I took the above panoramic view, spanning something like 120 degrees, in a local park towards the end of last year. The sun was almost on the horizon to the southwest, at right of frame. The moon was well risen in the southeast, framed by the little red box in the image above. After taking the panorama, I zoomed in for the enlarged view of the moon shown in the inset, to demonstrate the apparent problem. The moon is higher in the sky than the sun, but its illuminated side is pointing slightly upwards, rather than being orientated, as one might expect, with a slight downward tilt to face the low sun.

This appearance is quite common, whenever the moon is in gibbous phase (between the half and the full), and therefore separated by more than 90 degrees from the sun in the sky. Every now and then someone notices the effect, and decides that they have to overthrow the whole of physics to explain it. I could offer you a link to a relevant page, but I won’t—firstly, I don’t like to send traffic to these sites; secondly, you might be driven mad by the experience and I’d feel responsible.

Actually, the illuminated part of the moon is pointing directly towards the sun; it just doesn’t look as if it is. So (as with my previous post “Why Do Mirrors Reverse Left And Right But Not Up And Down?”) the title of this post is an ill-posed question—it assumes something that isn’t actually so.

Here’s a diagram showing the arrangement of Earth, moon and sun in the situation photographed above:

The Earth-bound observer is looking towards the setting sun. Behind and above him is the moon, its Earth-facing side more than half-illuminated. The sun is so far away that its rays are very nearly parallel across the width of the moon’s orbit. In particular the light rays bringing the image of the setting sun to the observer’s eyes are effectively parallel to those shining on the moon—the divergence is only about a sixth of a degree.

But we know that parallel lines are affected by perspective. They appear to converge at a vanishing point. The most familiar example is that of railway lines, like these:

But there’s a problem with this sort of perspective. To illustrate it, I took some photographs of the top of the very low wall that surrounds the park featured in my first photograph:

The views look north and south towards two opposite vanishing points. The surface of the wall is marked with the remains of the old park railings, which were sawn off and removed during the Second World War. These provide a couple of reference points, which I’ve marked with numbers. The parallel sides of the wall appear to diverge as they approach the camera towards Point 1; and they appear to converge as they recede from the camera beyond Point 2. But what happens between 1 and 2?

I used my phone camera again to produce this rather scrappy and unconventional panorama, looking down on the top of the wall and spanning about ninety degrees:

The diverging perspective at Point 1 curves around to join the converging perspective at Point 2. It’s mathematically inevitable that this should happen—what’s surprising is that we’re generally unaware of it. In part, that’s because our normal vision spans a smaller angle than we can produce in a panoramic photograph; but it’s also because our brains are very good at interpreting the raw data from our eyes so that we see what we need to see. In this case, as we scan our eyes along the length of this wall, we have the strong impression that its sides are always parallel, despite the fact that its projection on our retinas is more like a tapered sausage with a bulge in the middle.

So: our brains are good at suppressing this “curve in the middle” feature of parallel lines in perspective, at least for simple local examples like railway lines and walls.

Now let’s go back to those parallel light rays coming from the sun and illuminating the moon. Like railway tracks, they’re affected by perspective. In the photograph below, the setting sun is projecting rays from behind a low cloud:

Although the rays are in fact parallel, perspective makes them seem to radiate outwards in a fan centred on the sun. I’ve written about these crepuscular rays in a previous post, and at that time suggested that whenever you see them you should turn around and look for anticrepuscular rays, too:

These converge towards the antisolar point—the point in the sky directly opposite the sun—and they’re produced by exactly the same perspective effect. Which means solar rays have to do the same “diverge, curve, converge” trick as the sides of my park wall. Unfortunately, crepuscular rays tend to fade into invisibility a relatively short distance from the sun, and to reappear as anticrepuscular rays only a relatively short distance from the antisolar point. So we can’t visually track their grand curves across the sky.

But we can see the effect of that perspective curvature when the low sun illuminates a gibbous moon. Here’s a diagram of a sheaf of parallel solar rays, as they would appear when projected on to the dome of the sky:

Perspective makes the sun’s rays diverge when the observer looks towards the sun, but converge when the observer turns and looks at the antisolar point. Because the sun is sitting on the horizon, all the rays in my diagram above are not only parallel to each other, but also to the horizon. And because the gibbous moon is more than ninety degrees away from the sun, it’s illuminated by rays that are apparently converging towards the antisolar point on the horizon, rather than spreading outwards from the sun.

So the impression that that the moon’s illuminated portion doesn’t point towards the sun is a very strong one. This is because the scale of the moon-sun perspective is very much larger than the examples for which our brains have learned to compensate. The moon is the only illuminated object we see which is further away than a few kilometres, and our brains otherwise never have to deal with grand, horizon-spanning perspectives in illumination. So our intuitions tell us that the light rays illuminating the moon in the diagram above can’t possibly have come from the sun, since they’re apparently descending towards the antisolar point.

Standing in the open, observing the illusion, I find it impossible to mentally sketch the curve from sun to moon and see that it’s a straight line. Nothing that rises from one horizon and descends to the other horizon can possibly be a straight line, my brain insists, despite its cheerful acceptance that the straight, parallel sides of my park wall can appear to diverge and then converge in exactly the same way.

In the old days the approved way of demonstrating that there really was a straight line connecting the sun to the centre of the illuminated portion of the moon was with a long bit of string held taut between two hands at arm’s length. Placing one end of the string over the sun, and then fiddling with the other end until it intersected the moon, one could eventually produce a momentary impression that the straight line of the taut string really did alignment with the illuminated side of the moon. But it was all a bit unsatisfactory.

But now we have panorama apps on our phones. The one I use stitches together multiple images, and provides an on-screen guide to ensure that each successive image aligns with its vertical edge parallel to the image before—it forces the user to stay aligned in a single plane as they shift the viewing direction between successive frames. Usually, the object of the exercise is to scan along the horizon to obtain a wide-angle view of the scenery. But (as my odd little downward-looking panorama of the park wall demonstrated) it isn’t necessary to start the panorama with a vertically orientated camera aimed at the horizon.

So, back in the park and shortly after I took the image at the head of this post, I aimed my phone camera at the moon, and tilted it sideways so that it aligned with the tilted orientation of the moon’s illuminated portion. Then I triggered my panorama exposures and followed the on-screen guides—which led me across the sky in a rising and falling arc until I arrived at the setting sun!

Here’s the result:

So now perspective makes the horizon appear to curve implausibly, while the illuminated portion of the moon quite obviously faces directly towards the sun.

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I published my original “We Are Stardust” post some time ago, introducing the infographic above, which shows the cosmic origins of the chemical elements that make up our bodies, according to mass. At that time I concluded that Joni Mitchell should actually have sung “We are 90% stardust,” because that’s the proportion of our body weight made up of atoms that originated in the nuclear fusion processes within stars. The remaining 10% is almost entirely hydrogen, which is left over from the Big Bang.

The original post got quite a lot of traffic, largely courtesy of the Damn Interesting website. But it also prompted one correspondent to ask, “But what proportion of our atoms comes from stars?” Which is an interesting question, with an answer that requires a whole new infographic.

If you want to know more about the background to all this—how various stellar processes produce the various chemical elements, and the function of those elements in the human body—I refer you back to my original post.

This time around, I’m just going to take the various weights by element I used in my last post, and divide them by the atomic weight of each element. There’s a wide range of atomic weights among the 54 elements on my list of those present in our bodies in more than 100-microgram quantities. The heaviest atoms in that group, like mercury and lead, are more than 200 times heavier than the lightest, hydrogen. So each microgram of hydrogen contains 200 times more atoms than a microgram of mercury or lead. And that skews the atomic make-up of the human body strongly towards the lighter elements, and particularly to those lighter elements that are common components of our tissues.

Most of our weight is water, which consists of hydrogen and oxygen, making these two elements the most common atoms in our bodies. The carbohydrates and fats in our tissues also contain hydrogen and oxygen, along with a lot of carbon, which is our third most common atom. Proteins contain the same three elements, along with nitrogen and a little sulphur. And in fact the four elements hydrogen, oxygen, carbon and nitrogen, all relatively light and all relatively common, account for almost all the atoms in our bodies. The seven kilograms of hydrogen in a 70-kilogram person accounts for 62% of all that person’s atoms. Oxygen accounts for 24%, carbon 12%, and nitrogen 1%. That leaves just 1% for the fifty other elements on my list.

The calcium and phosphorus in our bones and dissolved in our tissues account for a further 0.5%. The only other elements present at levels greater than 0.01% are the sulphur in our proteins, and the sodium, magnesium, chlorine and potassium which are dissolved as important ions in our body fluids. Everything else—the iron in our haemoglobin, the cobalt in Vitamin B12, the iodine in our thyroid glands—accounts for just 0.003% of our atoms.

Hydrogen is the major element left over from the Big Bang, so our atoms are dominated by that primordial element. Oxygen comes almost entirely from core-collapse supernovae, and so is the main representative of that stellar process in our bodies, along with significant amounts of carbon and nitrogen. But most of our carbon and nitrogen was blown off by red giant stars, and those two elements account for most of our atoms from that source. In fact those three sources—the Big Bang, core-collapse supernovae and red giant stars—provided almost all our atoms:

If you compare the graphic above to the one at the head of this post, you can see how the balance has shifted strongly towards hydrogen (with its very light atoms) and away from oxygen (with atoms sixteen times heavier than hydrogen). And the even heavier atoms from Type Ia supernovae are so rare I can’t now add them visibly to my graphic.

So perhaps Joni Mitchell should have sung, “We are 38% stardust.”

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We are stardust
We are golden
And we’ve got to get ourselves
Back to the garden

Joni Mitchell “Woodstock” (1970)

A few months ago I ran into the periodic table above, detailing the cosmological origins of the chemical elements. And it occurred to me that I could quantify Joni Mitchell’s claim that “we are stardust”. How much of the human body is actually produced by the stars? But before I get to that, I should probably explain a little about the various categories indicated by the colours in the chart above.

As the Universe expanded after the Big Bang, protons and neutrons were formed from quarks, and there was a brief period (a few minutes) when the whole Universe was hot and dense enough for nuclear fusion to take place—just enough time to build up the nuclei of a couple of very light elements, but not enough to produce anything heavier in any significant quantity.* So what came out of the Big Bang was hydrogen, helium and a little lithium—the first three elements of the periodic table. The rest of the chemical elements that make up the solar system, the Earth, and our bodies were produced by fusion reactions inside stars.

The first stars formed about 100 million years after the Big Bang, and were composed entirely of hydrogen, helium, and a little lithium. Conditions at that time favoured the formation of large stars, which burned through their nuclear fuel quickly. A series of fusion reactions in such massive stars generates energy, producing progressively heavier atomic nuclei until the “iron peak” is reached—beyond that point, the creation of heavier nuclei requires an input of energy. When the star exhausts its energy reserves in this way, it collapses and then explodes as a core-collapse supernova (an “exploding massive star” in the table above). This propels the elements synthesized by the star out into space, and also drives a final burst of additional fusion as shock waves sweep outwards through the body of the star, producing a few elements that lie beyond the iron peak, as far as rubidium. The remnant of the supernova’s core collapses into a neutron star, or perhaps even a black hole.

These elements produced by the first supernovae seed the gas clouds which later condense into subsequent generations of stars. The lower-mass stars which appeared later in the Universe’s lifetime (such as the sun) are unable to drive internal fusion as far as the iron peak, and stall their fusion processes after producing only the relatively light nuclei of carbon and oxygen. They evolve into red giants stars (“dying low-mass stars”, above), puff off their outer layers, and then subside to become white dwarfs. But during this process they are able to breed higher-mass elements from the metal contaminants they inherited from the first supernovae. Neutrons from the star’s fusion reactions are absorbed by these heavy nuclei, gradually building them up into ever-heavier elements with higher atomic numbers, as some of the neutrons convert to protons by emitting an electron (beta decay). This building process finally “sticks” when it gets to Bismuth-210, which decays by emitting an alpha particle (two neutrons and two protons), rather than an electron. So, perhaps counter-intuitively, the gentle wind from low-mass stars in their red-giant phase enriches the interstellar medium with heavier atoms than does the spectacular explosion of a supernova.

But once a low-mass star has finished its red-giant phase and settled to become a white dwarf, it may (rarely) manage to turn into a supernova. For this to happen, it must have a companion star orbiting close by, which expands and spills material from its own outer envelope on to the white dwarf’s surface. Once a critical mass of material accumulates, a runaway fusion reaction takes place and blows the white dwarf apart in what’s called a Type Ia supernova. The nature of the fusion reaction is different from what occurs in the core-collapse supernovae, so Type Ia supernovae (“exploding white dwarfs”, above) eject a slightly different spectrum of elements into space—in particular, they don’t create any elements beyond the iron peak.

The processes described so far account for almost all the elements up to bismuth. To produce heavier elements requires a massive bombardment of neutrons, building up nuclei faster than they can decay and thereby pushing beyond the “bismuth barrier” I described earlier. Such torrents of neutrons occurs when neutron stars collide. As their name suggests, neutron stars contain a lot of neutrons, and if two of these supernova remnants are formed in close orbit around each other they may eventually collide. This unleashes a massive blast of neutrons, which bombard the conventional matter on the surface of the neutron stars, building up heavy radioactive elements before they have a chance to decay, and ejecting these products into space.

Finally, a few of the lightest elements are formed by cosmic rays (particles generated during supernova explosions). When these rapidly moving particles strike carbon or oxygen nuclei in space, they can break them into lighter fragments. This process accounts for almost all the beryllium and boron in the Universe, and some of the lithium.

Here on Earth, we have a few more processes that contribute to the mix of chemical elements around us. There’s natural radioactive decay, which is slowly converting some chemical elements into slightly lighter ones. And there are artificial radioactive elements, which we produce in our bombs and nuclear reactors. But these are essentially minor processes in the scheme of things, and I feel safe to ignore them here.

It should come as no surprise that our bodies are made up primarily of the most common chemical elements in the Universe—that is, hydrogen from the Big Bang, and those elements from early in the periodic table which are most frequently spewed out by dying or exploding stars. Indeed, apart from the noble gases helium, neon and argon, an adult human body contains significant traces (more than 100 micrograms) of every element from hydrogen to the “iron peak” elements that represent the limits of equilibrium stellar fusion processes. And a surprising number of these elements have biological roles.

The water that makes up the bulk of our bodies is composed of hydrogen and oxygen. The fats and carbohydrates of our tissues consist of hydrogen, oxygen and carbon, and our proteins add nitrogen and a little sulphur to that mix. Calcium and phosphate are the major structural components of our bones, and sodium, potassium, magnesium and chlorine are present as dissolved ions in our body water, regulating the activity of our cells.

In smaller quantities, there are elemental micronutrients that we must have in our diet in small quantities to stay healthy. Iron, an oxygen-carrying component of haemoglobin and myoglobin, is the major such micronutrient. Iodine is required for thyroid function, cobalt is a component of Vitamin B12, and chromium is present in a hormone called Glucose Tolerance Factor. To these we can add manganese, copper, zinc, selenium and molybdenum, all of which are required for the function of various enzymes. People eating anything approximating a normal diet obtain all these latter elements in adequate quantities, but they must be carefully provided for patients reliant on intravenous feeding in Intensive Care Units.

A few elements seem to produce deficiency syndromes in experimental animals that are fed very carefully controlled diets. Silicon, vanadium, nickel and tin fall into this group, but their biological role, and relevance to humans, is unknown. And then there are the elements which are known to be present in the human body, but appear to have no function—they’re probably just in our tissues because they’re in our food. Some are industrial contaminants, like mercury and lead; but some, like lithium and boron, are probably just part of the natural environment.

Estimates of the elemental make-up of the human body vary. I’ve used the figures quoted by John Emsley in his marvellous book Nature’s Building Blocks: An A-Z Guide to the Elements, and found 54 elements that are present in a 70-kilogram human in quantities that exceed 100 micrograms.

Summing the proportions of all these elements that come from different cosmological sources, I was able to produce this infographic:

(There are no prizes for identifying the original source of the human outlines used.)

Hydrogen, although the most common element in the human body, is also the lightest. So it accounts for just 10% of our weight, and that is, to a good approximation, the only component of our bodies that originated in the Big Bang, because we contain helium and lithium in only tiny quantities.

So the rest was produced, directly or indirectly, in stars. The oxygen in our body water and in our tissues accounts for the large majority of that, originating from core-collapse supernovae. Carbon and nitrogen in our tissues makes up much of the remaining mass, mainly coming from the stellar wind produced by low-mass stars in their red-giant phase. And the rare white-dwarf explosions, Type Ia supernovae account for just 1% of our weight, producing a significant fraction of the calcium and phosphorus in our bones, and some of the important ions dissolved in our body water. Neutron star mergers, even rarer than exploding white dwarfs, are responsible for a few trace elements, most notably almost all the iodine in our thyroid glands. And cosmic-rays from supernovae account for the production of the (apparently biologically inactive) boron and beryllium in our bodies, as well as a little of the lithium.

So to be strictly accurate, Joni Mitchell should have written “We are 90% stardust”.

* One of the earliest publications on this topic was a Letter to the Editor entitled “The Origin Of Chemical Elements” (Physical Review 1948 73: 803-4). The authors were Ralph Alpher, Hans Bethe and George Gamow. Alpher was a PhD student at the time, and Gamow was his supervisor. Alpher’s dissertation was on the topic of what’s now called Big Bang nucleosynthesis—the process of nuclear fusion during the first few minutes of the Big Bang. Bethe was a physicist working in the field of nuclear fusion in stars, but had made no contribution at all to Alpher’s work. He was only included as an author to allow Gamow to make a pun on the Greek alphabet—Alpher, Bethe, Gamow; alpha, beta, gamma. Gamow must have been delighted when their letter was published in the April 1 edition of Physical Review.

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A couple of weeks ago, I reviewed three books about the activities of 161 (Special Duties) Squadron, RAF, during the Second World War. For this post, I want to talk specifically about the cover of Hugh Verity’s memoir and personal history of 161 Squadron, We Landed By Moonlight (Revised Edition), published in 2000 by Crécy. It’s a marvellous book, as a source of both anecdote and historical record, and we should all be grateful to Crécy for keeping it in print—but it’s an odd cover.

The first thing that struck me about it is that the Westland Lysander on the cover is sporting South-East Asia Command roundels and flashes, putting it a very long way from 161 Squadron’s base in the south of England. The image in fact comes from this photograph of Lysander V9289, of 357 (Special Duties) Squadron, RAF, operating in Burma. Right kind of aircraft, right kind of duties, wrong continent. But that’s fair enough, given that 161 Sq. Lysanders flew almost entirely at night on secret missions, so tended to go unphotographed unless they actually crashed.

The Lysander image has been composited with a fine full moon, to produce an atmospheric cover image. (In fact, there are two versions of this cover from Crécy, both using the same Lysander and moon images—you can find the other in my link from the book title, above.) And it’s that moon image that really got me puzzling, and inspired this post. Here it is, in a larger and more contrasty version:

It definitely doesn’t look like our own familiar full moon:

But it doesn’t look like a random painting, either. And I found it naggingly familiar. At first, I wondered if it was a photograph of some other moon in our solar system, but then I began to recognize significant features. The curve of Mare Nectaris below the three linked blotches of Mare Serenitatis, Mare Tranquilitatis and Mare Fecunditatis settled it—it is a photograph of our Moon.

But there are three things wrong with it.

One is that it is mirror-reversed. The real Moon looks like this:

The second is that it is pretty much lying on its side. By my estimate, the north-south axis is tilted at about twenty degrees to the horizontal:

Now, you can see the Moon in this orientation, if you catch it just after moonrise in the tropics, when it’s moving almost vertically towards the zenith. But the farther north or south you go, the more tilted is the Moon’s trajectory as it rises, and the closer to vertical is its axis. Even when the Moon is as far into the northern sky as it ever gets, it can never be seen in that orientation anywhere in France, which was the 161 Sq. stamping ground.

But that’s a nitpick, really, because the striking thing about this view is that you can never see it from Earth. The paired dark blotches about halfway towards the upper rim of the Moon, above, are Mare Marginis and Mare Smythii, which (as the former name implies) sit right on the edge of the Moon’s disc when seen from Earth. The photograph used on Crécy’s cover has actually been taken by a spacecraft, somewhere over about 60ºE lunar longitude.

Once I’d figured all this out, I realized why the image looked naggingly familiar. This view is a classic, because it’s what successive Apollo astronauts saw as they departed the Moon towards Earth. At the conclusion of their lunar mission, they fired their main engine as they orbited over the far side of the Moon, and then came looping around the eastern hemisphere, pulling away in a long orbit back towards Earth. The same view was photographed several times, by several relieved astronauts, but I think the Moon on Crécy’s cover is this one:

It’s one of a series of departure photographs taken by Apollo 11 on 22 July 1969. If Verity landed his Lysander by that moonlight, he was a very long way from home!

Note: Crécy’s current edition of this classic book features the Lysander from the Shuttleworth Collection, which is painted in early 161 Sq. markings, landing in a field of poppies.

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