiven for saying to me: “What’s a few tens of thousands of kilometers, and less than a tenth of a second, among friends”? True, there’s still not a dramatic discrepancy between your time measurements and those of George. Does that mean time dilation’s not that interesting, after all? At this point, all I can say is: Hold on to your hats! In the next blog, we’re going to pull out all the stops and crank up Shirley’s speed to 299,999 kilometers per second, just one kilometer less than the speed of light itself.

299,999 kilometers per second is virtually twice as fast as Shirley was going in today’s blog. That extra speed is going to do nothing to George’s measurements, of course. It’ll be the same old boring two seconds for him. However, anybody care to speculate what time you’re going to record on your stopwatch?

At Last! Some Time Dilation!

2010-01-11T18:04:00.000-08:00

In our last blog, we discovered that when Shirley is moving and you are stationary, in your frame of reference the light pulse has to trace out a path up and down the hypotenuse of two right-angle triangles, and so the total distance traveled is more than in George’s frame of reference, where the pulse simply goes up and down in a straight line. As Shirley was traveling only 1 kilometer per second, the additional distance was too short for you to reliably measure the increase in time it took for the pulse to cover it. However, as we left our story, you and George were about to perform the time measurement experiment again, this time with Shirley moving smartly along at fully half the speed of light (150,000 km per second). After your customary hundred or so repetitions of the experiment, George is climbing the walls with boredom, as he keeps measuring the same two seconds for the light pulse to go up and down Shirley’s shaft. Your watch, though, is measuring a shade over 2.3 seconds.

Say what? How can the same event be taking different amounts of time? The difference can’t be blamed on George. He’s been well trained to toggle the searchlight on and off at the exact point he and Shirley pass over top of you. Thus, you have supreme confidence that you and he are starting your stopwatches at the same time. So where is the extra 0.3 seconds and change coming from?

The figure to the right shows where. In the second it takes the light flash to reach her ceiling, Shirley moves exactly 150,000 kilometers along the line to your right, stretching Line B to 150,000 kilometers in length. In George’s frame of reference, the pulse only has to go up and down a path that is the equivalent of Line C. For you, it has to climb up and down the two Line A’s, the hypotenuse of the two right-angle triangles. Looking at the shape of these triangles, it is clear that your “version” of the light pulse has to cover considerably more distance than George’s version.

How much more? Again, let’s work through the standard Pythagorean equation to find out. Remember, to get the length of line A, you multiply the length of Line B by itself (square it), multiply the length of Line C by itself, add the resultant values together, and take the square root of that summed value. I’ll spare you having to go to your calculator: 150,000 (Line B) squared equals exactly 22 billion, five hundred million kilometers (again, see what happens when you square an already huge number?). We know already that squaring Line C gives 90 billion kilometers. Adding these two big values together yields 112 billion, five hundred million kilometers. The square root of that number is just slightly under 336,000 kilometers. Therefore, both Line A going up and Line A going down are close to 336,000 kilometers long, for a total distance of almost 672,000 kilometers.

Whoa! That is roughly 72,000 kilometers longer than the 600,000 kilometers the light flash travels from George’s perspective! Clearly, your version is going to have to take more time to cover that extra distance. How much more? Easy. Light travels at 300,000 kilometers per second, so covering an additional 72,000 or so kilometers takes your pulse about 0.236 seconds of additional time.

The figure illustrates another important way to imagine this situation. For both you and George, the pulse travels at exactly the same rate: 300,000 kilometers per second. For George, that’s enough to get all the way to the ceiling in just one second. For your version of the pulse, traveling up Line A, one second of travel time puts the pulse 300,000 kilometers up the line. However, as you can see from the position of the pulse along the line, that’s only 88% of the way to the top; the pulse is still 36,000 kilometers short of the ceiling. Covering that extra distance is simply going to take more time. There’s no getting around it.

Let’s pause for a minute to take stock. Two different observers, you on the ground and George onboard Shirley, are measuring the duration of an identical event and finding that it takes different amounts of time to complete! The general conclusion from your measurements is irrefutable. When you determine the length of an event that occurs in the frame of reference of a spaceship (or anything else) that’s moving very fast with respect to you, the event covers a longer duration than it does when the event is measured from within the moving frame of reference itself (in this case, measured from “the moving frame of reference itself” means performing the measurement from onboard Shirley). Mathematically, it’s the increasing length of side “B”, brought about by Shirley’s rapid motion, that is “causing” the whole thing. The longer Line B becomes, the longer Line A is, and the more distance the light pulse has to cover.

But guess what? There’s actually a slight problem with this analysis. Recall that the average of the values you measured from your stopwatch was a little over 2.3 seconds, almost a tenth of a second longer than the 2.236 seconds you should have measured if the light pulse was traveling “only” 72,000 extra kilometers (36,000 km on the way up and 36,000 km on the way down). You’re understandably quite eager to sweep this discrepancy under the rug, attributing it to just measurement error perhaps, since it’s pretty small. But I can tell you now, blog readers, this difference is not to be trifled with. As we’ll discover in the next blog, it holds within it the key to the most mind-blowing aspect of the whole time dilation phenomenon!

Would you care to speculate in today’s comment section about where this discrepancy is coming from, and why it is so important? Or just wait? If you want to take a shot, here’s a hint in the form of a question: During the 0.236 seconds that it takes the light pulse to cover that extra 72,000 kilometers, what is Shirley doing?

Ladies and Gentlemen: Pythagoras!

2010-01-09T15:07:00.000-08:00

Picking up exactly where we left off, we’ve now solved the problem of how to ensure that you see the light pulse even though, when the pulse makes it back to Shirley’s floor, the old girl is two kilometers down the line to your right. At risk of belaboring the point, the solution is illustrated (again) in the figure above. The pulse travels up the cylinder to the top while Shirley travels one kilometer away to the right; it travels back down while she moves another kilometer to the right. Meanwhile, the glass in Shirley’s floor has reconfigured itself to reflect the light right back to your eyes. Your stopwatch automatically subtracts the extra time it takes the pulse to cover the distance from Shirley back to you.

The white arrows inside Shirley, showing the path of the light pulse from George’s perspective, illustrate the other important fact that I briefly touched on last time. When the pulse of light reaches the floor, in George’s reference frame it has traveled along a perfectly straight path up to the ceiling and back down, just as it did when Shirley wasn’t moving. This is because George is moving right along with Shirley. It would be just like if George shone a flashlight on the ceiling of a moving train from within the dining car. Both the flashlight and the dining car are moving together, at the same speed, so the beam from the flashlight would go directly up and down. Viewed another way, from George’s perspective, Shirley is perfectly stationary.

From your perspective, though, Shirley is displaced two kilometers when the flash returns to the floor. That simple geometric fact dictates one of the linchpins behind General Relativity and the whole phenomenon of time dilation: From where you’re sitting (actually, from where you’re lying, since you’re flat on the ground), the light pulse cannot have traveled straight up and down! To see the path it’s had to take, look at the really narrow triangle on the left side of the figure to the right. When the light pulse reaches her ceiling, Shirley is physically one kilometer down to your right. To get to the ceiling, therefore, the pulse HAS to have traveled up the slanted side (the hypotenuse) of a very, very, narrow (very acute) right-angle triangle. Then, after the pulse is reflected, it has to travel down the hypotenuse of an identical (but mirror-image) triangle to get back to the floor. As the figure illustrates, the hypotenuse of the two right-angle triangles, which I’m going to label Line A, is ever so slightly longer than the straight line that defines the path that the pulse follows for George. I’ve labeled the line in the triangle that is the equivalent length to George’s path as Line C. That leaves the side of the triangle created by Shirley’s motion down the line to your right as Line B.

Now here’s the crux of the issue. The speed of light is the same everywhere, in all frames of reference. Since the light pulse has to cover more distance for you than it does for George, the pulse has to take more time to complete its journey. How much more time is determined by how much longer Line A is than Line C. Employing some simple grade-school geometry, let’s now compute that distance.

To work it out, we need to use the handy old Pythagorean theorem that you hopefully remember from boring high school math courses. In plain English, the Pythagorean theorem provides a way to calculate the length of Line A based on the lengths of Lines B and C. As shown in the formula below, you multiply the length of Line B by itself, and then multiply the length of Line C by itself, and then add (sum) those two values together. Line A’s length is just the square root of that sum:

For the triangle we’re concerned with, the length of Line B is just the distance Shirley travels in one second, or one kilometer. Line C is the same distance that the light pulse covers from George’s perspective, or 300,000 kilometers. 300,000 multiplied by itself is a ridiculously large number, 90 billion in fact. On the other hand, as far as Line B is concerned, multiplying the number one by itself equals… LOL… one! The sum of these two numbers is 90 billion and one. Obviously, being so extremely close to 90 billion, the square root of 90 billion and one is virtually identical to the square root of 90 billion itself, or 300,000. Thus, the extra distance along Line A compared to Line C is miniscule.

Even thought these calculations may seem quite obvious, working through them enables me to draw your attention to a couple of things that will become important later on. First, note how the math jives completely with the physical shape of the triangle in the figure. My illustration doesn’t begin to do justice to the small size of the slope of line A; in reality, the slope is so small that if they were superimposed, A would overlay C almost perfectly. Clearly, when line C is super long and B is super short, virtually all of the length of the hypotenuse (Line A) is determined (though I actually prefer the connotations of the word “donated”) by the much longer line C.

Turning this around, and comparing the figure to the math, the visual impression that Line A is almost equal in length to Line C jives perfectly with the Pythagoran theorem. When you square a number that’s huge to begin with, you end up with a humungous number; when you square a very small number, the result stays pretty small. When you add them together (and then compute the square root), the big number (the long line) contributes virtually everything to the result, while the small number (the short line) contributes virtually nothing.

Forgive me, patient readers because, at this point, I’d like to take a short detour from our main thread to more thoroughly illustrate how the relative lengths of B and C determine what proportion of the length of Line A is controlled by the one line compared to the other. In the triangle we’ve been discussing up to now, Line B is very short compared to Line C. But now, please direct your attention to the triangles depicted in the figure below, where Lines B and C are equal in length (both 300,000 kilometers long). Since now, Line B equals the length of Line C, B squared and C squared are both 90 billion, and when you sum the two, for a total of 180 billion, and compute the square root, you find that they contribute exactly equal amounts to the length of side A (which turns out to be 424,264 kilometers). In other words, of Line A’s total length, B and C both “donate” exactly half, 212,132 kilometers, or just over 71 percent of their own lengths. The crucial take-home message: Once their lengths are equalized, Lines B and C contribute equal amounts to the length of Line A.

Don’t believe me? Pick some examples of your own, say a triangle where B and C are just one kilometer long, run the numbers through the theorem, and confirm it for yourself! And suppose you went back to our target triangle, where Line B is a tiny fraction of Line C, and used the Pythagorean theorem to calculate the length of Line A while gradually increasing the length of Line B. What you’d find, of course, is that the longer Line B got relative to Line C, the more Line B would “donate” to the length of Line A.

The importance of what happens when Line B approaches (and then exceeds) the length of Line C won’t become obvious until later blogs. For now, let’s return to our main thread, where Shirley is traveling only one kilometer a second, and Line B is miniscule compared to Line C. As I mentioned earlier, the tiny extra distance “donated” to Line A by the presence of Line B takes the light pulse such a tiny amount of extra time to traverse that it is way below the sensitivity of you and your stopwatch to measure. For all extents and purposes, both you and Astronaut George are still in the same “time zone”, and Einstein has still not really shown up to the party.

But that’s about to change. In the next blog, you and George are going to crank up Shirley’s speed to something far more substantial: fully half the speed of light (150,000 kilometers per second). That’s just absurdly fast, almost 2,000 times faster than as the fastest spacecraft ever launched. It is so fast, in fact, that in just the single second it takes for the light pulse to travel up to Shirley’s ceiling, the old gal travels 150,000 kilometers along the line to your right (and creates a Line B that’s 150,000 kilometers long). Add the additional second it takes the pulse to return to the spacecraft floor, and Shirley will travel a total of 300,000 kilometers away to your right!

What will such a blinding speed do to the shape of our all-important triangles? What will it do to the relative lengths of Lines B and C? If you get really ambitious, you could easily run the numbers through the Pythagorean theorem yourself, and then compute the distance the light pulse will now have to travel. What value will you, stationary on the ground, register on your stopwatch for how long it takes the pulse to complete its journey? And what value will George register?

Stay tuned!

Shirley's Magic Bottom

2010-01-08T17:50:00.001-08:00

Let’s recap. You are trying to measure the duration of an event, the journey of a pulse of light up and down the shaft of your spacecraft, Shirley. In the current version of the experiment, you’re stationary, looking up from the ground, while Shirley is moving at one kilometer per second along the line formed by your outstretched right arm. You and George start your stopwatches at the same time, when George toggles the searchlight on and off while Shirley is directly above you. The question is, do you and George measure the same amount of time for the light flash to travel up to the ceiling and back?

Before tackling that question, we first have to deal with an obvious problem that rears its ugly head now that Shirley is moving. We know that from George’s perspective it takes the flash two seconds to travel up and down Shirley’s shaft. By the time the flash gets back to the floor, therefore, Shirley is no longer directly above you; she’s two kilometers down the line to your right. That’s no problem for George; being inside Shirley, he still sees the flash as he always has. But obviously, you can no longer see it because, when it passes through the glass floor, you’re not there! As things, stand, you don’t have any event on which to stop your stopwatch.

To deal with that little complication, I’d like to take some poetic license and add a rather advanced capability to Shirley’s bottom. Specifically, the molecules of the glass that make up her ass automatically realign such that, when the light pulse hits the floor, it is reflected at just the right angle to travel directly back into your eyes. That way, both you and George see the return flash of light, and you both have an event to stop your stopwatches with.

Physically, this situation is illustrated in the figure above. Since George is inside the spacecraft, he’s moving right along with the pulse. From his perspective, the pulse goes straight up to the ceiling and straight back down, just like it did when Shirley (and George) weren’t moving. The white arrows going straight up and down inside Shirley illustrate the path that the pulse takes for George. Meanwhile, when the flash reaches the floor it is steered immediately along the dotted line right back to you. In this way, you continue to see the flash, too.

OK, how does the value on your stopwatch compare to the value on George’s? Light takes only a tiny amount of extra time, 1/150,000 of a second to be exact, to cross the two kilometers separating you from Shirley. Although the flash is reaching you just a teensy, tiny bit later than it reaches George, the difference is far too small to show up with a measuring device as crude as a human operating a stopwatch. Still, you’re a stickler for accuracy. You know that, compared to George, the light has had to travel a longer distance, which means that you are not… quite… measuring the duration of the same physical event that George is. But your smart stopwatch comes to the rescue! You simply program the watch to automatically subtract the additional time required for the flash to travel from Shirley to you (recall that your watch is already automatically subtracting your brain processing time).

Does the reconfigurable glass floor solve all the measurement problems that accompany the fact that Shirley is now moving? Possibly not, because something else about this whole situation is starting to gnaw at you. The figure provides a strong hint to the problem. Now that Shirley is moving, what is the exact path through space that the light has to take from your perspective in order to get to the ceiling and back? Is it possible that this path is different from the path that it takes from George’s perspective (which is just straight up and down)?

If so, how is it different? And why? The more you take a crack at answering that question, and “stay ahead” of the blogs, the less pain you’re going to have later on when Einstein finally crashes our little party! Anybody care to tackle this issue in the comments section?

Introducing Astronaut George!

2010-01-07T20:29:00.000-08:00

So, have you been doing your homework, faithful Whabblog readers? Have you been performing the thought experiment repeatedly, and always getting the same two second result (once your smart stopwatch automatically subtracts your “brain processing” time)? If you have, then chances are you’ve gotten thoroughly sick and tired of the whole, dare I use the word, tedious, activity, because the outcome is always the same! Your spreadsheet never shows anything but two-second entries.

To spice things up a little, you decide to hire an astronaut accomplice. Since “accomplice” kind of sounds like you’ve decided to rob a bank or something, perhaps “collaborator” or even “confederate” is a better label. Your “confederate”, who for entirely pernicious reasons is actor George Clooney, agrees to lie on the floor of the spacecraft in your stead. Therefore, it is now Astronaut George who toggles the searchlight on and off, and measures how long it takes to see the pulse of light on its return from the ceiling. When George does this, he gets the same two-second result you did (of course, his stopwatch is smart enough to automatically subtract his brain-processing time, too).

Meanwhile, where have you gone? Not far, because you want to continue to measure the travel time of the light flash right along with Astronaut George. Therefore, you arrange for Shirley to be tethered just a couple of feet off the ground – just high enough that you can lie flat on the ground underneath the transparent floor, and stare straight up through the floor in the direction of the ceiling (just like George is doing inside Shirley). And just like George, you’re going to start your stopwatch at the exact point when he flicks the searchlight on (remember, the searchlight is bidirectional, so you immediately see the light coming through the floor), and you’re going to push your stopwatch again, halting it, when you see the return flash. Of course, when the reflected flash reaches Shirley’s transparent floor, it only has to pass through the glass and travel another couple of feet to reach you – a negligible distance. Thus, not only do you both start your stopwatches at the same time, you stop them at the same time, too. Both stopwatches register exactly the same two-second duration for the light pulse to complete its journey.

Still, now it’s both of you, not just you yourself, who keep getting the same result, and two people are now getting restless and bored. The thought experiment, with its predictable two-second result on both stopwatches, is still rather tedious. This is all fine and dandy, you’re probably beginning to think, but what about Einstein?

Relax. It’s always darkest just before the dawn! Shirley is a spaceship, remember, so why not repeat the identical experiment with you remaining motionless in the same position on the ground, but with Shirley now in motion? To accommodate this slightly more complex situation, you and Astronaut George work out the following arrangement. Lying on the ground, you stretch your arms straight out to your left and right, so your body forms the shape of a cross. George takes control of Shirley and, still hovering at the same short distance above the ground from you, maneuvers her to a point over on your left, directly along the line formed by your outstretched left arm. Next, George accelerates Shirley to a constant speed, so that she (and he) move steadily towards you along the straight line. Still holding the same constant speed, Shirley passes right over top of you and then continues down the line formed by your outstretched right arm.

For your first time measurement experiment with Shirley in motion, you decide to have her move at a steady one-kilometer per second along the line. Although that’s actually pretty darn fast, about the speed of a supersonic military jet, it pales in comparison to the speed of the light itself: One kilometer per second is only 1/300,000 of THAT ridiculous speed! Still, it’s fast enough that Astronaut George has to take great pains to toggle the searchlight on and off at the exact instant Shirley passes over you. That way, you still see the flash of light through the floor of the spacecraft when George toggles the searchlight on, and you both START your stopwatches in unison.

But do you both still see the light flash return to the spacecraft floor at the same time? Do you both stop your stopwatches together, and measure the same two seconds for the event to happen? That is the question I’ll tackle in the next blog. In the meantime, feel free to speculate on the answer!

Tripping the light fantastic with Shirley

2010-01-06T21:02:00.000-08:00

In this second installment, let’s plunge right into a “thought experiment”, shall we, which is quite apropos since thought experiments were one of Einstein’s favorite investigative tools. Suppose you all suddenly became obsessed (and I mean seriously obsessed) with measuring precisely how long it takes events to occur (in other words, you went seriously crazy over measuring time). To satisfy this new interest, you construct a rather unique spaceship, shaped like a thin cylinder fully 300,000 kilometers (186,000 miles) in length. A structure of that length is kind of hard to imagine, I know. With the floor of the ship resting on the earth’s surface, the top is way out in space, about two thirds of the way to the moon. Nevertheless, because we can imagine anything in a thought experiment, please imagine that you manage to build it. Once construction is complete, you christen your ship “Shirley”, after a friend of mine in Nova Scotia.

Compared to the incredibly long distance from floor to ceiling, Shirley isn’t very wide: her rounded floor is just large enough to hold one passenger and a strong directional searchlight, like the ones used to send beams of light dancing across the sky at movie premieres. Imagine further that the floor is made of a layer of thick glass, fully strong enough to support a human passenger and the searchlight, but completely transparent (hint: the floor is glass so that somebody lying directly beneath the spacecraft can easily see the light from the searchlight. But I’m getting ahead of myself). Shirley’s searchlight is bidirectional, by the way: When switched on, it sends a beam of light both straight up and straight down through the floor.

300,000 kilometers above the floor, Shirley’s roof consists of a fully reflective mirror. The mirror is there for one purpose: to enable you to measure how long it takes for a brief flash of light, emitted by the searchlight, to travel the distance from the floor of the cylinder to the top, reflect off the mirror, and travel back down to the floor. With this goal firmly in mind, you proceed to enter Shirley and lie down on the glass floor, facing straight up. Right beside you is the bi-directional searchlight. At risk of belaboring the obvious, when the searchlight is powered on, the light it produces will travel in a perfectly straight line up the length of the cylinder, reflect off the mirror, and come straight back down. When the flash returns to the floor, it is still plenty strong enough to see the reflection.

You measure the light’s travel time by performing two simple tasks in quick succession. First, using your left hand, you toggle the searchlight on and off, creating just a brief flash of light. In your right hand, you hold a standard stopwatch, the kind that you start and stop by pushing the same button on the top. Now (we’ll assume you get a lot of practice at this) at exactly the same time as you flick the searchlight on, you press the button to start the stopwatch. Then, when you see the light flash reflected from the ceiling, you press the button again. Voila: the amount of time showing on your stopwatch is the journey time for the light! Quite excited at your own cleverness, you perform this little experiment over and over, entering the result from your stopwatch in a spreadsheet each time.

So what do you find? The figure to the right illustrates the situation and provides the straightforward answer. The mirror in Shirley’s ceiling is 300,000 kilometers away from you and the searchlight. By astonishing coincidence, light happens to travel at a speed of almost exactly 300,000 kilometers per second (or about 650 million miles per hour; rather fast, in other words). Therefore, the flash of light takes precisely one second to travel up to the mirror, and another second to make the return journey, for a total round-trip time of two seconds. In the figure, notice that the light pulse (the red squiggly line) spreads out slightly on its way up, so it reflects off of every part of the mirror. Thus, you see part of the reflection on the way down. And when you look at the stopwatch, two seconds is the value you should register.

So far, there’s nothing very complicated about this thought experiment, is there? The time to complete a journey by anything, light included, is simply the distance traveled divided by the speed of the thing doing the traveling. The total distance up and down the cylinder is 600,000 kilometers; light travels at 300,000 kilometers a second, so the total travel time is 600,000 divided by 300,000, or exactly two seconds.

Of course, human physiology enters the picture and muddies the waters a little bit. As you do the measurement over and over and write down the result in your spreadsheet, you quickly come to see that the values differ by small amounts, and the average of all those times is a little more than two seconds. This is because, before you can press the button to stop the stopwatch, your eye has to be stimulated by the light flash, and your brain has to respond to that stimulation by issuing a command to your finger to press the button. These activities have a variable duration, so they delay your button press by a slightly different amount each time you repeat the experiment. If you are like most people though, on average, the delay will be about two-tenths of a second (trust me; I’m a psychologist)!

Suppose that you were able to subtract the physiological delay from the value showing on your stopwatch. You would be left with the actual two seconds of time it took for the flash to complete its journey and return to Shirley’s floor. From now on, let’s imagine that the stopwatch is actually a pretty smart device, smart enough to automatically compute your “brain processing” time and subtract it from the value showing. That way, you always get a pure and totally accurate measure of the time it takes the light pulse to travel up and down Shirley’s shaft.

Are you with me so far? That is, have you got Shirley’s dimensions, and the path taken by the light pulse, so firmly established in your head that it is quite obvious why the pulse is taking exactly two seconds to complete its journey?

Excellent! In the next blog, we’ll take the thought experiment in a direction that I hope will pique your interest. Until then, feel free to speculate in the comments section on what you think I’m going to do next.

Hint: You’ve deliberately put your measurement apparatus inside a spaceship!

A blog beginning!

2010-01-05T19:13:00.000-08:00

Over the last 500 years, painstaking work by generations of scientists has produced an enormous “bank” of scientific knowledge. From the elegant yet simple standard model of physics that explained the fundamental building blocks of matter, to the periodic table of the elements that unlocked the secrets of chemical reactions, to the plate tectonic theory that explained earthquakes, volcanoes, and mountain building, and finally to the big bang theory that uncovered the origin and evolution of the cosmos, each scientific discipline has contributed a key piece of the overall picture. Today, 10 years in to the 21st century, that picture has meshed into an extraordinary consistent body of knowledge about the nature and history of the universe.

Unfortunately, there’s far too much knowledge in the bank for any one person to grasp more than a small fraction of it, and scientists are making new deposits at a faster clip than ever before. Making matters even worse, most of the knowledge in the "bank of reality" is expressed in a specialized language, often involving advanced and esoteric forms of mathematics, that erects an insurmountable barrier between the bank and most nonspecialists.

Last January, a dear friend named Karl Baker, who I met through my husband Jim, died unexpectedly from complications of colon surgery. Barely into his 60’s when he met this decidedly untimely death, Karl had already established himself in my eyes as an “uncommonly uncommon” person. For example, although Karl lacked any formal training in the hard sciences, he had a keen interest in cosmology, the study of the nature, history, and fate of the universe. So strong was his curiosity that he actually researched the topic enough to write a lengthy and very interesting essay on the Big Bang.

Shortly after Karl died, I decided to write a blog to mark and honor his unusual willingness to wrap his mind around an esoteric scientific concept. Little did I know that this decision would mushroom into a series of blogs, complete with figures, diagrams, and even (horrors!) a little simple mathematics (very simple, I promise). But, mushroom it certainly did!

My goal with these blogs is admittedly rather ambitious. I aim to slowly, carefully, and cautiously familiarize you with time dilation, a key aspect of Albert Einstein’s special theory of relativity. Why did I think this particular deposit in the Bank of Reality was worth unpacking to try to make it accessible to nonscientists? There are actually several reasons. First and foremost, in my humble opinion time dilation ranks up there with Newton’s laws of gravity and Maxwell’s equations explaining electromagnetism as one of the most important additions ever made to our understanding of physical reality. Simply stated, if you come to understand time dilation, you’ll have mastered one of the most important deposits in the bank. Second, there are good old-fashioned bragging rights to consider. Einstein is a famous figure even outside of scientific circles, but almost no one understands anything about his theories, and so no one has a clue as to why he’s so revered. Imagine the satisfaction that would come with living the rest of your life, smug with the realization that you “get” an important aspect of one of Albert Einstein’s theories! Imagine the thrill of being able to explain it to your children, grandchildren, or your students?

And, last but not least, learning about time dilation is actually fun! The details are so eerie, so bizarre, and ultimately so breathtaking, that they seem like science fiction. Yet, time dilation rests on a simple set of real-world conditions, so concrete that they can be grasped with just the slightest use of elementary mathematics.

We’ll get into the nitty-gritty of time dilation in the next blog. Today, I just wanted to introduce you to what it is I’m going to be writing about, and why. I’d really like the blog to be an interactive activity because, well, isn’t that the point of blogging? So, the question is, are you willing to be interactive, too? Are you ready to take me to task in the comments section as soon as you hit a section or an explanation that’s too dense to understand, so that I can address the confusion and keep you from being left behind? Are you ready to guess what the solution is to the little mystery or conundrum that I’ll pose at the end of each blog (and then answer in the next)? Ok, then. To kick things off, I’d love to know: have you ever given Albert Einstein’s scientific contributions any serious thought? To what his scientific contributions actually were, I mean, and to what he actually discovered? Go ahead, blog readers. Share away!