# Pints And Pounds

A pint’s a pound the world around.

A pint of water’s a pound and a quarter.

There’s something odd going on there, isn’t there? I learned that British mnemonic at primary school, and I can still vividly recall my first encounter with the American version—in a Robert Heinlein juvenile science fiction novel, Have Space Suit—Will Travel (1958). His protagonist, for reasons which need not detain us here, needs to fill a room with water. So (in that elaborate Boy-Scout Heinlein way) he measures the room using a combination of his feet, a dollar bill and a quarter coin. Armed with the number of cubic feet he has to fill, he estimates the flow rate of his water-source using an empty can:

The can looked like a pint and a “pint’s a pound the world ’round” and a cubic foot of water weighs (on Earth) a little over sixty pounds.

Up to that point I had been agog, but then the ould fella lost me—I could only conclude that he was really hopeless at arithmetic.

It turns out that a pint’s not a pound the world around at all—only in the USA, and a few other countries in South and Central America and the Caribbean. But that’s a few more than play in the “World Series” baseball championship (USA and Canada for that one), so I suppose we can’t complain.

This is a fundamental difference between the measurement standard in the United States (US customary units) and the United Kingdom (British Imperial units)—we have volume and weight units with the same names, but different magnitudes. And that’s what this post is about.

First, pounds. Our word pound comes from Latin libra pondo, which takes a bit of explaining. The Romans used the word libra to designate a set of balance scales (as in the name of the zodiacal constellation), but also one of the standard weight used on those scales, which we now call the “Roman pound”. Pondo is the adverb from pondus, “weight”, so libra pondo means “a pound in weight”. Unfortunately, we derived the word pound from the pondo bit, but derived the abbreviation, lb., from the libra bit.

The Roman pound weighed about 0.329 kg, and variants of that became established all across the old Roman Empire. In Britain alone there were multiple versions of the pound, used in different regions and for different purposes.

One of the more important pounds in England was the Saxon pound, equivalent to about 0.35 kg. A Saxon pound of silver was enough to make 240 silver pennies. An Anglo-Norman penny was called a sterling, perhaps from an Old English word meaning “bearing a star”, because of the star stamped on some of these coins*. So 240 silver pennies were a pound of sterlings, which is the origin of the unit of British currency, the pound sterling. (And older British readers will recall that, in pre-decimal times, there were still 240 old pennies in a pound sterling.) Like the pound weight, the pound sterling took its abbreviation from Latin libra—the £ symbol is just an ornate capital L. A reference for this important coinage weight was kept in the Tower of London, which accommodated the Royal Mint for five hundred years—and so it acquired the name tower pound.

Another pound took its name from the twice-yearly fairs held in the French city of Troyes. There seems to have been a standard weight used in Troyes to measure out precious metals and stones, and the Troyes fairs were so important for international trade that the troy pound was adopted as a standard weight in Britain, too.

The troy pound is still in use today for weighing precious metals, and it’s equivalent to about 0.373 kg. It’s divided into twelve troy ounces, and our word ounce derives from Latin uncia, “a twelfth part”. So when you read on the reverse of a South African Krugerrand coin that it contains “1OZ FINE GOLD”, that’s a troy ounce.

Apothecaries adopted the troy ounce as a standard weight for measuring drugs, subdividing it into grains, scruples and drachms, which I might write about on another occasion.

But the pound that is most familiar today, in the USA and UK, is the avoirdupois pound, of about 0.454 kg. That word avoirdupois is a corrupted version of seventeenth-century French aver de pois, “goods of weight”. It’s divided into sixteen ounces, in a departure from the etymology of “ounce”. This was used to measure all large weights, and it’s the pound that became the standard adopted by the British Weights and Measures Act of 1824, which gave the world (or at least the British Empire) British Imperial units.

The American and British pounds are identical. Unfortunately, there are problems with larger multiples. An American hundredweight is (intuitively enough) one hundred pounds. But elsewhere in the world, a hundredweight is generally equal to 112 pounds. In both systems, it takes twenty hundredweight§ to make a ton, which makes the American ton smaller than a British Imperial ton, too. So if we want to be precise, we need to talk about short hundredweights and short tons (907.185 kg) for the American version, and long hundredweights and long tons (1016.05 kg) for the Imperial version.

Now, pints. After the above discussion, you’ll be wearily unsurprised to learn that Britain also once hosted a selection of different pint measures, only one of which eventually found its way into the British Imperial system.

The volume of the pint is tied to that of the gallon—eight pints to the gallon. No-one seems to be entirely sure of the origin of either word.

Between about the fifteenth and nineteenth centuries, England (and later Britain) used an ale gallon of 282 cubic inches (4.621 litres), a wine gallon of 231 cubic inches (3.785 litres), and a Winchester gallon of 268.8 cubic inches (4.405 litres). The first two are self-explanatory; the Winchester gallon was used to measure pourable dry stuff, like grain, and was originally defined as the volume occupied by eight Troy pounds of grains of wheat, taken from the middle of the ear. (The Scots and Irish had their own definitions for their own gallons, but that’s probably a diversion too far, even for me.)

The Weights and Measures Act of 1824 did away with all these gallon variants, and defined an imperial gallon as being the volume occupied by ten avoirdupois pounds of water at 62°F, which turns out to be 277.42 cubic inches (4.546 litres). Which is all fine, except you’ll note that it happened some time after the United States became an independent country.

The USA decided to base the US customary liquid gallon on the English wine gallon of 231 cubic inches, which corresponds to almost exactly eight pounds of water. With eight pints to the gallon, the ten-pound imperial gallon accounts for the British mnemonic at the head of this post, and the eight-pound US gallon accounts for the American version.

And that’s why pints and pounds (not to mention hundredweight and tons) are so confusing.

* The Anglo-Norman silver penny seems to have been considered of good quality as currency—hence the development of the adjective, sterling, to denote something of good quality—a person of “sterling character”, for instance.
Drachm is pronounced, and sometimes spelled, “dram”. As a small measure, it gave us the word dram for a small drink, though no-one would thank you for a literal dram of whisky, being an eighth of an ounce—equivalent to a couple of thimblefuls.
In the words of the Oxford English Dictionary: The best modern spelling is the 17th c. averdepois; in any case de ought to be restored for du, introduced by some ignorant ‘improver’ c 1640–1650.
§ The plural of hundredweight is hundredweight.
The grains of wheat that defined the Winchester gallon are the same grains that apothecaries used as their smallest measure of weight. The measurement system of Winchester bushels, pecks, gallons and quarts was so named because the standard reference measures were kept in the city of Winchester.
The USA also adopted the Winchester bushel as a measure for dry goods—thereby adopting two standards of volume measurement that the British government very soon legislated out of existence. But I’m sure that wasn’t because of any sort of peevishness on the part of the Brits. Almost entirely sure.

# Transit Of Earth

At the meeting of the Society in November last I mentioned that on the day of the next opposition of the planet Mars, the Earth and Moon, as seen from Mars, would cross the Sun’s disk, a phenomenon which has not happened since the year 1800, and I stated the chief circumstances connected with the case.
[…]
The last occasion when the Earth and Moon crossed the Sun’s disk for Mars occurred on November 8, 1800. The two next transits, near the opposite node, will take place at the times of the oppositions in May 1905 and May 1984.

Only a minute to go; getting down to business. For the record: year, 1984; month, May; day, 11, coming up to four hours thirty minutes Ephemeris Time … now.
[…]
There it is … there it is! I can hardly believe it! A tiny black dot at the edge of the Sun … growing, growing, growing …
Hello Earth. Look up at me, the brightest star in your sky, straight overhead at midnight.

Arthur C. Clarke “Transit of Earth” 1971

We’re familiar, here on Earth, with the fact that the planets Venus and Mercury occasionally pass between the Earth and Sun, showing up as tiny shadow discs against the solar surface. I’ve previously written about one such transit of Mercury.

Less familiar, but intuitively obvious once you think about it, is the fact that the Earth and Moon will occasionally cross the solar disc if viewed from the planet Mars, our next neighbour outwards in the solar system.

My first quote, above, is the earliest record I can find of someone discussing this phenomenon, back in 1879, in anticipation of the transit of Earth that would occur that year, late on 12 November.

My second quotation is from the short story “Transit Of Earth”, in which Arthur C. Clarke has a dying astronaut, stranded on Mars, talk us through the last of the four transits mentioned in the MNRAS notice—that of 11 May 1984. The story was first published in Playboy, in January 1971, back in the day when claiming that you subscribed to the magazine just for the stories and articles was almost plausible. When the story was subsequently collected in The Wind From The Sun (1972), Clarke mentioned in the preface that his detailed description of the event was based on an article by Jean Meeus, published in the Journal of the British Astronomical Association (1962) 72: 286. My picture at the head of this post accurately depicts the same event, with the disc of Earth, trailed by the Moon, silhouetted against the Sun.

I can’t find a freely accessible copy of Meeus’s original work on-line, but no matter—he revisited the topic more recently, in an article co-authored with Edwin Goffin, and dedicated to Arthur C. Clarke*: Journal of the British Astronomical Association (1983) 93: 120.

The Sun, Earth and Mars roughly align once every 780 days (on average), as the Earth, in its faster orbit, catches up with the slower-moving Mars. (This is called the synodic period of Mars, for reasons that would provide material for an entire other post.) But Mars doesn’t see a transit of Earth every time that happens, because the orbits of the two planets are tilted relative to each other. It’s by less than two degrees, but that slight inclination usually means that the Earth, as seen from Mars, passes either north or south of the Sun, rather than across its disc. The only places that a transit can occur is the line along which the two orbital planes intersect, which is called the line of the nodes. (See my post about Keplerian Orbital Elements for more on that.) Here’s a diagram showing the shape and size of the two orbits, with the line of nodes marked:

So there are two opportunities for Mars to align precisely enough with Earth for a transit to occur. One is called the descending node, where Mars passes through the plane of the Earth’s orbit heading in a southerly direction. Earth passes through that end of the line of nodes in May, and if Mars is at or near its descending node at that time, a hypothetical Martian could glimpse Earth passing across the solar disc. I’ve marked such a May alignment on my diagram. The other opportunity is at the other end of the line of nodes, at (you’ve guessed it) the ascending node, which Earth reaches in November.

But if there’s an Earth transit one May, there certainly won’t be one the following year. The 780-day duration of the synodic period between Earth-Mars alignments means that, after one year, Earth and Mars will be on almost opposite sides of the Sun, and on the next year Mars will be well past its descending node before Earth arrives in the right vicinity. So for a repeat rendezvous, we need to wait out an interval that is both a round number of years and a round number of synodic periods. That happens after 79 Earth years, which is almost (but not quite) equal to 37 synodic periods. So transits of Earth generally occur in pairs separated by 79 years. And we can see that periodicity in Marth’s calculations at the head of this post—a pair of November (ascending node) transits in 1800 and 1879, and a pair of May (descending node) transits in 1905 and 1984.

Why don’t these transits repeat in another 79 years? Because the match between 79 years and 37 synodic periods isn’t exact. If Mars is exactly at its node for one transit, on the next transit alignment, 79 years later, it will pass through the node three days early. This changes its view of Earth. Here are the May 1905 and 1984 (descending node) transits, with the Moon’s orbit also shown:

In 1905, Mars had not yet reached Earth’s orbital plane, and so had a perspective looking slightly “down” on the Earth, so that the Earth and Moon traversed the southern half of the Sun’s disc. In 1984, Mars was already through the Earth’s orbital plane at the time of transit, so Earth and Moon crossed the northern half of the Sun. You can see how, after another 79-year cycle, the Earth would pass north of the Sun, missing it entirely.

The same applies, in reverse, to transits at the ascending node. Here are the November 1800 and 1879 transits:

This time, Mars is slightly below Earth’s orbital plane in 1800, so that the Earth traverses the northern half of the solar disc. By 1879 Mars is sitting above the Earth’s orbital plane at the time of transit, and the Earth transits the southern part of the Sun.

Notice, also, that the width of the Moon’s orbit is so large compared to the apparent diameter of the Sun that the Earth and Moon sometimes have completely separate transits—one of them leaving the solar disc before the other enters, as happened in 1800.

Once one of these 79-year pairs has occurred, it takes a long time for Earth and Mars to fall back into a good enough alignment for a repeat performance. The pairs normally repeat in a cycle of 284 years, which is a pretty good match for 133 synodic periods. So the usual pattern at one node looks like this:

If we add in the other node, the cycle becomes more complicated:

The asymmetry between the two nodes is a little surprising. One might imagine that the 79-year pairs for each node would fall neatly in the middle of the 205-year gap at the other node. But, again, the transitions need to be whole numbers of years (plus a half, for the difference between May and November) which also produce the correct synodic alignment at the opposite node. And this is complicated by the asymmetry of Mars’s orbit. If you look again at my orbit diagram, you can see that the line of the nodes is almost at right angles to the line of the apsides, joining Mars’s closest approach to the Sun (perihelion) with its farthest excursion (aphelion). Mars moves faster when its closer to the Sun, so it completes the half orbit from descending node to ascending node faster than the return journey from ascending node to descending—the difference is about 80 days.

But even the 284-year cycle isn’t precise. So the location of the 79-year pairs drifts slowly with each 284-year repeat, until one of the pair gets very close to the northern or southern rim of the Sun. Here’s what happens at the descending node in a few hundred years’ time:

The 2473 and 2552 transits are part of the normal 284-year cycle, but they’re so far north they’ve opened up space for an early transit to sneak in, in 2394. So the 79-year pair has become a 79-year triplet, disrupting the usual progression:

The early transit in May 2394 intrudes between the normal 79-year November pair, producing a rapid-fire alternating sequence at 25½ and 53½ year intervals, followed by another “normal” 25½-year gap before the regular May pairing in 2473/2552. But the early transit in 2394 establishes a new rhythm of 79-year pairs, with the next pair appearing in 2678 and 2757.

So such early transits have the potential to disrupt the neat alternation between May and November transits at intervals of 79, 25½, 79 and 100½ years. But Meeus and Goffin’s data provide an illustration of how that disruption can sort itself out. Here’s a long run of Earth transits starting in the year 459:

Because this sequence predates the shift to the Gregorian calendar in 1582, the transits are occurring at the end of April and end of October, rather than mid-May and mid-November; but the 79, 25½, 79, 100½ pattern is ticking along nicely, until an early ascending-node transit occurs in October 664, kicking off a triplet like the one we’ve just seen. This briefly causes a shift to a 79, 21½, 79, 104½ pattern as the descending node transits continue in their regular rhythm—until an early descending-node transit in 1337 resynchronizes April with the time-shifted November transits, and restores the 79, 25½, 79, 100½ rhythm that we’re currently enjoying.

So Meeus and Goffin’s 3000-year dataset seems to suggest that the current pattern is the standard pattern, disrupted only for a few centuries each millennium when first one node and then the other experiences an early transit. But, actually, over the longer term the current stable situation proves to be unusual—because the line of the nodes and the line of the apsides move slowly, and in opposite directions. This means that, in a few millennia, Mars’s perihelion will be near its ascending node, and its aphelion near the descending node. Because Mars will be moving fast through the ascending node, the time window in which an Earth transit can occur will be correspondingly narrowed, while the window at the descending node will be wider. Looking at John Walker’s Quarter Million Year Canon of Solar System Transits we discover that by the sixth millennium, Earth is reaching Mars’s ascending node in December, and its descending node in June—and there are three or four June transits for every December transit!

And rolling back to the fourth millennium BCE, we find the situation reversed, with Mars’s perihelion near the descending node, and aphelion at the ascending node, so that October transits are three or four times more common than April transits.

So even patterns that last for millennia count as fleeting, by the standards of the solar system.

* Interestingly, all three have asteroids named after them: 1722 Goffin, 2213 Meeus and 4923 Clarke.
Notice that the 1337 transit didn’t kick off a triplet—there was no transit in 1495, because the Earth just barely missed the solar disc in that year.

# Banks et al.: Why Do Animal Eyes Have Pupils Of Different Shapes?

Human eyes have round pupils, but there is considerable variation in the animal kingdom, from the vertical slit pupil of a cat, to the horizontal slot of a goat, as pictured above.

So Martin S. Banks and his colleagues asked the question “Why?” in an article published in Science Advances in August 2015.

They started by collecting data on 214 terrestrial species, classifying them on their foraging mode (herbivorous, active predator, ambush predator), their time of activity (diurnal, nocturnal or polyphasic) and pupil shape (vertical slit, sub-circular, circular or horizontal slot). Then they plotted the data on a chart, and observed something interesting:

Ambush predators were more likely to have vertical slit pupils than were herbivores, whereas herbivores were more likely to have horizontal slots. And nocturnal animals were more likely to have vertical slits or subcircular pupils (vertically orientated ellipses) than animals that operate in the daytime. When they did the statistics on this, they came up with a p-value that is the most statistically significant result I’ve ever seen in my life: p<10-15. So there’s obviously something going on. And Banks et al. had a potential explanation to offer.

First, we need to think about depth of field, a concept familiar to anyone who has more than a passing interest in photography. When we focus our camera lens (or our eyes) on an object, objects in the foreground and background of that object appear blurred, because they’re out of focus. The range of distances over which this blurring is so slight as to be invisible is called the depth of field. This depth-of-field blurring is more pronounce if the camera aperture (or eye pupil) is wide—so narrow pupils will give a deep depth of field, while dilated pupils have a shallow depth of field. And depth of field also gets shallower as we focus on objects closer to us.

The blurring happens because each ray of light coming from an out-of-focus object is projected on to the camera sensor (or the retina of the eye) as a little image of the pupil. (Whereas light from an in-focus object is brought to a sharp point focus at the surface of the sensor/retina.) So eyes with circular pupils produce out-of-focus images that are, in effect, composed of lots and lots of little circular overlapping spots of light.

But a vertical slit pupil will project an out-of-focus version of itself on to the retina, producing an image that is much more blurred vertically than horizontally. Which means that vertical slit pupils have a greater depth of field for vertically orientated edges than for horizontal edges. Here’s a simulation of what an out-of-focus cross would appear like, to an animal with a vertical slit pupil:

For a given level of light, the pupil must dilate or constrict to control the amount of light reaching the retina. If we compare a slit pupil with a circular pupil of equal area (which therefore would admit an equal amount of light), we can see that the vertical slit pupil is much narrower in the horizontal direction, and longer in the vertical direction. So an animal with a vertical slit pupil trades off good depth of field for vertical structures with poor depth of field for horizontal structures.

Why would evolution come up with such a trade-off? Banks et al. think it helps with depth perception. One way in which we judge distance is by triangulating our gaze—the amount by which our eyes converge on a target tells us how far away it is. And because our eyes are set horizontally, we triangulate best on narrow vertical structures, for which we can confidently bring together and superimpose the images from each eye. A horizontal line is correspondingly harder to triangulate, because there’s ambiguity about the “correct” superposition.

And the increased depth-of-field blur for horizontal structures is also a potential aid to depth-perception, say Banks et al. It produces a shallow depth of field which allows the predator to focus its eyes very precisely on its target. So we have to imagine the predator converging its gaze rapidly on its target, using its deep vertical depth of field, and then fine-tuning its distance estimate using its shallow horizontal depth of field.

But is there any evidence that predators actually use these visual cues? Banks et al. point out that small predators have their eyes nearer the ground and focus on more nearby prey, which will give them a reduced depth-of-field compared to larger predators. The two hypothesized advantages for vertical slit pupils should therefore be more advantageous for smaller predators, driving evolution more strongly towards equipping smaller predators with slit pupils. And so it proves to be:

We evaluated this prediction by examining the relationship between eye height in these animals and the probability that they have a vertically elongated pupil. There is indeed a striking correlation among frontal-eyed, ambush predators between eye height and the probability of having such a pupil. Among the 65 frontal-eyed, ambush predators in our database, 44 have vertical pupils and 19 have circular. Of those with vertical pupils, 82% have shoulder heights less than 42 cm. Of those with circular pupils, only 17% are shorter than 42 cm.

But the most striking support evidence they advance is the fact that birds almost all have circular pupils, which might reflect the large heights from which they usually observe. The only birds known to have vertical slit pupils are the skimmers. These birds catch fish by flying fast and low just a few inches above a lake surface, with their lower jaws dipped into the water, ready to snap shut on any unsuspecting fish. An ability to precisely judge the distance of obstacles ahead might well be an advantage.

What about the horizontal pupils of herbivores? These will have the reverse properties of the vertical slits—good depth of field for horizontal structures, at the expense of poor depth of field for vertical structures.

How could that help a prey animal? Banks et al. point out that light rays entering our eyes from the peripheries tend to be poorly focussed—something called astigmatism of oblique incidence. We tend not to notice that, because our peripheral vision is more involved in detecting movement than in trying to resolve images. But prey animals generally have their eyes placed on the sides of their heads, so that they can observe a wide field of view. This means that they have a very limited field of binocular vision in front of their noses, and that’s all mediated by peripheral vision. By prioritizing deep depth of field for horizontal structures, their pupils reduce the effect of astigmatism of oblique incidence in important parts of the visual field—straight ahead, when galloping full-tilt to escape predators, and also at the “corner of the eye” looking backwards, so a slight shift of the head can assess where a pursuing predator is. Or, as the authors put it:

We conclude that the optimal pupil shape for terrestrial prey is horizontally elongated. Such a pupil improves image quality for horizontal contours in front of and behind the animal and thereby helps solve the fundamental problem of guiding rapid locomotion in a forward direction despite lateral eye placement.

And they have a test for this hypothesis, too. If horizontal slot pupils are to be advantageous in this way, they should stay horizontal, despite the animal’s head position, rotating to maintain their orientation as the animal bows and raises its head—something called cyclovergence. So Banks and his colleagues went off to farms and zoos and shot video footage of some prey animals raising and lowering their heads—sheep, goats, white-tailed deer, horses and moose—and they all exhibited strong cyclovergence. How cool is that? Rest assured that I will be closely observing the next sheep I encounter.

There’s a lot more to the paper, including analysis of other pupil shapes, like those of geckos and dolphins, which constrict to multiple small apertures in bright light, and an investigation of how often different pupil shapes have emerged during the evolution of cats and dogs. If I’ve whetted your appetite, head off to look at the original paper, accessible in full here.

# How Apollo Left Earth (And Returned): Part 2

This is the long-delayed second post in my discussion of the departure and return orbits of the Apollo missions. If you haven’t read the first post, you can find it here—it’ll give some useful background to what follows.

The diagram at the head of this post shows a plot of Apollo 11’s departure trajectory, superimposed on a chart of the inner Van Allen Radiation Belt—the one that presents the greatest radiation hazard to astronauts on their way to the Moon. It’s a graph of distance against geomagnetic latitude, and it’s orientated so that the geomagnetic equator is horizontal. The white dots mark intervals of one hour after Translunar Injection. You can see how Apollo 11’s orbital tilt lofted it over the northern fringes of the inner VAB, thereby avoiding the densest area of radiation near the geomagnetic equator. And its return orbit did the same in reverse:

The white dots now mark hourly intervals before the “entry interface” at 400,000 feet altitude—the point at which the Apollo capsule began to significantly interact with Earth’s atmosphere.

It’s often said that all the Apollo lunar missions followed this same sort of trajectory, to avoid passing through the central part of the Van Allen Belts—I’ve even made that claim myself, previously. And certainly some of them did. Here are the departure trajectories of the three other missions that swept north of the VAB, with Apollo 11 for comparison:

As my graph title indicates, all these missions had Translunar Injections over the North Pacific. As I explained in more detail in my post How Apollo Got To The Moon, a TLI in the northern hemisphere was used when the Moon was situated south of the celestial equator. The more southerly the Moon, the more northerly the TLI, and you can see that reflected in the trajectories above—Apollo 15 had the most northerly TLI, and its trajectory is the one that begins to head south most rapidly.

Here’s a map of all the Apollo Translunar Injections:

The green grid lines show the orientation of the Earth’s geomagnetic field, which contains the Van Allen Belts, and the yellow tint indicates the region between forty degrees north and south geomagnetic latitude—the approximate limits of the dangerous inner VAB. The labelled coloured dots mark the position at which each mission’s S-IVB stage engine was ignited, and the tip of each coloured arrow is the position at which, ten seconds after engine shut-down, Translunar Injection was deemed to take place. You can see how Apollos 8, 11, 12 and 15 all have TLIs in the northern hemisphere. (Apollo 11 actually fired up the S-IVB while it was still in the southern hemisphere, but TLI was achieved north of the equator.) But there’s another cluster, containing Apollos 10, 13, 14 and 16, with TLIs south of the equator. As you can see from my map, while the “North Pacific” group were heading towards the fringes of the VAB as their trajectories rose away from Earth, the “South Pacific” group were on trajectories that were heading towards the middle of the VAB. Apollo 17 constitutes a “North Atlantic” group all on its own, but has the same problem as the South Pacific TLIs—it has the densest part of the VAB ahead of it, rather than behind it.

I’ll show you the departure trajectories of these missions in a minute, but there’s something else worth noting before I leave the TLI map. The dotted black line marks the ground track of the typical Earth Parking Orbit from which the Apollo missions departed—their second pass over the Pacific, leading into the third pass over the Atlantic. This is the track followed if the mission took off on schedule, right at the start of the launch window. Under those circumstances, the Saturn V would launch somewhat towards the northeast, on an azimuth of approximately 72° east of north. If there were launch delays, the launch azimuth would shift progressively towards the east, and eventually towards the southeast, reaching a maximum of 108° just before the launch window closed. Here’s an aerial view of the Apollo launch pads at Cape Kennedy, with that range of azimuths marked:

This shift in the launch azimuth resulted in Earth Parking Orbits with progressively more westerly ground tracks, allowing TLI to always take place on the opposite side of the Earth from the Moon, despite the launch delay and associated eastward rotation of the Earth. (To be more accurate, TLI was orientated with reference to the Moon’s position at the time Apollo would arrive there, three days after TLI. For more on that, see How Apollo Got To The Moon.)

You can see that most Apollo missions launched along the “standard” ground track. Apollo 14 had a short launch delay, so ended up a little to the west of standard. Apollo 17 had a very long delay, and its TLI shifted all the way across the Atlantic from the planned position. The westerly shift for Apollo 15 has another explanation, however—it was the first of the heavier “J” mission launches, and its launch window was made deliberately narrower. It launched on time, but in a more easterly direction—an azimuth of 80°. This ensured that the Saturn V got more of a boost from the west-to-east rotation of the Earth, making it easier to propel the heavier load into orbit. (Subsequent “J” missions went back to the standard launch window.)

So here’s a plot of the South Pacific (and North Atlantic) TLIs:

You see the problem—all of these missions were obliged to track through denser regions of the inner Van Allen Belt. Apollo 14, with the most southerly TLI, actually speared right through the most concentrated radiation. It was the speed of transit that protected these astronauts from an excessive radiation dose, rather than the orientation of their trajectory.

Now, if we look at the return trajectories, we can see that the Apollo missions all reentered the atmosphere on a west-to-east trend, but otherwise came in from all sorts of directions:

The dotted lines connect the atmospheric interface (coloured and labelled dot) with splashdown (black cross).

The variety of return trajectories is even clearer when we look at a geomagnetic plot:

All the South Pacific / North Atlantic TLIs, which took an increased radiation exposure on the way out, come back through only the fringes of the inner VAB. (Apollo 16, in particular, hooked “under” the Earth at high southern latitudes, accounting for that mysterious kink in the geomagnetic plot of its return trajectory.) Of the North Pacific TLIs, which departed through the fringes of the VAB, Apollo 11 and Apollo 15 returned in the same manner. Apollo 8 penetrated more deeply on the return than on departure, and Apollo 12 speared through the higher radiation region.

While there are on-line tools, like SPENVIS, which will calculate spacecraft radiation exposure from a model of the Van Allen Belts and a set of orbital elements, it’s pretty futile to attempt this for the Apollo missions, since we can’t accurately model the radiation shielding effect of the complicated structure of the Command/ Service Module, let alone the additional partial shielding provided by the S-IVB stage and Spacecraft Lunar Module Adapter, to which the CSM was still attached during its outward passage through the VAB. And, given the variability of the VAB radiation environment, we’d also need to make some educated guesses about the real proton and electron flux to which the spacecraft would have been exposed.

But we know that most of the astronauts’ radiation exposure came during the small number of minutes during which they passed through the VAB, with a lesser contribution mounting up throughout the rest of the mission, from exposure to cosmic rays and secondary radiation from the lunar surface.

So we would anticipate that the longer-duration missions (15, 16, and 17) would have more cumulative exposure to radiation; and that the North Pacific TLIs (8, 11, 12, 15) would have lower VAB exposure on departure than the other missions. Apollo 14, with its very southerly TLI, would receive a significant VAB dose on departure, and Apollo 12, with a return trajectory close to the geomagnetic equator, would sustain the highest VAB dose on return.

All the astronauts wore film badges to record their overall mission radiation dose. The readings, averaged over all three astronauts, were published in a NASA publication, Biomedical Results of Apollo, in 1975. I’ve graphed them below:

So it all hangs together. You can see how Apollo 14 received the highest mission dose, because of its departure trajectory. And Apollo 12 had the highest dose of the North Pacific TLIs, because of its return trajectory. Of all the South Pacific TLIs, Apollo 13’s was closest to the equator, and so sustained the least VAB exposure; it was also a shorter mission overall (the explosion on board meant that the mission simply looped around the Moon and came straight back to Earth), and so has the lowest dose of all the South Pacific TLIs.

Being able to plot the near-Earth trajectories of the missions makes sense of what is otherwise a rather random-seeming scatter of radiation doses.

# The Solar System’s Place In The Milky Way: Part 2

At the end of my previous post on this topic, I left you with a diagram of the solar system’s orientation and approximate trajectory in its orbit around the Milky Way galaxy. Below, we’re looking past the solar system towards the galactic core. The plane of the galaxy runs horizontally across the image, north is at the top, and the solar system is travelling around the galaxy in a clockwise direction when viewed from galactic north. For more on all that, consult my first post via the link above.

This time, I want to give some more detail of the exact direction in which the solar system is moving, to the extent that this is known. And this will bring me to my gripes about Wikipedia‘s discussion of the topic, as I promised last time.

Although my arrow marks the general direction followed by the Sun and other stars in its vicinity, in practice this movement is reminiscent of a swarm of bees—they’re all heading in the same direction, but they’re all moving relative to each other as well. Astronomers find it useful to break this orbital motion down into two components—they separate out the general trend of motion from the smaller random relative movements.

To do this, they invoke something called the Local Standard of Rest, which (somewhat paradoxically) is actually in motion around the galaxy. It applies only to our local solar neighbourhood, and it moves at a speed such that a star at rest in the LSR would follow a circular orbit around the galaxy. There’s some uncertainly about the LSR’s velocity, which depends on getting an accurate measurement of the Sun’s distance from the centre of the galaxy, among other things. But it seems to be somewhere around 220 km/s.

Within the LSR, individual stars have their own random motions, typically of a few kilometres per second—these small deviations from the perfect local circular velocity mean that each star is following its own, slightly elliptical, slightly tilted orbit around the galaxy. The relatively small additional velocity component of each star is called its peculiar motion—“peculiar” because it’s unique to that star, not because it’s odd in some way.

Here’s how it fits together:

While stars have very different peculiar motions within the LSR, they’re actually all heading in pretty much the same direction around the galaxy.

The Sun has its own peculiar motion, too. If we add up all the peculiar motions of stars in our local neighbourhood, they don’t average down to zero—the residual velocity is the result of the Sun’s motion relative to the other stars. In practice, this is tricky to work out—we have to sample only stars that belong to the “thin disc” of the galaxy, which share similar orbits with the Sun, and also allow for some complicated dynamics that I won’t go into here. Because of this trickiness, estimates of the Sun’s peculiar motion vary considerably. But my twenty-year-old copy of the fourth edition of Allen’s Astrophysical Quantities gives some fairly typical values—9 km/s towards the galactic centre, 7 km/s towards galactic north, and moving around the galaxy 12 km/s faster than the LSR—giving it an overall velocity (relative to the LSR) of about 16.5 km/s directed towards a point near the star Mu Herculis.

This point, marking the direction in which the Sun is moving relative to the other stars in its neighbourhood, is called the solar apex. (The astronomer William Herschel called it “the apex of the Sun’s way” in the eighteenth century, and you’ll still see that phrase floating around the internet.) There’s a solar antapex, too, on the other side of the sky in the constellation Columba.

Over the years, the constellation of Hercules has become pretty much peppered with solar apex candidates, as each new study samples different stars and makes different dynamical allowances. But here’s the thing to remember about the solar apex—it’s the point towards which the Sun is moving relative to our local stars. But all those local stars, and the Sun, are also whooshing around the galaxy, with the Local System of Rest, at 220 km/s. So the solar apex is not the Sun’s direction of travel in its orbit around the galaxy. As my diagrams above show, the Sun’s peculiar motion will slightly modify the velocity it inherits from the LSR, but not by much.

Which brings me to my first gripe about the Wikipedia page I mentioned in my first post. (Because Wikipedia changes from time to time, my link goes to a copy of the relevant page capture on 19 October 2023, just so you can see what I’m talking about.) Here’s the vexatious text:

The apex of the Sun’s way, or the solar apex, is the direction that the Sun travels through space in the Milky Way. The general direction of the Sun’s Galactic motion is towards the star Vega near the constellation of Hercules, at an angle of roughly 60 sky degrees to the direction of the Galactic Center.

Well, no, the solar apex is very much not “the direction the Sun travels through space in the Milky Way”, or “the general direction of the Sun’s Galactic motion”. Nor is the solar apex particularly near Vega, though Vega is easier to pick out, for the uninitiated, than is Hercules.

Once we factor in the relatively high velocity of the Local Standard of Rest, the Sun is heading pretty much at right angles to the direction of the galactic centre. Here’s a map of the relevant bit of sky, with significant locations superimposed on a base map from In-The-Sky.org:

Around the big label “SOLAR APEX” in Hercules, I’ve dotted a few estimates of the solar apex from different sources—my old Allen’s Astrophysical Quantities (2000), Schönrich et al. (2010) and Ding et al. (2019). The grey horizontal line marks the plane of the galaxy, and the black cross is at 90 degrees galactic longitude—the direction in which the Sun would be headed if it were at rest in the LSR, in a perfectly circular orbit around the galaxy. The point marked “True direction of motion” is what happens when we added 220 km/s of tangential velocity to Ding’s solar apex—the large orbital velocity dominates over the relatively small northward and inward components of the Sun’s peculiar motion. So forget Vega—we’re headed towards Deneb, in the constellation Cygnus.

But, you’ll see, we’re heading slightly inwards and rising northwards out of the galactic plane as we go. Back in 1986, the astronomer Frank Bash wrote a chapter in the textbook The Galaxy And The Solar System, entitled “The Present, Past and Future Velocity of Nearby Stars: The Path of the Sun in 108 Years”, in which he used the current orbital motion of the Sun to predict its future movement around the galaxy. In astronomical terms, this is a positively prehistoric reference, but Bash’s calculations are still commonly quoted, despite the fact our understanding of the galaxy has moved on considerably since he wrote. And I haven’t been able to find a more recent reference. What follows is therefore probably accurate in its broad picture, but wrong in the details.

So (Bash writes), the Sun is moving gently inwards, but will reach its closest approach to the galactic centre (perigalacticon) in 15 million years’ time, at which point it will be at about 99.5% of its current distance. It will then move outwards, reaching a maximum distance (apogalacticon) of about 114.5% after completing half an orbit of the galaxy—so about 135 million years from now. And it’ll continue to move between those extremes, tracing out a rosette pattern around the galactic centre, unless it has some sort of catastrophically close encounter with another star.

Meanwhile, its velocity will loft it northwards at a steadily slowing rate until (14.6 million years from now) it reaches a peak at about 77 parsecs (250 light years) above the galactic plane, after which it will fall back to, and through, the galactic plane, making an excursion to a similar distance on the southern side. And so on. Bash calculated that a full cycle would take about 66 million years. The exact number of north-south oscillations per galactic orbit depends on the vertical mass distribution within the plane of the galaxy—Bash’s figures suggest about 3½ cycles per orbit, but you’ll see other figures quoted.

And that concludes my post about the movement of the solar system within the galaxy. But I now need to briefly return to that annoying Wikipedia page, to highlight another bit of misinformation. Having just assured us we’re all heading for Vega, the page goes on to contradict itself:

The Solar System is headed in the direction of the zodiacal constellation Scorpius, which follows the ecliptic.

Scorpius is a long way from Vega. It’s aligned with the centre of the galaxy—in fact, you can see it in the background of my diagram of the solar system’s orbital motion, above—pretty much at 90 degrees to the Sun’s real direction of orbital travel. Where on Earth does that piece of misinformation come from?

It comes from a 2011 article in National Geographic, headed:

Solar System’s “Nose” Found; Aimed at Constellation Scorpius

A NASA craft has uncovered the solar system’s “nose,” which points in the direction our sun is moving through the Milky Way, a new study says.

The study cited actually says no such thing—the whole bit about the “direction our sun is moving through the Milky Way” seems to be a bit of improvisation by the journalist who wrote the National Geographical article.

This is all about the shape of the heliosphere, which is defined by the location of the heliopause—the surface at which the solar wind of particles radiating outwards from the Sun runs into the gas and dust of the interstellar medium. As the Sun ploughs through the interstellar medium, the heliosphere experiences an opposing drag, and develops a “nose” and a “tail”, like this:

The “nose” of the heliosphere therefore points in the direction from which the interstellar medium is flowing past the Sun—and that “nose” is aimed at the constellation Scorpius. But if the solar system were actually “moving through the Milky Way” in that direction, we’d all be falling towards the centre of the galaxy, rather than orbiting around it. So this is just another case of relative velocities—the interstellar medium is also in orbit around the galaxy, and what we’re seeing is the result of the “peculiar motion” of the interstellar medium relative to the LSR.

# Annular Solar Eclipse

As this post goes live, it’s only a few days until an annular solar eclipse, like the one pictured above, will sweep across the Americas on 14 October 2023.

Annular eclipses get their name from Latin anulus, “small ring”, which refers to the ring of sunlight that’s visible around the lunar disc, as shown in the image at the head of this post. They’re the poor relations of total eclipses, in which the lunar disc entirely obscures the Sun.

October’s annular eclipse will follow the track shown in red in the map below.

The Moon’s shadow crosses the Earth from west to east, moving at the Moon’s orbital velocity, about a kilometre per second—equivalent to about three and a half hours to cross the full width of the Earth. The noticeable tilt in the eclipse track above, from northwest to southeast, is because of the inclination of the Earth’s equator to the plane of the Moon’s orbit. You can see the same effect in the track followed by the total eclipse of 9 March 2016, here seen in time-lapse from a Japanese weather satellite stationed above Indonesia:

Notice that the tilt in the March eclipse track is the reverse of what we see in the October eclipse map, above. This is because the Earth is more or less on opposite sides of the Sun in March and October.

The large shadow you can see in the video above is the region in which the Moon’s disc partially obscures the solar disc (a partial eclipse). Only right in the middle of that large patch of shadow does the Moon traverse the exact centre of the Sun’s disc. For a short distance either side of that central eclipse (anything from a few kilometres to something more than a hundred) we can still see either a total eclipse, or the ring of light that characterizes an annular eclipse, but the shift in position will make the eclipse lopsided and of shorter duration.

So we’ve got a narrow region in which we can observe either of the two kinds of central eclipse, annular or total, and a much broader region in which we can see a partial eclipse. These regions correspond to separate areas of the Moon’s shadow cone, which I’ve labelled below:

The blue lines mark off the Moon’s entire shadow—if we stand anywhere within that spreading cone, we’ll see the Moon overlapping the solar disc, to a greater or lesser extent. Within the red lines, the entire lunar disc is superimposed on the Sun. These lines mark off two shadow regions—the umbra, within which the Moon completely obscures the Sun; and the antumbra, from which we can see the Moon outlined against the larger solar disc. The umbra is where we need to be to see a total eclipse; the antumbra is where annular eclipses happen. And the whole surrounding shadow area, called the penumbra, is the region from which partial eclipses can be observed.

So why do we see a mix of the two types of central eclipse?

It’s because of a remarkable coincidence—the Moon is about 400 times smaller than the Sun, but is also 400 times closer, making the two bodies appear to have almost the same size in the sky. So small variations in the relative distance of Sun and Moon can make the Moon look either bigger than the Sun, and able to block out all its light in a total eclipse, or smaller than the Sun, and able only to produce an annular eclipse.

Our distance from the Sun varies a little during the course of a year—we come closest (and the Sun appears largest) in early January; and we’re farthest away six months later, in early July. But the distance to the Moon varies more widely during the course of a month, and also over a longer period of a little over 200 days, in a way a described in detail in my recent post about “supermoons”.

So here’s a comparison of the apparent diameters of the Sun and Moon at maximum, mean and minimum:

In the bottom row, I’ve superimposed the lunar disc on the solar. In the middle, the mean lunar and solar discs match to within a hundredth of a degree. At left, the maximum moon serves to completely block the maximum sun (represented by the yellow circle around the edge of the moon); at right, the minimum moon is unable to block even the minimum sun. So when the Moon is at its closest to Earth (called perigee), it will always cause a total eclipse if it correctly aligns with the Sun; but when it’s at its farthest (apogee), it can never cause a total eclipse. (And, obviously, between those extremes we can get a mixture of annular and total eclipses.) Looking back at my shadow diagram, above, the Moon sometimes comes close enough to Earth for the tip of its umbra to reach the Earth’s surface, causing a total eclipse; and sometimes it’s so far away that the umbra doesn’t reach the Earth at all, and we experience an annular eclipse within the antumbra.

You’d perhaps think that, given how closely the mean sizes of Sun and Moon match, we’d have equal numbers of annular and total eclipses. But that turns out not to be the case, because the Moon moves more quickly near perigee, and more slowly when it’s near apogee. So it actually spends more time at distances greater than its mean distance. Which means we end up with more annular than total eclipses.

Looking at NASA’s Five Millennium Catalog of Solar Eclipses, we can see that there are a whole lot of eclipses in which the Moon’s shadow either just misses or barely grazes the Earth’s surface, but if we concentrate on “pure” annular and total eclipses, during which it’s possible to stand exactly in the centre of the Moon’s shadow with a full eclipse track on either side, there are 3827 annular solar eclipses in the 5000-year catalogue, and only 3121 totals.

And, given that the apparent diameter of the Sun is at its greatest in January and least in July, we might expect to see a corresponding periodicity in the ratio of annular to total eclipses—total eclipses should be “easier” to achieve in July, when the solar disc is smaller. And so it turns out to be:

Finally, one other kind of eclipse is worth mentioning—the hybrid eclipse. These occur when the Earth is positioned almost exactly at the point at which the umbra becomes antumbra. In these circumstances, the curvature of the Earth becomes significant, so that the rim of the Earth can lie in the antumbra while the central bulge extends into the umbra. Like this:

So these eclipses start as annular eclipses when they first make contact with the Earth’s surface, evolve into total eclipses as they cross the Earth, and then turn back into annular eclipses just before they lose contact. They’re fairly rare, but not vanishingly so—there are 569 in the Five-Millennium Catalog. The most recent crossed Indonesia in April 2023, but we’ll need to wait until November 2031 for the next one, which will cross mainly open ocean.

* The steady speed at which the Moon’s shadow crosses the Earth is not reflected in the speed at which the shadow moves across the ground, however. The Earth rotates in the same direction as the shadow moves, and that reduces the ground velocity a bit, depending on latitude. And there’s a geometric effect just after the shadow contacts the Earth’s surface, and just before it leaves, when a small sideways movement of the shadow results in a large movement across the ground, because the Earth’s surface is at an acute angle to the axis of the shadow.

# Blue Supermoon: Part 2

We recently had a blue supermoon (on 31 August 2023). If you saw it, did you think it was super? Me neither.

In my previous post, I wrote about blue moons—what they are, why they happen—and in this post I aim to do the same for supermoons.

Supermoons happen when a full moon occurs at a time when the Moon is a little closer to the Earth than usual. The Moon’s orbit is slightly elliptical—its distance from the Earth varies between a close approach (called perigee) of about 363000 km, and a farthest excursion (apogee) of about 405000 km. The line connecting the perigee and apogee, forming the long axis of the orbital ellipse, is called the line of the apsides, for obscure reasons that I explained in my post “Keplerian Orbital Elements”. It turns out that the Moon’s orbit gets a little more elliptical when the line of the apsides is pointing at the Sun. This is a tidal effect, caused by the Sun’s gravity tugging on the Moon, in the same way the Moon’s gravity tugs on the Earth to raise the ocean tides. For brevity, I’m going to refer to those episodes when the apsides align with the Sun as apsidal alignments.

So there’s a rhythmic variation in the Earth-Moon distance, like this:

The short-period cycle is just the time it takes the Moon to move from one perigee to the next—this is called the anomalistic month*, and it lasts about 27.55 days. The longer-period cycle is that of the recurring apsidal alignments, which drive more extreme perigees and apogees. If the line of the apsides always pointed in the same direction, then that cycle would take half a year, repeating whenever either perigee or apogee points towards the Sun. But, just to make matters more complicated, the line of the Moon’s apsides is rotating slowly, completing one revolution every 8.85 years. So apsidal alignments actually happen at intervals of about 206 days.

In my post about blue moons, I introduced the lunation, the time between two successive full moons, which averages 29.53 days. So a lunation is about two days longer than an anomalistic month. (This happens because, by the time the Moon makes one orbit from perigee to perigee, the Earth has moved around the Sun through an angle of about 27°, which means that the Moon has to orbit for another couple of days to “catch up” with the changed angle of illumination.)

So now I’ll plot full moon dates on my previous diagram:

If you choose a full moon near perigee, and follow the progress of successive full moons across the diagram, you can see how they come progressively later, relative to perigee, with each anomalistic month that passes. But after 14 lunations and 15 anomalistic months, the full moon returns to approximately the same relationship with perigee—because 14×29.53 and 15×27.55 are both approximately 413 days. And, remarkably and (I think) coincidentally, that’s almost the same as two cycles of apsidal alignment, 412 days.

So you can see from the diagram that successive full moons creep only slowly past perigee—once we have one supermoon, we tend to get a season of them. It varies somewhat according to the definition of “supermoon” you use, but by one definition we’re currently passing through a four-supermoon season extending from the start of July to the end of September 2023. And then, after 14 lunations and 15 anomalistic months, we’ll get another four-supermoon season, from mid-August to mid-November in 2024—the slippage of about a month-and-a-half from year to year of course reflecting the amount by which 413 days exceeds the calendar year.

Strictly speaking, there’s another, invisible cycle of supermoons taking place, precisely out of phase with the one I’ve just plotted—that’s the cycle of new supermoons, which can be contrasted with the full supermoons I’ve just been talking about. These are (as you’ve guessed) new moons that occur near perigee, and you can imagine their cycle as a sinusoid that reaches perigee in the empty gaps between the full-moon perigees on my diagram. No-one talks much about new supermoons, because they’re dark and therefore invisible, so the word supermoon on its own usually designates the full-moon version. Then there are the micromoons, a horrible name that designates full or new moons occurring at apogee. You can see a couple of full-micromoon seasons on my diagram above, spanning January-February 2023 and February-March 2024. New micromoons fill in the gaps between the full micromoons, and are perhaps even less popular than new supermoons.

So far, I’ve given no indication of how close to perigee a full moon must be to qualify as a supermoon. And that’s because definitions vary. The first thing to know is that the term supermoon didn’t come from astronomers—it was invented by an astrologer called Richard Nolle in 1979, in the now blessedly defunct Dell Horoscope magazine, and used in the context of doom and disaster, summarized in the marvellous phrase geocosmic shock window. Because supermoons (full and new) are associated with unusually high tides, called perigean spring tides, Nolle chose to associate them with all sorts of potential disasters, none of which have actually materialized in any statistically defensible way. Nolle wrote that the name supermoon described “a new or full moon which occurs with the Moon at or near (within 90% of) its closest approach to Earth in a given orbit”. I find this phrasing a bit impenetrable, but when Nolle gives a worked example, it’s evident that he takes the distance between apogee and perigee, and if the full or new moon occurs when the moon is 90% or more of the way from apogee towards perigee, it’s a supermoon. Although Nolle’s original definition stipulated “a given orbit”, by 2011 he had revised this, using the most extreme apogee and perigee for a given year. This shift in definition somewhat reduces the number of supermoons that can occur in a year. Astrophysicist Fred Espenak, who uses Nolle’s original definition, counts four full supermoons in 2023 (the July-to-September grouping I mentioned above; Nolle’s own list classifies only the central two, in August, as “super”. Some other sources forget all about the apogee/perigee thing, and instead choose a simple distance cut-off—for instance, timeanddate.com use a cut-off of 360000 kilometres, and list only two full supermoons for 2023.

But why are supermoons so unexceptional in appearance? The difference in distance looks extremely impressive on my graphs above, after all. But if I ensure that my vertical axis starts from zero, to give a correct impression of how the distance from Earth varies, it looks like this:

That rather modest wobble means that a typical supermoon isn’t hugely different from an average moon:

We can see that difference when they’re presented side by side, but it’s not particularly striking when we’re viewing a supermoon in isolation.

A supermoon is also brighter than usual, reflecting light in proportion to its apparent diameter squared. A figure that gets trotted out every supermoon season is that the full moon will appear 30% brighter than it does when it’s at its farthest away (a micromoon). Which is certainly true, but our eyes are very good at adjusting to differing light levels. Outdoor sunlight is a hundred times brighter than a brightly lit room indoors—but we just don’t perceive the difference unless we walk from one to the other. So our clever eyes actually make it impossible for us to pick up on a comparatively trivial 30% difference, given that we’re unable to make a direct comparison, but have to work from memory of previous full moons.

And that’s why I find it difficult to get excited about supermoons. (Except, of course, for the lovely mathematical patterns they generate.)

* The name anomalistic refers to the way in which an object’s position in orbit is measured, using its angular distance from closest approach, which is called its anomaly. So from one perigee to the next, the Moon moves through a full 360 degrees of anomaly. Why this angle is called the anomaly is a complicated story that I covered in my post “Keplerian Orbital Elements”.

# Blue Supermoon: Part 1

On 31 August 2023 we’re going to have a blue supermoon, which will be neither particularly blue, nor particularly super, though to read some of the media coverage of these events, you might expect to see something like the image above. So I thought I might write a bit about blue moons (this post) and supermoons (to follow).

So: a blue moon, in current usage, is the second of two full moons falling in the same calendar month—the full moon of 31 August is a blue moon because there has already been a full moon on 1 August.

The phrase once in a blue moon, meaning “very occasionally”, has been around since the nineteenth century. The current astronomical usage is more recent—it originated in 1946, with a notorious error in a Sky and Telescope article, and became popular only during the 1980s, when it was included as a question in the game of Trivial Pursuit. (I’ll write a little more about all that at the end of this post, if you’re still with me.)

To show how blue moons work, I need to introduce the idea of a lunation, which (for our purposes) is the time between two successive full moons. Although the moon looks full to a casual observer for a couple of nights each month, in astronomical terms there is a precise time at which the full moon occurs—the moment at which it’s on exactly the opposite side of the sky from the sun. The time between two of those full moons averages around 29.53 days, but can vary by six or so hours in either direction, as the moon responds to the gravitational tug-of-war between the Earth and the sun. And, given that the astronomical timing is precise, I need to specify the time zone I’m using in what follows—a full moon that occurs before midnight in one part of the world can easily occur after midnight (and perhaps therefore in a different month) somewhere else. But it seems to be customary to figure the blue moon calendar according to Greenwich Mean Time, or its astronomical equivalent, Coordinated Universal Time.

So there’s room, in any calendar month of 30 or 31 days, for an entire lunation to take place. If there’s a full moon very early in the month, there can be another at the end of the month. For a 31-day month, there’s a window of about 1½ days at the start, during which a full moon can occur while still leaving room for another one in the same month, 29½ days later. For a 30-day month, the window is just half a day—meaning that blue moons will be about three times more common in 31-day months than in 30-day months. (When Steve Holmes of the British Astronomical Association crunched the numbers, taking into account the variable duration of a lunation, he determined that any given 31-daymonth has a 1:250 chance of hosting a blue moon; for 30-day months it’s just 1:835.) But February, even in a leap year, is too short to accommodate even the shortest lunation, and is therefore the only month that never has a blue moon.

A “normal” year plays host to twelve full moons. But twelve lunations add up to only 354.36 days, which is almost eleven days less than an average calendar year of 365.25 years, and those “missing” eleven days have a couple of consequences.

Firstly, the date of the full moon in any given month will drift earlier in the month, by about eleven days, with each successive year. The first full moon of 2021 came on 28 January; in 2022 it fell on 17 January; in 2023 on 6 January. In 2024 it will leap back towards the end of the month, 25 January—which we can think of as the 5 February full moon of 2023 coming eleven days earlier in 2024, as a January full moon.

Secondly, if a full moon falls in the first eleven (or so) days of January, that allows room in the calendar year for a thirteenth full moon before the end of December. Thirteen full moons in a twelve-month year means there will be a blue moon in some month during such a year. So the late-January full moons of 2021 and 2022 “prevented” a blue moon; whereas the early January full moon of 2023 provided the opportunity for the blue moon of August. But there will be no blue moon in 2024, when the January full moon skips back towards the end of the month.

It seems like there’s a cycle here, with 13-moon years interspersed with 12-moon years. How often do we get a blue-moon year? We can make a rough-and-ready estimate by realizing that there are eleven days at the start of January during which a full moon will be associated with a blue-moon year, and twenty days in the rest of January when a full moon will prohibit a blue-moon year. So that suggests that a blue moon will appear in 11/31 of years, or about 35%. (Steve Holmes’s detailed calculations came up with a figure of 36.3%, so my quick estimate turns out not to be too shabby.)

I can put all this together in a chart:

The years run from 2011 to 2030, from top to bottom. The grey dots marking full moons form up into long diagonals, sweeping down and left through the years, illustrating the 11-day-per-year mismatch between twelve lunations and the calendar year. Blue moon are marked with larger, blue dots. Notice the interesting effect in 2018, when we had two blue moons—an early first full moon in January left room for a blue moon at the end of that month, which was so close to the end of the month that the lunation skipped February entirely and put the next full moon in early March, leaving room for another blue moon.

In blue-moon years, the earlier the first full moon of the year occurs, the earlier the month of the blue moon tends to be. 2018 had its first full moon on 2 January, giving us the January/March blue-moon pair. The first full moon of 2026 falls on 3 January, and brings a May blue moon. Whereas the first full moon of 2028 is on 12 January, pushing the blue moon to 31 December. (And if 2028 wasn’t a leap year, there wouldn’t be room for a blue moon at all.)

Another example of this “leap year effect” occurred in 2012, when the extra day in February moved the blue moon by two months from where it “would otherwise” have been. If the 9 January full moon of 2012 had happened in a non-leap year we’d have seen a full moon on 1 September instead of the blue moon of 31 August, then a full moon on the first day of October (rather than the last day of September), turning the full moon at the end of October into a blue moon.

So there’s a lot to see in my little twenty-year chart. I’ll just point out one final thing—take a look at the positions of the full moons in the top and bottom rows of the chart. They look pretty much the same! This approximate return of lunar phases to the same dates after nineteen years is called the Metonic Cycle. It turns out that 235 lunations are very similar in duration to 19 tropical years—years as measured by the turn of the seasons, which is what our modern Western calendar approximates with its occasional leap years. For instance, if you care to use my little chart to count off 235 full moons after the first full moon of 2011, which fell on 19 January, you’ll find yourself, 19 years later, at the first full moon of 2030, also on 19 January.

To fit 235 lunations into 19 years, we need to have twelve normal years and seven blue-moon years, since (12×12) + (7×13) = 235. Take a look at my chart again and, sure enough, you’ll find seven blue-moon years. (2018 contains two blue moons, to be sure, but the missing full moon in February ensures that the full-moon count is just thirteen.) And 7/19 = 36.8%, another close approximation to Steve Holmes’s more exact calculated figure, above.

The Metonic Cycle gets its name from Meton of Athens, a Greek mathematician of the 5th century BCE, who incorporated the cycle into a lunisolar calendar—that is, a calendar that tracks both the seasons and the lunar phases. People all over the world have used this sort of calendar, counting off the full moons as a guide to how the seasons are progressing, with three full moons in each “normal” season, and a four-moon season cropping up in seven years out of nineteen.

Well into the twentieth century, there were farmers who would keep an eye on the lunar cycle as a guide to their seasonal activities. In the United States they were aided in this practice by the yearly publication of various “farmers’ almanacs”. And one of those, the Maine Farmers’ Almanac, seems to have originated the idea of applying the name “blue moon” to the extra full moon in a four-moon season. So every year the almanac would publish a list of dates of full moons, together with seasonal names from Christian European tradition like Lenten Moon and Moon Before Yule, interspersed with the occasional Blue Moon, but without an explanation of why specific full moons had been chosen to be blue moons.

In 1946, an amateur astronomer called James Pruett tried to figure out the blue-moon rule used by the Maine Farmers’ Almanac, and in a Sky & Telescope article entitled “Once In A Blue Moon” he described the familiar Metonic Cycle:

Seven times in 19 years there were—and still are—13 full moons in a year. This gives 11 months with one full moon each and one with two. This second in a month, so I interpret it, was called Blue Moon.

Sky & Telescope went on to use Pruett’s definition, and the rest is history.

But in 2006, Daniel Olson, Richard Tresch Fienberg and Roger Sinnott pored over a lot of copies of the Maine Farmers’ Almanac, and figured out the underlying blue-moon rule that had actually been used, which is very different from Pruett’s version:

At last we have the “Maine rule” for Blue Moons: Seasonal Moon names are assigned near the spring equinox in accordance with the ecclesiastical rules for determining the dates of Easter and Lent. The beginnings of summer, fall, and winter are determined by the dynamical mean Sun. When a season contains four full Moons, the third is called a Blue Moon.
Why is the third full Moon identified as the extra one in a season with four? Because only then will the names of the other full Moons, such as the Moon Before Yule and the Moon After Yule, fall at the proper times relative to the solstices and equinoxes.

You can read their full Sky and Telescope article, with much more detail, here.

# Hill Lists: “On Top Of The World”

I haven’t written about hill lists for a while, and after writing about the classic Scottish hill lists, and dealing in separate posts with the Corbetts and the Donalds, I’m overdue to write about the third (and original) classic, the Munros. But instead, I’m veering off into the long grass with this one, which deals with a list covering the whole world, featuring 6464 separate peaks, all of which place a summit observer “on top of the world”, by strict geometric criteria.

The list is an offshoot of the work of Kai Xu, at Yale University, which he described in a paper entitled Beyond Elevation: New Metrics to Quantify the Relief of Mountains and Surfaces of Any Terrestrial Body. The paper offers four new descriptors for the way in which mountain peaks relate to the surrounding terrain: dominance, jut, submission, and rut, which together sound like a firm of sadomasochistic lawyers. You can find details of jut on Xu’s website devoted to the topic, but the On Top Of The World (hereafter, OTOTW) list is derived from the measure Xu calls submission.

Submission is defined in Xu’s paper as follows:

The submission of point p is the maximum height of any point on the planetary surface above the horizontal plane of p:
[…]
Submission measures how high the surroundings of a point rise above the point itself, yielding a value greater than or equal to 0 for any point on the planetary surface. As with dominance, submission only considers points within a local vicinity, as points very far away from p correspond to negative height values irrelevant to the calculation of submission.
[…]
A point with a submission equal to (or less than) 0 is known as a dominant point. A person standing at a dominant point is “on top of the world,” as no point rises above their horizontal plane.

The OTOTW list includes all those summits that are also dominant points, under Xu’s definition. Time for a diagram:

The summit in the middle of my diagram above (the one with the little observer perched on its top), is associated with a local horizontal plane that I’ve sketched in blue. Nearby hills fail to pierce this horizontal plane because they are too low. A higher peak at left is sufficiently far away that the curvature of the Earth prevents its summit piercing the horizontal plane. My little observer is therefore “on top of the world”.

Coming up with an exhaustive list of such summits requires the processing of a shed-load of topographic data, and also factoring in the lumpy shape of the geoid, the true shape of the Earth at sea level. You can find a nice map of Xu’s entire collection of OTOTW summits here.

It’s a fine thing to contemplate, but I thought I’d simplify the contemplation a little by honing down, very parochially, on the hills I know well—the twenty OTOTW summits in Scotland, shown on my map at the head of this post.

The first thing to notice is that the big hills drive out the small—the northern mainland of Scotland is dominated by eleven high summits, all of them of Munro status—that is, higher than 3000 feet (914 metres). Two of these Munros lie offshore, the highest points on the islands of Skye and Mull, but they’re near enough to the mainland to suppress the OTOTW aspirations of many west-coast hills.

The Southern Uplands, meanwhile, are dominated by the two highest hills in that region—Merrick in the west and Broad Law in the east.

The outlying islands are far enough from the Highland giants to generate their own OTOTW summits—Goatfell on Arran, Beinn an Oir in the Paps of Jura, An Cliseam on Harris, and Ward Hill on the island of Hoy, in the Orkneys. Even farther out, we get our final three summits—all low, but far enough from everything else to still reach OTOTW status—Ronas Hill in Shetland, Conachair on St Kilda, and Da Sneug on Foula.

On the mainland, some summits seem oddly close together—the Ben More / Ben Lawers pair; the trio of Ben Hope, Ben Klibreck and Ben More Assynt. These groupings are made possible by the fact that the hills involved have roughly similar heights. Lawers is just 40 metres higher than Ben More, and the 26-kilometre separation between the two is enough to drop Lawers (by my rough calculation) about 15 metres below the local horizontal plane drawn from Ben More’s summit. Ben Klibreck is 35 metres higher than Ben Hope, but 23 kilometres away, dropping it about six metres below Hope’s local horizontal.

And for those familiar with the Scottish hills and outlying islands, there are some surprising omissions. Ben Wyvis (1046m) stands in notable isolation, but doesn’t make OTOTW status—the summit of Sgurr Mor (1109m) is just high enough to break through Wyvis’s local horizontal. The little island of North Rona, 70 kilometres northwest of Cape Wrath, is low (just 108 metres), but also a long way from any high ground—surely it should qualify? But a distant glimpse of Foinaven (911m) on the mainland is enough to pierce Rona’s horizontal plane. (And Foinaven, in turn, falls victim to Ben More Assynt, farther to the south.) And the whole chain of islands of the Outer Hebrides is denied OTOTW status by sight of Sgurr Alasdair (and the other Skye Cuillins), until the terrain gets high enough, and far enough north, for An Cliseam to triumph.

Finally, there’s actually a twenty-first Scottish OTOTW summit that isn’t listed by Xu—the Atlantic islet of Rockall, which since 1972 has been officially (in the UK at least) part of Scotland. Over 300 kilometres from the nearest land, and just 17 metres high, absolutely nothing is visible above its sea horizon, making it an obvious shoo-in for On Top Of The World status. I suspect the omission from Xu’s list is because the topographic databases he processed in order to generate his data just don’t contain this tiny bit of remote real estate.

Note: CCCP stalwarts Steve and Rod contributed significantly to the discussion of hills that have surprisingly failed OTOTW status, and it was Steve who spotted Rockall as a missing qualifier.

# How Apollo Left Earth (And Returned): Part 1

This is the fifth in my occasional series of posts about the orbits followed by the Apollo spacecraft as they departed from (and returned to) the Earth. It’s a companion to, and expansion of, my old post “How Apollo Got To The Moon”, informed by a more recent series of posts that culminated in my deriving a set of orbital elements for Apollo 11’s departure towards the moon.

That series started with a post entitled “Keplerian Orbital Elements”, which introduced the various parameters used to describe an orbit—these are the numbers you need to plug into a piece of orbit-plotting software, like Celestia, so that it will display the spacecraft’s trajectory for you. (It’s what I used to prepare the diagram at the head of this post.)

Then I progressed to “Finding Apollo Trajectory Data”, in which I provided links to the original Apollo documentation, and described how to pull the necessary data from those sources.

Then I digressed into “The Advent Of Atomic Time”, as a way of introducing the difference between the GMT times listed in the Apollo trajectory documents, and the Terrestrial Time (TT) we need to use in order to correctly describe the Apollo orbits.

Most recently, I offered a fairly equation-intensive post entitled “Converting Apollo State Vectors To Orbits”, in which I drew together the principles established in the first three posts and gave a worked example, deriving the orbital elements of Apollo 11’s departure from Earth.

The logical progression, at this point, would be to subject you to another blizzard of equations, showing how to use those orbital elements to calculate the orbital position of a spacecraft at any given time, how to convert those positions to a ground track, and how to transform the geographical coordinates of the ground track into geomagnetic coordinates, so as to plot a trajectory relative to the Van Allen Radiation Belts. But I’m going to skip all that for now, and instead just show you some actual results.

The reason all this stuff about ground tracks and so on is important is because the static image at the head of this post can’t tell the whole story—because while Apollo 11 moved along its orbit (the red curve in the picture), the Earth and the Van Allen Radiation Belts rotated beneath it. This produced some interesting dynamics, which I can demonstrate in a little (30-second) video showing Apollo 11’s view of Earth during the first ten hours of its flight to the Moon, speeded up 1000 times. The animation was produced in Celestia, using the orbital elements I derived at the end of my previous post on this topic.

The journey begins at the moment of Translunar Injection (TLI), when the Apollo S-IVB stage finished its second burn, having accelerated the Apollo spacecraft into an orbit that would take it to the Moon. We’re looking straight down at the night-time Pacific, which fills the screen. But very soon our viewpoint shoots into daylight, travelling west-to-east over the United States, where it then seems to loiter for a while above the Caribbean, before we see the Earth apparently, and belatedly, start to rotate in its normal fashion beneath the retreating spacecraft. So our view of the Earth tracks quickly west-to-east, pauses, and then begins to drift slowly east-to-west.

What’s going on there? The reversal in relative motion is caused by the shape of Apollo’s elliptical orbit. It starts off travelling very fast from west to east, and almost parallel to the Earth’s surface, so that it overtakes the rotating Earth—the Apollo 11 astronauts in fact saw the sun rising in the east ahead of them as their orbit carried them into daylight over the USA.

But their orbital trajectory was rising and slowing, and as their velocity decreased it also became directed more away from the Earth (towards the Moon!) rather than parallel to its surface. Like this:

Eventually, as their trajectory carried them almost directly away from the Earth, they were able to watch it make its usual 24-hour rotation behind them. But there was an intermediate stage, between the extreme situation in which they overtook the Earth’s rotation, and the “normal” view they obtained later—for a while, their velocity approximately offset the Earth’s rotation, so their viewpoint loitered for an hour or two over the Americas.

We can see how this played out using a couple of graphs, prepared from the orbital elements of Apollo 11’s departure trajectory. Below, I plot the spacecraft’s velocity and flight angle during the first six hours of its translunar trajectory. (The flight angle is the angle between the trajectory and the local horizontal. A flight angle of zero corresponds to an orbit parallel to the Earth’s surface. A flight angle of ninety degrees indicates a vertical trajectory.)

Apollo’s velocity, marked in blue and plotted against the left axis, starts at almost 11 kilometres per second, but decays steadily under the influence of Earth’s gravity. Its flight angle (red, right axis) begins with a slight upward tilt of about seven degrees, but quickly progresses towards the near-vertical (over 70 degrees) as it draws away from Earth.

Putting these two factors together, we can plot the angular velocity of the Apollo spacecraft as it moves around its orbit and compare that to the constant angular velocity of the rotating Earth:

We can see how Apollo moved faster than the Earth’s rotation for about an hour after TLI, but for the next hour was moving only a little faster or a little slower than the Earth, so that it would appear to hang in the sky over the Earth’s surface for a while.

Here, in blue, is Apollo 11’s ground track for its first six-and-a-half hours after TLI. The circles mark off intervals of one hour along the track. In green, I’ve superimposed latitude and “longitude” lines for the Earth’s geomagnetic field.*

At the time of the Apollo 11 mission, the north geomagnetic pole was situated close to the entrance to the Nares Strait, between Greenland and Ellesmere Island, and the geomagnetic field was correspondingly tilted southwards over the Americas, taking the Van Allen Belts with it. The inner VAB, which contains the bulk of the dangerous proton radiation, lies mainly between magnetic latitudes forty degrees either side of the magnetic equator—I’ve shaded that region in yellow. As previously described in my post “How Apollo Got To The Moon”, you can see that Apollo’s departure orbit passed north of that critical magnetic latitude band during its first hour, and entirely avoided the region of most intense radiation near the magnetic equator.

Armed with Apollo’s orbital elements, we can get a better view of the spacecraft’s passage through the VAB by converting its geographical coordinates and distance from the centre of the Earth into magnetic coordinates. After doing that we can plot its orbital radius and magnetic latitude and superimpose that on a diagram of the Van Allen Belts. I’ve adapted and coloured the VAB diagrams from NASA’s Bioastronautics Data Book, Second Edition (1973).

Here’s the Apollo 11 trajectory relative to the electrons trapped in the VAB:

The electrons outline the inner and outer radiation belts—the intense inner VAB is show in red and orange, the green band is a region of relatively decreased radiation, and then we have the larger but less intense outer VAB in yellow-brown. Apollo 11 traversed the entire region in about an hour, but was in minimal danger from electrons, which are easily blocked by the structure of the spacecraft.

More dangerous was the energetic and penetrating proton radiation, largely confined to the inner VAB:

I’ve marked the “danger zone” along the Apollo trajectory in red, but you can see that it traversed only the fringes of the inner VAB, avoiding the core area of high radiation. With reference to the specific diagram above, I find that Apollo 11 was within the sketched limits of the proton VAB for a total of 14 minutes, starting three minutes after TLI. It’s important, though, to realize that the Van Allen Belts are very variable structures, so we shouldn’t read too much into specific radiation counts on specific charts. But we can get the general message from this diagram that the radiation dose to the Apollo 11 crew members was limited by both the speed with which they traversed the VAB, and by using an orbit that avoided the most intense regions.

I can now go back to my ground track diagram, and show the red “danger zone” section on that:

It illustrates, from a different viewpoint, how the Apollo departure trajectory exploited the tilt of the Earth’s magnetic field so as to pop northwards out of the VAB as soon as possible.

I’ve also marked another event along the early departure orbit—the Transposition, Docking and Extraction manoeuvre (TD&E), during which the Apollo Command and Service Module turned around, docked with the Lunar Module in its stowed position atop the Saturn S-IVB stage, and extracted it. (See my link for a more detailed explanation.) The whole procedure typically took about an hour, and you can see it started quite soon after Translunar Injection—about thirty minutes later, in this case.

Why the rush to get going with that? One reason, I think, was to make sure that the Apollo spacecraft and the spent S-IVB stage started moving apart as soon as possible, to avoid the danger of collision during later spacecraft manoeuvres. But it was also handy that the whole operation could be carried out during Apollo’s “loiter” over the Caribbean, within line-of-sight radio transmission of the United States, thereby avoiding the potential problems involved in relaying radio messages through the other NASA ground stations dotted around the globe.

All this happened in reverse when Apollo returned to Earth, with a “loiter” in the ground track taking place over the southern Indian Ocean. And this time Apollo was approaching the Pacific Ocean from the south, again exploiting the tilt of the VAB, except with an even more inclined orbit—close to 40° inclination to the Earth’s equator, compared to 30° on departure. Like this:

Six hours before entering the atmosphere, the spacecraft was 80,000 km above Western Australia. Its ground track then swung out over the southern Indian Ocean and loitered for a bit, before turning back, gathering speed and diving through the inner VAB while it recrossed Australia, heading for splashdown in the Pacific.

With the specific plot used here, they spent just nine minutes shooting through the outer rim of the inner VAB, then another three minutes in free flight above the Pacific before hitting the “entry interface”—the point at which the Earth’s atmosphere began to have a significant effect on their trajectory, at an altitude of 400,000 feet.

Many of the Apollo trajectories followed a similar pattern—north over the VAB on the way out, south under the VAB on the way back. But there were exceptions. That’s what I’ll write about next time.

* The geomagnetic poles define the overall tilt of the Earth’s magnetic field, and are different from the magnetic “dip” poles towards which your compass needle points. Scientific American has a discussion of the difference between the two magnetic poles here. The World Data Center For Geomagnetism in Kyoto provides some nice maps showing the recent wanderings of the north and south magnetic poles and geomagnetic poles. To calculate the magnetic latitudes of the Apollo trajectories, I used a little program called GM POLE, from the National Oceanic and Atmospheric Administration, which provides the coordinates of the geomagnetic poles on any given date between 1900 and 2015.