Tuesday, June 29, 2010

The Cloud in God's Eye


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.

Thursday, June 24, 2010

You'll love it at Leavitt's!


By the beginning of the 20th 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. 

Monday, June 14, 2010

Breaking the Inverse Square Law


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 20th 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.