# Pole Stars Of Other Planets?

When I wrote about Philip Latham’s juvenile science-fiction novel Missing Men Of Saturn (1953) recently, I pointed out that Latham had made an astronomically well-informed guess about a possible pole star for Saturn’s moon Titan. Latham (a professional astronomer) knew the orientation of Saturn’s rotation axis, which would have allowed him to deduce the location on the celestial sphere around which stars would appear to rotate in Saturn’s sky, in the same way they appear to rotate around our pole star, Polaris, in Earth’s sky. And it was a reasonable guess that Titan’s rotation axis would be moderately well aligned with Saturn’s, leading Latham to have his protagonist use the star Gamma Cephei to orientate himself during his exploration of Titan.

So I thought this time I’d write about the location of the celestial poles of other planets—that is, the two apparently stationary points on the celestial sphere around which the sky of each planet appears to rotate, as a result of the planet’s rotation. Whether or not a star will turn out to be close enough to these points to function as a “pole star” is another matter. We’re in fact astronomically lucky (literally) to have a bright star situated so close to Earth’s north celestial pole; there’s no corresponding star in the south. And, because of the slow precession of Earth’s rotation axis, the alignment with the star Polaris is only a temporary one.* At present the celestial pole is moving slowly closer to Polaris, but Jean Meeus tells us, in his book Mathematical Astronomy Morsels (1998), that this motion will soon end, and pole and star will start to drift apart again—the closest approach will occur in February 2102, which is pretty close to being tomorrow, in astronomical terms.

Latham’s assessment that a major moon of a gas giant would tend to have its rotation axis aligned with its parent planet has proved to be correct. We now know that the major satellites of Saturn are so aligned (out as far as Titan), as are those of Jupiter, Uranus and Neptune. Even the two tiny satellites of Mars share their rotation axis with their parent. So in what follows, you can consider that the pole position of a planet, as marked on my small-scale sky maps, also indicates the pole positions of many of its satellites.

But before showing you my sky maps of the various celestial poles, I need to clarify what is meant by “north pole” and “south pole”.

If we look at the Earth from above its north pole, it appears to rotate anticlockwise, like this:

This gives us two, not entirely consistent, ways of defining “north” for other planets. The first would be to define the north pole as being that rotation pole from above which the planet appears to rotate anticlockwise. This is sometimes called the “right-hand rule”, because if we imagine wrapping our right hand around the planet’s equator, with our fingers pointing in the direction of its rotation, then a “thumbs up” sign from this position will point towards the planet’s north pole. This is a nice generalizable rule, and I’ll come back to it at the end of this post, but it’s not the one adopted by the International Astronomical Union.

In 1970, the IAU defined a sort of “solar system north”, using the Earth’s north pole as the criterion. The hemisphere of sky that lies on the north side of the plane of the solar system, as judged by the orientation of Earth’s north pole, defines the north poles of all the other solar system planets and their major satellites. The “right-hand rule” and “solar system north” produce consistent results when a planet rotates anticlockwise as viewed from solar system north. But they produce opposite results if the planet appears to rotate clockwise when viewed from solar system north. The IAU defines such planets as having retrograde rotation, because they turn backwards when compared to the Earth. The opposite of retrograde is prograde or direct, and most (but not all) of the major bodies in the solar system have prograde rotation.

You might be wondering how the IAU defines “the plane of the solar system”. Though all the planets orbit in roughly the same plane, they interact gravitationally with each other, so the precise angles between the various orbits varies. But the total angular momentum of the solar system stays the same, and that defines an axis and corresponding plane called (appropriately enough) the Invariable Plane of the Solar System (henceforth, the IPSS).

So there’s a “north pole” in the northern sky associated with the IPSS, and it, with its southern counterpart, are the only poles that have an absolutely constant position on an astronomical time scale. Nearby is another north pole associated with the plane of the Earth’s orbit around the Sun—the ecliptic pole. It’s near the IPSS pole because Earth’s orbit is tilted only slightly relative to the IPSS. It will move very slowly as the Earth’s orbit slowly shifts under the gravitational disturbance of the giant planets. And we know that the Earth’s axis of rotation is tilted at 23½º relative to the plane of its orbit, so the Earth’s north celestial pole lies 23½º away from the ecliptic pole, in the constellation of Ursa Minor, near the star Polaris. As I’ve already mentioned, this pole too is in motion—it describes a wide circle around the ecliptic pole every 26,000 years.

The north IPSS pole and ecliptic pole lie in the constellation of Draco, within the curve of the “neck” of the imaginary dragon that is sketched out by the constellation’s stars. A little cluster of other north poles is gathered around them. Venus and Jupiter follow orbits that are minimally tilted relative to the IPSS, and have rotation axes that are only slightly inclined to their orbits, so their poles sit close to the IPSS pole. Mercury also has a minimal tilt relative to its orbital plane, but its orbital plane is tilted by more than six degrees relative to the IPSS, moving its pole a corresponding distance from the IPSS pole. In the constellation diagram below, I’ve also marked the north pole of the Sun, which is tilted by around six degrees relative to the IPSS. So all the north poles mentioned so far lie in Draco, in or around the loop of the dragon’s neck, but none of them near any particularly bright stars.

On the maps that follow, planets with direct rotation are marked in green, and retrograde rotators in red.

(You’ll notice that I’ve deviated from my habit of using Celestia to generate my astronomical and astronautical illustrations—all my plots this time are superimposed on star charts generated at In-The-Sky.org, which are clearer for my purposes today.)

Also plotted above are the poles of Saturn, Mars and Neptune, all significantly tilted relative to their orbital planes and the IPSS. Saturn’s north pole lies in a dark corner of Cepheus, actually a little closer to Polaris than to Gamma Cephei (marked with its name Errai on the chart). Mars has its pole inconveniently placed in the dark sky between Cepheus and Cygnus. Neptune’s pole has perhaps the most navigationally convenient location, on the left wing of Cygnus the Swan, almost on the line between second-magnitude Sadr and third-magnitude Delta Cygni.

In contrast to the other bodies plotted above, Earth’s Moon is marked with a dashed circle rather than as a single point. This is because, like Earth, the Moon’s rotation axis precesses around the ecliptic pole—but it does so in a mere 18.6 years. So on any given date the Moon’s north pole will be located somewhere around the circumference of a circle about three degrees across, centred on the ecliptic pole.

In the southern sky, the pattern of poles is the same, but inverted relative to their northern counterparts.

The poles of Mars and Jupiter lie in dark parts of the constellations of Vela and Puppis, respectively. Mercury has a good, but relatively faint, southern pole star in third-magnitude Alpha Pictoris. Saturn’s pole star, Delta Octantis, is even fainter. This leaves the cluster of poles around the ecliptic pole, all of which lie in the faint constellation of Dorado. They all have a good pole “star” in the form of the Large Magellanic Cloud (not shown on my chart) which extends south from Delta Doradus and into the neighbouring constellation, Mensa.

One planet is missing from the charts above—Uranus. This is because Uranus notoriously orbits lying pretty much on its side, the inclination of its equatorial plane to its orbital plane being variously given as 98º or 82º and retrograde. This puts its poles in the vicinity of the plane of the ecliptic, rather than the ecliptic pole.

On my charts below, the brown line is the ecliptic, as a stand-in for the IPSS. Uranus’s south pole (retrograde rotation) lies in a dark area of sky, but is positioned between the two bright constellations of Orion and Taurus, making it relatively easy to find, at least approximately.

Its north pole lies in Ophiuchus, conveniently close to the second-magnitude star Sabik (Eta Ophiuchi).

Finally, I promised I’d come back to that “right-hand rule” definition. While the “solar system north” standard works well for planets and their large satellites, the International Astronomical Union realized that smaller bodies can have rotation axes that precess or migrate so quickly that they could easily shift back and forth across the IPSS within a few years. Under the “solar system north” rule, this would mean that the body would swap its north and south poles, and shift between direct and retrograde rotation, over the same period. So in 2009 they officially fell back on the good old “right-hand rule” for dwarf planets, minor planets, their satellites, and cometary nuclei. To avoid running two conflicting definitions of “north” in parallel, they designate the right-hand-rule pole of these bodies the positive pole, with the negative pole lying in the opposite direction.

Which brings us to Pluto, everyone’s favourite ex-planet. Until 2009 it was officially a retrograde rotator, with its north pole in Delphinus and its south pole in Hydra. Now, under the new definition, its positive pole is in Hydra, and its negative pole in Delphinus.

These poles will serve for Pluto’s huge moon, Charon, too. But sadly, both poles lie in rather dim and undistinguished parts of the sky.

* When the Phoenicians were setting out on their voyages of discovery, three millennia ago, they had no pole star to guide them. The north rotational pole of the Earth lay in a fairly empty bit of sky north of the bowl of the Little Dipper.

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# Which Way Does Space Station V Rotate?

The “Phenomena” posts have been a little tied up with abstruse orbital mechanics and obscure revisions to lists of Scottish hills, of late, so I thought it might be time for a break from all that.

So this post is about something superficially trivial in the film 2001: A Space Odyssey, which has mildly annoyed me for the last fifty years.

People seem to particularly like compiling lists of mistakes in 2001: A Space Odyssey, presumably because it’s a classic film with pretensions to scientific accuracy, made by a famously exacting director advised by a famously knowledgeable science-fiction author. Who wouldn’t want to pick holes in that?

There are mistakes that are genuine scientific errors—I’ve written before about the dire physiological consequences that would have ensued if the Dave Bowman character had really tried to hold his breath when explosively decompressed. (Arthur C. Clarke later said that he would have advised against that, if he’d been present during the filming.) And there are mistakes that are simply technical malfunctions—like the glimpse of Bowman’s bare wrist we get when his spacesuit glove comes away from his spacesuit sleeve. And then there are “mistakes” which are self-evidently artistic decisions to step away from strict accuracy—the various planetary and solar alignments that herald great events, for instance, are clearly inconsistent with the astronomical positions of these bodies in previous scenes.

Finally, there are the continuity glitches, of which there are strikingly many in 2001, given what a notorious perfectionist Stanley Kubrick was.

So here’s the one that first caught my attention, when I watched the film from the front row of the top balcony of the Victoria cinema, back in 1970.

There are eight shots in which the giant wheel-shaped space station, Space Station V, appears in the film. I do love this thing, to the extent I’ve built a model of it. It rotates in order to generate “centrifugal gravity” at its rim, and it consists of two rings, one complete and one under construction. The docking port at the hub of the completed side is internally lit with white lights, while the inactive docking port on the other side is lit in warning red. So if we imagine approaching the station towards its active, white-lit hub (as the Orion III space shuttle in the film does), then we can state unambiguously whether it rotates clockwise or counterclockwise as seen from that vantage point.

When we first see it, twenty minutes in, to the accompaniment of Strauss’s Blue Danube waltz, it is rotating clockwise, by that definition:

I’ve marked the picture above with an arrow to indicate the rotation direction, and have noted the timing of the shot, in minutes and seconds, on my old “Deluxe Collector Set” DVD, from which I’ve made these muddy screengrabs.*

Then we get a series of shots involving the Orion III spaceplane on its way to the station, before seeing the station in the distance, with the approaching Orion III in the foreground and the moon beyond. Still clockwise.

Next shot, another hero shot of the space station, with the pursuing Orion III appearing later. This is the point at which, back in 1970, I said (rather loudly, I’m told), “Hang on a minute!” The station is now rotating anticlockwise.

Then a view from inside the cockpit of the Orion III. Still anticlockwise.

There follows a brief view of the instrument panel, in which a wire-frame model of the docking port continues the anticlockwise rotation, and then in the next shot we’re inside the station hub looking outwards. The stars outside are moving in clockwise circles, implying the station is rotating anticlockwise as viewed from inside, and therefore (dammit) clockwise when viewed from our stipulated vantage point.

Next shot, and we’re outside again, watching the spaceplane synchronize its rotation with the docking port. Anticlockwise again.

Then a view from the Orion III cockpit, now synchronized with the docking port ahead. The stars sweep past in clockwise circles beyond the station, so the station (and spaceplane) are rotating anticlockwise.

And this anticlockwise rotation is maintained in the final approach shot.

So that’s just a minor curiosity—there are many worse errors in many other films, after all.

Except …

Once you start noticing this stuff, you keep on noticing it. 2001 is positively stuffed with left/right switches, as well as odd 90-degree anomalies. Another gross example occurs when we see the Earth, low on the Moon’s horizon. (This is another of those occasion on which artistry has overruled scientific accuracy—the Moon would be higher in the sky as seen from Clavius and Tycho, where the action takes place.)

As the Aries 1B shuttle approaches the Moon, we see the Earth illuminated from the right. But when we see a shot of several astronauts watching the shuttle’s approach, the illuminated portion faces left. Back to the shuttle, and it’s shifted right again. The same thing happens as we follow the moon-bus across the lunar terrain—the Earth starts off illuminated from the right, then switches to the left, then back to the right again.

(In the Moon’s southern hemisphere, where the action takes place, the correct orientation at lunar sunrise, when the action takes place, is to have the Earth’s illuminated portion to the right, facing east.)

Later in the film, from shot to shot, we see the character Frank Poole not only reverse the direction in which he’s running around the Discovery centrifuge (twice), but the entire centrifuge (and Poole) becomes mirror-reflected. The bone thrown aloft by the man-ape Moon-Watcher, at the beginning of the film, reverses its rotation direction between the two shots that follow its trajectory. And there are multiple geometrical inconsistencies during the sequence set on the Aries 1B moon shuttle, relating to the orientation of the control cabin.

It’s all very odd, and I don’t pretend to have an answer, but stuff like the constantly shifting orientation of the Earth seems too egregious to be anything other than deliberate—it would have been easier to have that not happen. And Kubrick, of course, has history with this sort of thing—the geometry of the Overlook Hotel in The Shining (1980) is notoriously protean.

Perhaps there’s a clue to what it all means, embedded in the very first shot in which we see a left-right reversal. The Dawn of Man chapter of the film famously used front-projected African scenes to provide back-drops for the outdoor sequences featuring the man-apes (which were actually filmed on a sound-stage). One particular reddened sky provided the backdrop for the set depicting the area around the man-apes’ cave refuge, and was used multiple times, both as a sunset and a sunrise. But in one shot (and only one shot), the sky image is reversed—and that’s in the shot in which the man-apes (and audience) first discover that a giant alien monolith has materialized outside the cave during the night.

* I’ve been trapped in a Tommy Lee Jones Cycle for decades, now. His character Kay in Men In Black (1997) kept having to buy the Beatles’ White Album over and over again, as physical media improved. I’ve now owned 2001: A Space Odyssey on VHS, DVD, Blu-ray and 4K disc. I suspect the rising graph of increasing visual and auditory fidelity from new technology has now crossed the descending graph of my failing eyesight and hearing.
I’m indebted to Juli Kearns for pointing out this key reflection in her shot-by-shot analysis of 2001: A Space Odyssey on her Idyllopus Press website.

# Scottish Hill Lists: The Donald Revisions

This is the second in my planned series of posts dealing with the revision history of the three “classic” tables of Scottish hills—the Munros, Donalds and Corbetts, which I introduced in an earlier post. I also introduced the idea of topographic prominence, and a way of charting these hill tables in two dimensions by plotting height against prominence. If any of this is strange to you, I refer you back to the original post via my link above, for a quick tutorial.

Last time, I dealt with the Corbetts (hills between 2500 and 3000 feet in height, with a prominence of at least 500 feet), and pointed out a number of ways in which new topographic data can lead to a hill either being deleted from, or added to, a set of tables.

This time, it’s the turn of the Donalds, lowland hills higher than 2000 feet. In contrast to Corbett’s tables, which have pretty simple and strictly numerical entry criteria, Donald’s tables feature a combination of rather more complicated topographic criteria with some value judgements, sorting the tabulated summits into two major categories—“Tops” and “Hills”:

“Tops”—All elevations with a drop of 100 feet on all sides and elevations of sufficient topographic merit with a drop of between 100 feet and 50 feet on all sides.
“Hills”—Groupings of “tops” into “hills” except where inapplicable on topographical grounds, is on the basis that “tops” are not more than 17 units from the main top of the “hill” to which they belong, where a unit is either 1/12 mile measured along the connecting ridge or one 50-feet contour between the lower “top” and its connecting col.

(Donald’s “drop on all sides” is the equivalent of modern “prominence”, the term I use in my charts below.) Donald’s rules seeks to reflect something about the shape of the landscape—allowing a single high summit, the “hill”, to dominate a fairly tight cluster of lower “tops”. The criteria given above mean that the summit of a “top” can’t be more than 1⅓ miles (2.15 km) from its parent “hill”.

In modern discussions of these tables, the “hills” have come to be referred to as “Donalds”, while the “tops” are called “Donald Tops”. The tables, as originally published, contained 86 Donalds and 47 Donald Tops. Donald also listed five English hills, close to the Scottish border, which fulfilled his criteria, but he did not assign them numerical entries in the tables. And, in an appendix, he added 15 summits:

[…] not meriting inclusion as tops, but all enclosed by an isolated 2,000-feet contour. These have been included in order that the table may be a complete record of every separate area of ground reaching the 2,000-feet level.

These locations have sometimes been referred to as “Minor Tops”, and that marginal category has actually been the main focus for such revisions as have been made, the remainder of Donald’s tabulation being surprisingly resistant to major change. Indeed, the Donalds remained entirely unrevised for 45 years, through multiple editions of Munro’s Tables.

Then, in the editions of 1981 and 1984, the availability of better mapping led to a considerable expansion to the list of Minor Tops, from 15 to 28—this despite the promotion of three Minor Tops (Keoch Rig, Conscleuch Head, and the south-west top of Windlestraw Law) to full Top status. These three were presumably selected on the basis of “sufficient topographic merit”, since they all have prominences between 50 and 100 feet.

The 1980s editions also ushered in a decade of confusion on the double-humped ridge of Black Law—creating a rather dubious Top on its north-east summit in 1981, to complement the existing Donald on the south-west summit; then switching the Donald and Top around in 1984, as the north-east summit proved to be higher than the south-west … only to have the Top deleted again in 1997, on the grounds (presumably) that its 36-foot prominence falls far short of Donald’s minimum criterion. So Black Law appears twice on my Donalds chart, with one summit marked as deleted and the other appearing as an addition. (A similar, later, migration of the Donald summit of Meikle Millyea is also marked. This was long anticipated, but not confirmed to the SMC’s satisfaction until 2015.)

The 1984 edition is also responsible for the only “promotion” of a Donald Top—Carlin’s Cairn.

Donald would have counted no less than seven fifty-foot contours on the ascent of Carlin’s Cairn, shown below on the one-inch mapping of 1926.

This whopping prominence, half again higher than any of the other Donald Tops in the original tables (see my plot of the Tops, above), means that Carlin’s Cairn meets Donald’s 17-unit criterion only because it’s less than a mile from nearby (and higher) Corserine. Donald was presumably swayed towards making it a Top rather than a Donald because it’s quite evidently part of the northern ridge of Corserine; but the 1984 Tables editor presumably felt that the comparatively large re-ascent from the col made the 17-unit rule “inapplicable on topographic grounds”, and so bumped Carlin’s Cairn to full Donald status.

Two significant revisions occurred in the 1997 Munro’s Tables. The first was the abandonment of the Minor Tops—they were either promoted to full Donald Top status, if merited, or deleted. Only one, Notman Law, survived the cull. (At the same time, Donald’s unnumbered list of five English summits was also dropped.)

The second revision was altogether more dramatic—the inclusion of a whole new and previously unsuspected group of Donalds and Tops. The discovery of these “Lost Donalds” on the south side of Glen Artney was first reported in The Angry Corrie, in 1994. Although Donald never clearly described what he meant by “the Scottish Lowlands” when he published his tables, it’s clear from the lists themselves that they document the 2000-footers of the Central Belt and Southern Uplands. The northern edge of this lowland area is commonly understood to be the Highland Boundary Fault (HBF). And this fault runs along Glen Artney, placing the 2000-foot hills on its southern side squarely in the Lowlands. You can check this for yourself on the Geological Survey of Great Britain (Scotland) Sheet 39W—Artney and the HBF lie in the top left corner and the new Donalds (Uamh Bheag and Beinn nan Eun) and associated Tops (Meall Clachach and Beinn Odhar) are visible in the Ordnance Survey mapping below the geological overlay. All are labelled on my charts.

Finally, there’s the vexatious (to me, at least) matter of Auchope Cairn and Cairn Hill West Top. The first of these two Tops was introduced by Donald, and discarded in 1997. The second appeared as a numbered Top in 1981 and is still with us. As my chart above shows, both fail to meet Donald’s 50-foot threshold prominence for inclusion, scoring 30 feet and a laughable 16 feet, respectively. Both are, also, much farther than the 17-unit threshold from the nearest Donald summit, at Windy Gyle; even The Cheviot, which featured as an unnumbered summit in Donald’s original tables, is not close enough to these two hills to play “Donald” to their “Top”. A map of the current Donalds and Tops makes their bizarre status in this regard clear:

According to the 17-unit rule, the Donald Tops (open triangles) all lie within 1⅓ miles of their parent Donalds (filled triangles), forming dense clusters … except for Cairn Hill West Top, which sits in splendid isolation on the Scottish/English border. (Auchope Cairn is not marked, but lies only 700m north-west of CHWT, and would be superimposed upon it if plotted on my map.) Here’s the one-inch map of 1927, annotated with the position of Cairn Hill West Top (hereafter, CHWT):

So what’s going on? It appears that Donald was keen to include some indication of the highest point on the Scotland/England border, and Auchope Cairn was the closest named summit to that point. (There was a reluctance, in the early days of hill-tabulation, to include summits that were unnamed on Ordnance Survey maps.) The actual highest point was on the rounded shoulder of Cairn Hill, which has now been dubbed Cairn Hill West Top. This was marked by a 2422-foot spot-height on the Ordnance Survey six-inch map to which Donald would have referred; but that height had been inferred by sighting from a triangulation point about 800 feet to the south-west, with an altitude of 2419 feet.

So Donald provided Auchope Cairn with a rather gnomic footnote:

The highest point on the Union Boundary is (2,419) 2,422.

This footnote persisted until 1981, when the point was promoted to Top status under the newly minted name “Cairn Hill—West Top”. It was provided with a footnote that read:

The highest point on the Union Boundary. Not named on either O.S. or Bartholemew maps.

Auchope Cairn limped on in tandem to CHWT until it was finally deleted in 1997, presumably as surplus to requirements, leaving CHWT as an isolated anomaly—essentially a footnote with ideas above its station.

Note: My data source for this post is the Database of British and Irish Hills v17.2, combined with “The Donalds 1953-2021” dataset (version 3), both obtained from the DoBIH downloads page.

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# Scottish Hill Lists: The Corbett Revisions

In a previous post, I wrote about the three “classic” Scottish hills lists—the Munros (1891), Donalds (1935) and Corbetts (1952), and how these were brought together, in a publication commonly referred to as Munro’s Tables, by the Scottish Mountaineering Club in 1953.

As a way of displaying the topographic data for these hills, I also introduced the idea of plotting each summit’s height above sea level against its prominence, a measure of its height above the surrounding terrain.

For more about the classic lists, the concept of prominence, and the design of the chart above, please refer back to my previous post.

It was inevitable that the classic tables would be overtaken by improved cartography, since they were based on early topographic surveys that have now been much improved upon. And although the idea of freezing these tables into historical documents has been discussed, particularly in the early days of Hugh Munro’s table of 3000-footers, there was also a countervailing idea that the compilers themselves would have embraced any changes imposed by improved cartography—Munro, for instance, continued to update his own tables throughout his life. So the SMC has “maintained” the tables, by sporadically publishing revised versions of Munro’s Tables and the associated guidebooks. (The pace of revision has slackened off in recent decades, as Ordnance Survey mapping has become more definitive, and the remaining “problem” hills have been subjected to careful survey with Differential GPS.)

What I’m going to do in this post (and two more) is to discuss the process of revision that has taken place. I’m going to do it in reverse chronological order, starting with the Corbetts and finishing with the Munros.

The Corbetts are a nice simple list to start with, since they’re based on well-defined criteria—a height between 2500 and 3000 feet, and a prominence of greater than 500 feet—so they occupy a very precise area of my height-prominence chart.

What I’ve done below is to plot Corbett’s original list of summits, but with the height and prominence we know they have today. Any original summits that are no longer part of the current tables are marked with a black cross; any summits in the current tables which were not listed by Corbett are marked with a red plus sign:

There are three obvious ways that Corbetts can end up being added to, or removed from, the tables. Firstly, a survey can show that a Corbett actually attains a height of more than 3000 feet, moving it into the “Munro” territory of the chart; or a hill previously considered to be a Munro can turn out to be lower than 3000 feet, potentially qualifying as a Corbett. So I’ve marked examples of hills that have crossed the 3000-foot divide since Corbett’s original compilation. Ruadh Stac Mor officially graduated to Munro status in 1974; Beinn Teallach in 1990. Beinn an Lochain moved the other way in 1974.

Secondly, we can have similar transitions at the 2500-foot limit of the Corbetts. Again, I’ve marked examples—Cook’s Cairn was “demoted” in 1990; Beinn na h-Uamha graduated to Corbett status as recently as 2016.

Thirdly, hills can make the transition in or out of Corbett status if a survey carries them across the 500-foot prominence line. This has been a relatively common way in which we’ve lost and gained Corbetts, primarily because prominence has been historically harder to pin down, since the Ordnance Survey understandably devoted more attention to finding the altitude of summits than defining the lowest point of cols. The transitions at this boundary are too many to label clearly, but you can easily see the cluster of crosses and pluses on either side of the 500-foot prominence line. Most of these transitions occurred in the 1981 and 1984 editions of the Tables, in the light of improved mapping.

But what about those deletions that have extremely low prominence? The deletion I’ve marked as “Sgurr nan Eugallt (East Top)” has a prominence of only 87 feet. Surely the Ordnance Survey could never have mapped that as exceeding 500 feet?

Here’s the mapping situation when Corbett was compiling his list—below is the relevant bit of the Ordnance Survey’s one-inch “Popular” edition, published around 1950:

You can see that the summit of Sgurr nan Eugallt, as labelled, is surrounded by a loop of 2900-foot contour—this is the summit that Corbett originally listed in his tables, with a height of 2933 feet.* But to the northwest there’s a broad rounded dome, also surrounded by a loop of 2900-foot contour, to which no-one seems to have paid any attention for fifty years. Corbett’s original summit appeared in every edition of Munro’s Tables up to the most recent, in 1997. But then in 2002 the second edition of the SMC’s guide-book The Corbetts & Other Scottish Hills suddenly pointed out:

Note that the true summit lies 600 metres or so NW along the undulating ridge.

According to more recent surveys, that broad rounded dome turns out to rise to 898 metres (2946 feet), whereas Corbett’s original summit comes in at only 895 metres (2936 feet). So the name Sgurr nan Eugallt has now been moved to a new home 600 metres northwest, while Corbett’s original summit is relegated to being merely “Sgurr nan Eugallt (East Top)”, with its prominence measured only from the nearby col. I’ve marked both summits in my chart. The pair Meall Coire nan Saobhaidh and Meall na h-Eilde have undergone a similar transition, with the former originally being considered the higher of a pair of two neighbouring lumps, but the honour moving to the latter in 1981.

So the Corbetts illustrate five potential ways in which a new topographic survey can change a hill’s status—too high, too low, insufficiently prominent, more prominent than previously thought, and turning out to be lower than a nearby summit to which the honour is transferred.

But my chart doesn’t capture the full complexity of the revision history of the Corbetts—some summits have made double transitions. For example, Corbett originally listed Sgurr nan Ceannaichean with a height of 2986 feet. Then in 1981 it was bumped to Munro status, with a listed height of 915 metres (3002 feet), only to be demoted again in 2009 when a more accurate survey revealed a height of 2997 feet.

Whereas Beinn Talaidh on Mull has made the opposite journey. In 1952 the Ordnance Survey showed it falling short of Corbett’s lower threshold by just four feet. In 1981 it popped up in the revised Corbetts list, with a note clarifying that:

Highest point lies 25 metres south west of the [triangulation] pillar and is 2502 ft.

But by the 1997 revision it had fallen off the Corbetts list again, with a height of just 761 metres (2497 feet).

So that’s the Corbetts— which were a nice, well-defined group to start with, illustrating most of the considerations that drive table revisions. Next time I’ll deal with the Donalds, which are complicated by being divided into two categories, Hills and Tops.

Note: My data source for this post is the Database of British and Irish Hills v17.2, combined with “The Corbetts 1953-2016” dataset (version 4), both obtained from the DoBIH downloads page.

* You’ll see that the one-inch map I’ve reproduced is inconsistently marked. The height “2933” appears to refer to a spot-height in the col, rather than to the summit marked Sgurr nan Eugallt, but this spot-height lies below the 2900-foot contour. Larger-scale maps (to which Corbett would have referred) clearly place the 2933-foot spot-height at NG 931044, on the summit originally marked as Sgurr nan Eugallt, with the col dropping to 2894 feet at NG 928046. Interestingly, the old six-inch map of 1902, which shows spot-heights but no contours, plots a spot-height of 2941 feet at NG 927048, on what we now understand to be the “real” summit of Sgurr nan Eugallt! So either Corbett missed this, or it was not present on the maps he consulted.

(Be the first)

# Finding Apollo Trajectory Data

A while ago I wrote a post entitled “How Apollo Got To The Moon”, which featured a few orbit graphics generated in Celestia, like the one above (which shows the orientation of Apollo 11’s departure orbit relative to the most intense region of the Van Allen Radiation Belt). I got a few enquiries about the data I’d used to plot the spacecraft orbits, and this is a long-delayed response to those enquiries.

As a preamble to this post, see my post “Keplerian Orbital Elements”, which describes in detail the various numbers used to define an orbit’s size, shape and orientation—the numbers I needed to plug into Celestia in order to draw the orbit in the illustration above.

This post is going to be about the NASA data sources I used, which don’t provide the orbital elements directly, but instead provide tables of state vectors, specifying the spacecraft’s position and velocity at a series of times. To plot the orbits the Apollo missions followed when they departed from, and returned to, the Earth, I need two specific state vectors. For departure, I want the state vector at Translunar Injection (TLI), which was defined to take place ten seconds after the Saturn S-IVB stage shut down its engines, having launched the Apollo stack on its way to the Moon. For the return orbit, I want the state vector for the Command Module at the atmospheric entry interface, defined as an altitude of 400,000 feet. In my next post on this topic, I’ll describe how to convert these state vectors into orbital elements.

A commonly used Apollo data source is Apollo By The Numbers (2000), a NASA publication compiled by Richard W. Orloff. Its data tables are widely available—they even have their own web pages on the NASA History website. The data I’m currently interested in appear under Translunar Injection and Entry, Splashdown, and Recovery. There’s also a useful page of Earth Orbit Data. But Apollo By The Numbers has the disadvantage of being a secondary source, containing a number of copying errors. So I went back to the primary sources—mission documents prepared immediately after each Apollo flight. These are available on-line as scans of the original typewritten pages, but the data are scattered across multiple sites, and links sometimes turn out to be broken. Even the NASA Technical Reports Server is missing some items, and is so full of oddly indexed material that it’s sometimes difficult to find even the material that is present. So I’ve spent some time recently compiling a collection of specifically Apollo-related documents at the Internet Archive, which I’ll link to as required below.

For the Apollo departure orbits, I took my TLI state vectors from the Postflight Trajectories prepared by Boeing. For atmospheric entry, my main source was the Mission Reports compiled at what’s now the Johnson Space Center.

The Postflight Trajectories include a pair of appendices charting reams of state vectors for the launch, Earth orbit and Translunar Injection phases on an almost second-by-second basis. The most useful tables for the TLI state vector are B-VII and C-VII (data in metric and imperial units, respectively). These have the look of being about as primary as you can get, given that they have the appearance of photo-reduced computer printouts, in contrast to the typewritten document to which they’ve been appended.

The relevant position data are the geocentric distance (GC DIST), longitude (LONG DEG E) and geocentric latitude (GC LAT DEG N)—a distance and two angles, which provide the spacecraft’s position relative to the centre of the rotating Earth. The relevant velocity data are the heading (HEAD DEG), flight-path angle (FLT-PATH DEG) and space-fixed velocity (SF-VEL)—again, two angles and a speed completely specifying the spacecraft’s velocity in three dimensions. The heading is the compass course, in degrees east of true north, along which the spacecraft is travelling; the flight-path angle is the angle above or below the local horizontal (parallel to the surface of the Earth directly below) in which it’s travelling; and the space-fixed velocity is its speed relative to a non-rotating Earth. This last value takes the spacecraft’s velocity relative to the surface of the Earth and augments it by the local rotation velocity of the Earth. That combined velocity determines how far the spacecraft will travel in its long elliptical orbit towards the moon—all the Apollo missions (and space missions in general) launched towards the east, to take advantage of the Earth’s rotation to give the spacecraft an extra boost.

These six numbers, together with the time and a knowledge of the Earth’s mass, are all that’s needed to derive an elliptical orbit that will be valid for the first few hours of Apollo’s departure from Earth.

But first we need to translate the time, given in seconds in the first column of the table above, into a format that’s meaningful in terms of orbital mechanics. The time given is what’s called the Range Time, or Ground Elapsed Time (GET)—the time since launch. Section 1 of the Postflight Trajectory report for Apollo 11 tells us that the mission launched on 16 July 1969, at 08:32:00 Eastern Standard Time. That corresponds to 13:32:00 Greenwich Mean Time.* To make that date and time useful for plotting orbits, we need to convert it into a single number, the Julian Day (JD). My link gives you the necessary formulae to do that, but there are plenty of on-line calculators, too. There’s a suitable simple one here, which takes input in the form of the date and Greenwich Mean Time.* It only accepts a whole number of seconds, but all the Apollo missions launched on a whole number of seconds—most, like Apollo 11, on a whole number of minutes. I’ve nevertheless set the output to give the number of Julian Days to eight decimal places, to accommodate the TLI Range Time, which is quoted to a thousandth of a second in the table above.

The calculator tells us that launch (Range Zero) occurred on Julian Day 2440419.06388889. To that we need to add the Range Time of 10213.030 seconds (first dividing it by 86400, the number of seconds in a day). That gives us a total of 2440419.18209525 which, in the jargon of orbital mechanics, is the epoch of TLI.

The Postflight Trajectory documents also contain a summary table of the conditions at Translunar Injection, which include a smattering of Keplerian orbital elements—the inclination, descending node and eccentricity—which will provide good cross-checks on my own calculations.

Unfortunately, there’s no similar data source for the state vectors at the time of atmospheric entry—the Postflight Trajectory tables end with the start of the Transposition, Docking and Extraction manoeuvre, on the way to the Moon. The only reports consistently providing relevant data are the Mission Reports§, which span the time from launch to splashdown, and these seem to be the source for the tabulated data in Apollo By The Numbers. Here’s the table of entry conditions from the Apollo 11 Mission Report:

The “miles” in the table are in fact nautical miles, the equivalent of 400,000 feet. The table provides a time, longitude, velocity, flight-path angle and heading angle, but lacks the geocentric latitude and distance that I need. Instead it gives a geodetic latitude and an altitude.

The time is the Range Time, again, and I can convert it to Julian Days in the same way I did for TLI above. The mission elapsed time of 195h03m05.7s corresponds to 8.127149 days, giving me an entry epoch of 2440427.191038.

To calculate the missing geocentric latitude and distance requires some research into the finer points of Apollo coordinate systems, and a bit of geometry. To find the geocentric latitude of a point, we (figuratively) draw a line from the point to the centre of the Earth, and measure the angle between that line and the plane of the equator. To find the geodetic latitude, we drop a line at right angles to the local horizontal plane, and measure the angle that makes with the plane of the equator. On a perfectly spherical Earth, these two latitudes would be exactly the same, but because the Earth bulges at the equator, they’re slightly different. Here’s a diagram, with the flattening of the Earth greatly exaggerated:

The angle labelled ψ is the geocentric latitude, which is what we need for orbital mechanics; the angle labelled ϕ is the geodetic latitude, which is the latitude generally quoted in atlases and other geographical reference sources. The geocentric distance is the line r, connecting the centre of the Earth to the spacecraft, and the altitude is h, the distance between the surface of the Earth and the spacecraft, measured at right angles to the local horizontal plane.

Of course, the Earth’s surface isn’t a smoothly curving ellipsoid as in the diagram—but for the purpose of calculating geodetic latitude it’s treated as such. Various standard ellipsoids have been used by cartographers over the years, and by consulting the Project Apollo Coordinate System Standards, we can find out that the standard ellipsoid used throughout the Apollo missions was the Fischer “Mercury” Ellipsoid (1960). This ellipsoid was defined as having an equatorial radius (symbolized by a) of 6378166 metres. Its polar radius (b) was defined according to the ratio (ab)/a, called the flattening (f), which Irene Fischer determined to be 1/298.3. And that’s all I need in order to calculate the geocentric latitude and distance, using the geodetic latitude and altitude.

First, I need to work out the eccentricity (e) of the Fischer ellipsoid:

e=\sqrt{f\cdot\left ( 2-f \right )}

Then I need the length of R, which runs from the Earth’s axis to its surface in my diagram above, and is called the prime vertical radius.

R=\frac{a}{1- \left ( e\cdot sin\left ( \phi  \right )\right )^{2}}

The distance p, between the Earth’s axis and the spacecraft, measured parallel to the equator, is then:

p=\left ( R+h \right )cos\left ( \phi  \right )

And z, the distance between the equatorial plane and the spacecraft, measured parallel to the Earth’s axis, is:

z=\left [ R \left ( 1-e^{2} \right )+h \right ]sin(\phi )

Then the geocentric latitude is:

\psi =atan\left ( \frac{z}{p} \right )

And the geocentric distance is:

r=\sqrt{p^{2}+z^{2}}

Plugging in the Apollo 11 data from the table above, I get a geocentric latitude of -3.17 degrees, and a geocentric distance of 6500.02 kilometres. So now I have my state vector for atmospheric entry in the same format as for Translunar Injection, and I’m ready to calculate the orbital elements. Which I’ll do next time I return to this topic.

* Actually Universal Time, the successor to Greenwich Mean Time for astronomical time-keeping; but GMT was still the standard during the Apollo missions.
This is ludicrously optimistic, of course, given the uncertainty of some other numbers that will feed into the eventual calculation—but I prefer to do my rounding at the end, rather than in the middle.
Similar summary data appear in the Mission Reports, as well as in the Saturn V Launch Vehicle Flight Evaluation reports from what’s now the Marshall Space Flight Center. The Mission Report data generally seem to be derived from the Postflight Trajectories, though there are some departures from that. The Flight Evaluation data are often very slightly different from the Postflight Trajectory data—I think because of differing emphases on radar tracking and telemetry in the two reports from different organizations with different responsibilities. But none of the other sources provide anything as immediately useful as the state vector tables in the appendices to the Postflight Trajectories. (In the main, Apollo By The Numbers seems to take its data for TLI and Earth Parking Orbit from that source, though not entirely consistently.)
§ A number of earlier missions have Entry Postflight Analysis reports, which provide a more precise estimate of the entry state vectors than the Mission Reports, but this document was abandoned for Apollo 13 and doesn’t seem to have been reinstated.
This eccentricity is exactly the same property of the ellipsoid as the eccentricity that specifies the shape of an elliptical orbit, described in my post “Keplerian Orbital Elements”.

# Scottish Hill Lists: The Classics

If you’ve spent any time at all reading The Oikofuge, you’ll have gathered that I’m quite interested in hills—climbing them, looking at other hills from their summits, understanding their names and their place in history, landscape and land-use. What you won’t have seen me mention very often is the plethora of classifications that have been imposed on the Scottish hills over the years, starting with Sir Hugh Munro’s table of 3000-footers published in 1891, and culminating in the ongoing GPS-assisted activities of the good people over at the Database of British and Irish Hills.

It’s not that I’m uninterested in these tabulations, or the various parameters they’re based on. I have, after all, actually prepared a (long-obsolete) set of mountain tables all of my own. And the maps that accompany my various walk reports show the summits colour-coded according to their classification—you can find the key to the colours used in the FAQ section of the blog. And a glance at my annual CCCP reports will reveal a definite tendency to clamber up any 3000-footer that happens to be nearby, that being something of a raison d’être for the Crow Craigies Climbing Party.

But I don’t structure my walking activities around trying to “complete” any particular hill list—indeed, it’s only in the last few years I’ve attempted to reconstruct a list of all the summits I’ve visited in fifty years of hill-walking. And that process has led me to think a bit more about hill-lists in general, and how they came to exist. So I thought I’d write something about that. For this post, I’m going to start with the classics—the three Scottish hill-lists that dominated the mental landscape of hillwalkers back in the 1970s when I was first venturing out on to the summits.

## Munros

Like the others in this trio, Munro’s list first appeared in the Scottish Mountaineering Club Journal, and has been curated by the Scottish Mountaineering Club ever since. His “Tables Giving All The Scottish Mountains Exceeding 3,000 Feet In Height” [SMCJ 1(6): 276-314] lists 538 Scottish “tops” that exceed the height limit. Munro then innocently instigated a century-long argument by separately enumerating those peaks he felt could “fairly be reckoned distinct mountains”. These were the 283 summits that came to be designated “Munros” in his honour; the remaining 255 tops on his list would then become the “Munro Tops”.

Munro worked on revising his list in the light of new mapping, and the SMC planned to issue his Revised Tables as part of their new General Guide-Book. This eventually saw the light of day in 1921, issued in instalments after a delay necessitated by the First World War—and, unfortunately, after Munro’s death in 1919. This list, consisting of 276 Munros and 267 Munro Tops, is perhaps the closest we can get to the “historical” Munros—Munro’s list largely devoid of input from other hands.

Unfortunately, Munro left no guidance on how he had decided whether one of his “tops” counted as a “mountain”. He certainly seems to have considered that large jagged mountains, like Beinn Eighe or An Teallach, could consist of only one “mountain” summit (the highest point) together with several mere “tops”; while he tended to scatter the “mountain” designation rather more profligately on rolling plateau land, like the Monadlaiths. This apparently unequal distribution of “mountains” relative to the difficulty of ascent would be the source of many later arguments, and I’ll come back to that when I write about the occasionally vexed topic of table revisions, in a later post. (There will be charts.)

## Donalds

Munro’s 3000-footers are restricted entirely to the Scottish Highlands, and that may have been the inspiration for Percy Donald’s publication in 1935 of “Tables Giving All Hills In The Scottish Lowlands 2,000 Feet And Above” [SMCJ 20(120): 415-438]. This was a list of the highest summits in the Lowlands and Southern Uplands, thereby complementing Munro’s Highland-centric list.

After the pattern of Munro, Donald provided a list of 133 “tops”, and further classified 86 of these tops as “hills”. Unlike Munro, he attempted to provide some formal reasoning for his selection. But, again after the pattern of Munro, a degree of personal choice was permitted to creep in. His list of “tops” comprised:

All elevations [over 2000 feet] with a drop of 100 feet on all sides and elevations of sufficient topographical merit with a drop of between 100 feet and 50 feet on all sides.

Donald here introduces the idea of “drop”—the vertical distance between a hill’s summit and the highest connecting col. This is nowadays frequently called “topographic prominence”, and has become a key concept in modern hill lists. It can be formally defined as the summit’s height above the lowest contour which encircles the summit without enclosing any higher summit. But Donald then muddies the waters by making a subjective judgement with regard to the “topographical merit” of those 2000-foot eminences with drops in the 50-to-100-foot range.

In order to decide which of his “tops” were also “hills”, Donald gathered his tops together into groups, and nominated the highest of each group to be the “hill”, and the remainder to be “subsidiary tops” of that hill. His method of defining a group of tops is generally referred to by the SMC and other commentators as a “complicated formula”, but regular readers of The Oikofuge will no doubt recognize it as being a really simple formula. Donald measured the horizontal distance between adjacent tops along their connecting ridge, and measured the drop of the lower top by counting 50-foot contours between its col and summit. Each twelfth of a mile horizontally, and each 50-foot interval vertically, constituted one “unit”, and a “hill” could lay claim only to such subsidiary tops as fell within 17 units of its summit. As far as I can tell, Donald didn’t offer a justification for this particular formula, but it’s evident that his 17 units translate to about half-an-hour’s walking for someone setting a slightly more leisurely pace than the one stipulated by Naismith’s Rule. His “hills” are now called “Donalds” in his honour, and the subsidiary tops are “Donald Tops”. (In an appendix to his main tables, Donald also listed fifteen summits enclosed by isolated loops of 2000-foot contour, which are sometimes referred to as Minor Tops, but are of largely historical interest for reasons I’ll mention when I write about the revision history of these tables.)

## Corbetts

The final member of the classic table trio (or triptych, as the SMC would no doubt style it) arrived in 1952, with the publication of J. Rooke Corbett’s awkwardly entitled “List Of Scottish Mountains 2,500 Feet And Under 3,000 Feet In Height” [SMCJ 25(143): 45-52]. To a considerable extent, this was a continuation of Corbett’s work tabulating the hills of England and Wales that rise to more than 2500 feet—a list he referred to as the “Twenty-Fives”, and had published in the Rucksack Club Journal in 1911.

Sadly, Corbett had died in 1949, and his tables were passed on to the SMC by his sister. The SMC appears to have been initially somewhat bemused, to judge from the foreword written by John Dow, who describes Corbett’s list of 219 summits as “incomplete”, stating that:

[…] reference to the maps—e.g., 1-in. Ordnance Sheets 42, 43, 49, etc.—makes it clear that numerous heights of equal “merit” to those listed have not been shown.

However, it soon became clear to the SMC that Corbett had in fact completed his tables—the apparent omissions were because he had, like Donald, applied a “drop threshold” below which summits failed to qualify for inclusion. Unlike Donald, he had not then applied any further, subjective judgements. When Corbett’s tables were republished in 1953, Dow’s revised foreword stated:

There was no indication in Corbett’s papers as to the criterion he adopted in listing the heights [ie, summits] included, but it seems clear that his only test was a re-ascent of 500 feet on all sides to every point admitted, no account being taken of distance or difficulty. No detailed check has been made, but the 500 feet qualification has obviously been exhaustively applied and rigidly adhered to […]

It is left as an exercise for the interested reader to figure out how this criterion could be deemed to have been “rigidly adhered to” in the absence of a “detailed check”, but a topographic prominence of 500 feet has been a stipulated qualification for Corbett-hood ever since.

These three tables were brought to together in a single publication in 1953: Munro’s Tables And Other Tables Of Lesser Heights on the cover, but more grandly styled Munro’s Tables Of The 3000-Feet Mountains Of Scotland, And Other Tables Of Lesser Heights on the title page. Its cover features at the head of this post.* It went through numerous editions and revisions over the course of the next four decades, until the most recent edition, in 1997, changed the title to the less judgemental Munro’s Tables And Other Tables Of Lower Hills. No matter: almost everyone refers to the publications as just “Munro’s Tables”.

A plot of the Munros, Corbetts and Donalds (according to the current lists) reveals some interesting features of their distribution:

The Munros (in red) are confined by the nature of Scottish topography to the region north of the Highland Boundary Fault. Most are on the mainland, but two of the Inner Hebrides (Skye and Mull) host Munros. And we can see how Donald’s decision to confine his own tables to the Lowlands and Southern Uplands creates a complementary distribution of Donalds (in orange). Corbetts (in yellow) are spread all across Scotland—fringing the Munros in the north, reaching into several more islands, and mingling with the Donalds in the south. And because Donald set no upper limit to the height of his hills, there is in fact an overlap between the Corbetts and the Donalds—seven Donalds reach above 2500 feet with sufficient prominence to also qualify as Corbetts.

It’s informative, too, to plot the same hills on two axes according to their height and prominence:

In the absence of large areas of ground below sea level in Scotland, no hill can have a prominence greater than its summit’s height above sea level; and the only summits with prominence equal to their height are the highest points of islands. So I’ve plotted the Island Line on my chart, and labelled the three Munros and four Corbetts that lie on it.

The Corbetts cluster neatly, bounded by the 3000-foot contour above, the 2500-foot contour below, the Island Line to the right, and 500-foot prominence to the left. The Donalds sprawl a bit more—bounded by the 2000-foot contour below, but spilling into Corbett territory above, with seven orange triangles superimposed on the corresponding yellow Corbett plots. The Donalds all lie to the right of Donald’s 100-foot prominence cut-off; the Donald Tops (all bar one) lie to the right of his 50-foot lower limit. The reason for that anomalous Donald Top of negligible prominence will be explained (or at least, elucidated) when I write about later revisions to the tables.

Finally, the modern Munros list appears fairly well-behaved, too, with all the Munro Tops having prominences less than 500 feet, while the Munros themselves have prominences greater than 100 feet. This was not always so—it’s a product of later table revisions. And there’s another anomaly on the chart, in the form of a single Munro with negligible prominence. That something else I’ll explain in a later post on this topic.

Note: My data source for this post is the Database of British and Irish Hills v17.2, obtained from the DoBIH downloads page.

Resources: The original tables are slightly awkward to get at, being buried in large pdf scans of various volumes of the SMC Journal. And, once got at, the tables of Munro and Donald turn out to be difficult to read, the former having been printed in landscape orientation, the latter as double-page spreads. I’ve therefore prepared a little compendium of the relevant publications for these three sets of tables, rotating Munro’s landscape pages and merging Donald’s double pages for ease of consultation. The result is available on the Internet Archive here, to browse or download.

* It’s easy to be misled by the colophon that appears at the start of every edition of “Munro’s Tables” claiming a first edition in 1891, and two subsequent editions in 1921 and 1933. These dates refer to the first three publications of Hugh Munro’s tables, initially in the SMCJ and then in the two editions of the SMC’s General Guide-Book. The single publication commonly referred to as “Munro’s Tables”, containing the tables prepared by Munro, Corbett and Donald, didn’t (indeed couldn’t) come into existence until 1953, and I’m not sure why the SMC tries to push its publication history back into a time before the works of Donald and Corbett even existed.

This can lead to curious behaviour from walkers intent on “bagging” both Corbetts and Donalds. I once met a man on White Coomb who told me, in solemn tones, that he had been “forced” to climb the hill twice, because it was “once for the Donald and once for the Corbett”.

# R.A.J. Matthews: Tumbling Toast, Murphy’s Law And The Fundamental Constants

In what follows we model the tumbling toast problem as an example of a rigid, rough, homogeneous rectangular lamina, mass m, side 2a, falling from a rigid platform set a height h above the ground. We consider the dynamics of the toast from an initial state where its centre of gravity overhangs the table by a distance δ0

Robert A.J. Matthews published this seminal bit of applied physics in 1995. The journal reference is European Journal of Physics 16(4): 172-6, and you can access the full paper at ResearchGate, here. For his efforts, he was awarded an Ig Nobel Prize in 1996.

Matthews was the first (but by no means the last) to use mathematical physics to explore the popular claim that “dropped toast always lands butter-side down”. The usual “explanation” invoked for this perceived rule is Murphy’s Law—“If anything can go wrong, it will”—but Matthews sought to show that there were sound physical principles underlying the phenomenon.

He starts by dismissing the common physical explanations offered to account for this, principally airy claims relating to off-centre mass or aerodynamics effects created by the butter. He also dismisses those experiments that have claimed to disprove the rule—it’s unsurprising that buttered toast hurled randomly into the air* shows no particular preference for the side on which it alights, but this hardly reproduces the normal process by which toast falls.

Matthews starts with a static rectangle of toast, as described in the quotation at the head of this post. When its centre of mass moves beyond the edge of the table, it begins to tip over under the force of gravity. With any angle of tipping beyond zero (horizontal), gravity also produces a force that tries to slide the toast farther over the edge of the table. This is initially opposed by friction with the table edge, but eventually translates into a sliding motion. Gravity continues to accelerate the rate of rotation until the combination of sliding and rotation lifts the trailing part of the toast away from the table edge. Thereafter, the toast falls freely, and now rotates at a constant rate (neglecting air friction) until it hits the ground. If the toast rotates more than 90º but less than 270º on its way to the ground, it will strike butter-side down. Matthews appears to ignore the <90º regime during his initial analysis, presumably because toast falling from table height is observed to always rotate farther than that before hitting the floor.

In the quotation at the head of this post, Matthews sets the half-length of the toast to a, and the length by which the centre of gravity overhangs the edge of the table to δ. From these he defines an “overhang parameter”, η, equal to δ/a. The critical overhang parameter at which the tipping toast loses contact with the table edge is η0, and the tipping angle at which this occurs is φ. With g representing the acceleration due to gravity, he derives an equation for the constant angular velocity of the free-falling toast, ω0:

\omega _{0}^{2}=\left ( \frac{6g}{a} \right )\left ( \frac{\eta _{0}}{1+3\eta _{0}^{2}} \right )sin\phi

The time, τ, it takes the toast to fall to the floor under gravity can be estimated using an approximation of the total distance it falls:

\tau =\sqrt{\frac{2(h-2a)}{g}}

And if the toast is to successfully rotate through “butter-side down” and into “butter-side up” during this time then:

\omega_{0}\tau > 270^{\circ }-\phi

So that’s the story. Toast tips, slides, rotates free of the table edge, and then falls with a constant rate of rotation until it hits the floor after some elapsed time determined by the height from which it falls. If it rotates fast enough, or falls from high enough, it will manage to land butter-side up. But there will be a critical range of rotation rates and heights which will carry the toast into a butter-side-down impact.

The overhang parameter η0 is critical—if the toast has high enough friction with the table edge it will maintain contact with the edge for longer, allowing its rotational velocity to build up more before it falls free, maximizing the chance of a butter-side-up impact. Matthews derives a rather splendid formula for the minimum value of η0 which will generate sufficient rotational velocity for a butter-side-up landing.

\eta _{0}> \frac{2(h/a-2)\left ( 1-\sqrt{1-\frac{\pi ^4}{12(h/a-2)}} \right )}{\pi ^{2}}

(I’ve somewhat rearranged the equations in his paper, here, but the above is equivalent to those he provides.) For a table height h = 75cm and half-length of toast slice a = 5cm, it turns out that η0 has to be greater than 0.06.

Experiments involving bread, toast and kitchen Contiboard ensue, and Matthews finds that toast has a characteristic η0 of just 0.015, with untoasted bread only a little higher at 0.02. In his words:

This implies that laminae with either composition do not have sufficient angular rotation to land butter-side up following free-fall from a table-top. In other words, the material properties of slices of toast and bread and their size relative to the height of the typical table are such that, in the absence of any rebound phenomena, they lead to a distinct bias towards a butter-side down landing.

In fact, we can work out the minimum table height above which falling toast will have time to rotate far enough to land butter-side-up:

\frac{h}{a}=2+\frac{\pi ^{2}\left ( 1+3\eta _{0}^{2} \right )}{12\eta _{0}}

Plugging in the previously derived numbers yields an inconvenient minimum height of three metres.

Matthews then explores the effect of the horizontal velocity with which the toast departs the table edge—if fired over the edge with sufficient speed, the toast would have little time to start tipping over, would gain correspondingly little rotational velocity, and might stay relatively horizontal all the way to the floor. (That is, it would stay in the <90º rotation regime.) He concludes that the normal range of speeds with which toast is nudged off tables or tipped off plates is insufficiently high to prevent the butter-side-down landing.

Finally, there’s a section dealing with the fundamental constants of nature. In it, he builds on a paper by William H. Press, “Man’s size in terms of the fundamental constants” (American Journal of Physics, 48(8): 597-8), which you can find as a pdf here. Distilling down a more detailed argument, Matthews concludes that the upper height limit, LH, for humans is constrained by the ratio of the strengths of the electromagnetic force (which holds our bodies together) and the gravitational force (which breaks us if we fall from too great a height). If we got much taller than LH, we’d frequently sustain disabling or life-threatening injuries from simple trips and falls. After pushing around some equations, he concludes that:

L_{H}<\sim 50\times \left ( {\alpha /\alpha _{G}} \right )^{1/4}\alpha _{0}

Where α is the electromagnetic fine structure constant, αG the gravitational coupling constant for protons, and α0 the Bohr radius. These arguments are at best order-of-magnitude estimates, but Matthews plugs in the numbers and finds a surprisingly reasonable maximum figure of three metres for LH.

Matthews concludes that the frictional properties of toast set a limit on its rotation rate when falling from an edge, while the basic constants of the Universe set a limit on how tall humans are, which in turn sets a limit (about half LH) on how high useful tables are.

Our principal conclusion is a surprising one, given the apparently quotidian nature of the original phenomenon: all human-like organisms are destined to experience the ‘tumbling toast’ manifestation of Murphy’s Law because of the values of the fundamental constants of the universe. As such, we have probably confirmed the suspicions of many regarding the innate cussedness of the universe.

What to do? Reducing the size of the toast to match the scale of our tables is one solution, but the required size of ~2.5cm squares is (as Matthews remarks) “unsatisfactory”. He proposes instead the counterintuitive solution of speeding the toast on its way, to limit its opportunity to build up rotational velocity—flick it briskly over the edge, or snatch the supporting plate away, backwards and downwards.

So now you know.

* The BBC’s QED strand conducted just such an experiment in 1991.
Matthews’ work provoked a flurry of additional publications investigating the problem of tumbling toast. Analysis of video suggested that the free-falling toast rotates faster than Matthews predicted, probably because he had neglected the kinetic friction that occurs during the sliding phase. For more on the topic, take a look at the following:
Bacon ME, Heald G, James M. “A closer look at tumbling toast” American Journal of Physics (2001) 69(1): 38-43
Borghi R. “On the tumbling toast problem” European Journal of Physics (2012) 33: 1407-20

# Keplerian Orbital Elements

1. All planets move in elliptical orbits, with the sun at one focus.
2. A line that connects a planet to the sun sweeps out equal areas in equal times.
3. The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.

Kepler’s Laws of Planetary Motion (formulated 1609-1619)

Okay, this is probably a bit niche, even by my standards, but it’s part of a longer project. I eventually want to write some more about the Apollo spacecraft, and the orbits they followed on their way to, and return from, the Moon. And the problem with that is that (for various good reasons) NASA didn’t document these orbits with a list of “orbital elements” that would allow the spacecraft trajectories in the vicinity of the Earth to be plotted easily. Instead, the flight documentation includes long tables of “state vectors”, listing the position and velocity of the spacecraft at various times—these are more accurate, but unwieldy to deal with. So in a future post I’m going to write about how to extract orbital elements from a few important state vectors. But first I need to describe the nature and purpose of the orbital elements themselves. Which is what I’m going to do in this post, hopefully enlivened by explanations of how the various orbital elements came by their rather odd names.

But first, the “Keplerian” bit. Johannes Kepler was the person who figured out that the planets move around the sun in elliptical orbits, and who codified the details of that elliptical motion into the three laws which appear at the head of this post. In doing that, he contributed to a progressive improvement in our understanding, which began with the old Greek geocentric model, which placed the Earth at the centre of the solar system with the planets, sun and moon moving in circles around it. This was replaced by Nicolaus Copernicusheliocentric model, which placed the sun at the centre, but retained the circular orbits. Kepler’s insight that the orbits are elliptical advanced things farther. (Next up was Isaac Newton, who provided the Theory of Universal Gravitation which explained why the orbits are ellipses.)

So Keplerian orbits are simple elliptical orbits.* They’re the sort of orbits objects would follow if subject to gravity from a single point source. In that sense, they’re entirely theoretical constructs, because real orbits are disturbed away from the Keplerian ideal by all sorts of other influences. But if we look at orbits that occur under the influence of one dominant source of gravity, and look at them for a suitably short period of time, then simple Keplerian ellipses serve us well enough and make the maths nice and simple. (And that’s what I’ll be doing with my Apollo orbits in later posts.)

Before going on, I’ll introduce a bit of necessary jargon. Henceforth, I’ll refer to the thing doing the orbiting as the satellite, and the thing around which it orbits as the primary. In Kepler’s original model of the solar system, the “satellites” are the planets, and the primary is the Sun; for my Apollo orbits, the satellites will be the spacecraft, and the primary is the Earth. Kepler’s First Law tells us that the primary sits at one focus of the satellite’s elliptical orbit. Geometrically, an ellipse has two foci, placed on its long axis at equal distances either side of the centre; only one of these is important for orbital mechanics. Pleasingly, focus is the Latin word for “fireplace” or “hearth”, so it seems curiously appropriate that the first such orbital focus ever identified was the Sun. Kepler’s Second Law tells us, in geometrical terms, that the satellite moves fastest when it’s at its closest to the primary, and slowest when it’s at its farthest. I’ll come to the Third Law a little later.

The Keplerian orbital elements are a set of standard numbers that fully define the size, shape and orientation of such an orbit. The name element comes from Latin elementum, which is of obscure etymology, but was used as a label for some fundamental component of a larger whole. We’re most familiar with the word today because of the chemical elements, which are the fundamental atomic building blocks that underlie the whole of chemistry.

The first pair of orbital elements define the size and shape of the elliptical orbit. (They’re called the metric elements, from Greek metron, “measure”.)

For size, the standard measure is the semimajor axis. An ellipse has a long axis and a short axis, at right angles to each other, and they’re called the major and minor axes. As its name suggests, the semimajor axis is just half the length of the major axis—the distance from the centre of the ellipse to one of its “ends”. It’s commonly symbolized by the letter a. The corresponding semiminor axis is b.

To put a number on shape, we need a measure of how flattened (or otherwise) our ellipse is—so some way of comparing a with b. For mathematical reasons, the measure used in orbital mechanics is the eccentricity, symbolized by the letter e. This has a rather complicated definition:

e=\sqrt{1-\frac{b^{2}}{a^{2}}}

But once we’ve got e, we can easily understand why it’s called eccentricity, because the distance from the centre of the ellipse to one of its foci turns out to be just a times e. Our word eccentricity comes from Greek ek-, “out of”, and kentron, “centre”. So it’s a measure of how “off-centre” something is. And multiplying the semimajor axis by the eccentricity does exactly that—tells us how far the primary lies from the geometric centre of the ellipse.

For elliptical orbits, eccentricity can vary from zero, for a perfect circle, to just short of one, for very long, thin ellipses. (At e=1 the ellipse becomes an open-ended parabola, and at e>1 a hyperbola.)

Before I move on from the two metric elements, I should mention another concept that’ll be important later. The line of the major axis, which runs through the centre of the ellipse and the foci (marked in my diagram above), has another name specific to astronomy and orbital mechanics. It’s called the line of the apsides. Apsides is the plural of Greek apsis, which was the name of the curved sections of wood that were joined together to make the rim of a wheel. The elliptical orbit is deemed to have two apsides of special interest—the parts of the orbit closest to the primary (the periapsis) and farthest from the primary (the apoapsis), and these are joined by the line of the apsides.

Then there are three angular elements, which specify the orbit’s orientation in space. They’re specified relative to a reference plane and a reference longitude. A good analogy for this is how we measure latitude and longitude on Earth. To specify a unique position, we measure latitude north or south of the equatorial plane, and longitude relative to the prime meridian at Greenwich. For orbits around the Earth, like my Apollo orbits, the reference plane is the celestial equator, which is just the extension of the Earth’s equator into space. The reference longitude is called the First Point of Aries, for reasons I won’t go into here—it’s the point on the celestial equator where the sun appears to cross the equator from south to north at the time of the March equinox, and I wrote about it in more detail in my post about the Harvest Moon.

The first angular element is the inclination, symbolized by the letter i, which is the angle between the orbital plane and the reference plane. The meaning of its name is blessedly obvious, because it’s the same as in standard English.

Following its tilted orbit, the satellite will pass through the reference plane twice as it goes through one complete revolution—once heading north, and once heading south. These points are called the nodes of the orbit, from Latin nodus, meaning “knot” or “lump”. The northbound node is called the ascending node, and the southbound node is (you guessed) the descending node—names that reflect the “north = upwards” convention of our maps. The angle between the reference longitude and the ascending node of the orbit, measured in the reference plane, is called the longitude of the ascending node, symbolized by a capital letter omega (Ω), and it’s our second angular element.

Those two elements tell us the orientation of the orbital plane in space—how it’s tilted (inclination) and which direction it’s tilted in (longitude of the ascending node). Finally, we need to know how the orbit is positioned within its orbital plane—in which direction the line of the apsides is pointing, in other words. To do that job, we have our third and final angular element, the argument of the periapsis, which is the angle, measured in the orbital plane, between the ascending node and the periapsis, symbolized by a lower-case Greek omega (ω). The meaning of argument, here, goes back to the original sense of Latin arguere, “to make clear”, “to show”. That sense of argument found its way into mathematical usage, to designate what we’d now think of in computing terms as an “input variable”—a number that you need to know in order to solve an equation and get a numerical answer.

Those five elements exactly define the size, shape and orientation of the orbit, and are collectively called the constant elements. In addition to those five, we need a sixth, time-dependent element, which specifies the satellite’s position in orbit at some given time. (The specified time, symbolized by t or t0, is called the epoch, from Greek epoche, “fixed point in time”.) There are actually a number of different time-dependent elements in common use, but the standard Keplerian version is the true anomaly, which is the angle (measured at the primary) between the satellite and the periapsis. Different texts use different symbols for this angle, most commonly a Greek nu (ν) or theta (θ).

To understand why it’s called an “anomaly”, we need to go back to the original geocentric model of the solar system. Astronomers knew very well that the planets didn’t move across the sky at the constant rate that would be expected if they were adhering to some hypothetical sphere rotating around the Earth. Sometimes Mars, Jupiter and Saturn even turned around and moved backwards in the sky! These irregularities in motion were therefore called anomalies, from the Greek anomalos, “not regular”. And there were two sorts of anomaly. The First or Zodiacal Anomaly was a subtle variation in the speed of movement of a planet according to its position among the background stars. The Second or Solar Anomaly was a variation that depended on the planet’s position relative to the Sun. Copernicus explained the Second Anomaly by placing the Sun at the centre of the solar system, because he realized that much of the apparent irregularity of planetary motion was due to the shifting perspective created by the Earth’s motion around the Sun. The First Anomaly persisted, however, until Kepler’s Second Law showed how it was due to a real acceleration as a planet moved through periapsis, followed by a deceleration towards apoapsis. Because this “anomaly” was a real effect linked to orbital position, the word anomaly became attached to the angular position of the orbiting body. And if you’re wondering why it’s called the “true” anomaly, that’s because there are a couple of other time-dependent quantities in use, which are computationally convenient and which are also called “anomalies”. But the true anomaly is the one that measures the satellite’s real position in space.

And those are the six standard orbital elements, together with their odd names. However, we generally need to know one more thing. Kepler’s Third Law applies to all orbits—the larger the semimajor axis, the longer it takes for the satellite to make one complete revolution, with a cube-square relationship. But for a given orbital size, the time for one revolution also depends on the mass of the primary. A satellite must move more quickly to stay in orbit around a more massive primary. So we need to specify the orbital period of revolution (variously symbolized with P or T) if we are to completely model our satellite’s behaviour. The word comes from Greek peri-, “around”, and odos, “way”.

So—six elements and a period. That’s what I’ll be aiming to extract from the Apollo documentation when I return to this topic next time.

* Parabolic and hyperbolic “orbits” are, strictly speaking, trajectories, since they don’t follow closed loops. The word orbit comes from the Latin orbis, “wheel”—so something that is round and goes round.
Periapsis and apoapsis are general terms that apply to all orbits. Curiously, they can have other specific names, according to the primary around which the satellite orbits. Most commonly you’ll see perigee and apogee for orbits around the Earth, and perihelion and aphelion for orbits around the Sun. See my post about the word perihelion for more detail.

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# Fata Morgana

As the weary traveller sees
In desert or prairie vast,
Blue lakes, overhung with trees,

Fair towns with turrets high,
And shining roofs of gold,
That vanish as he draws nigh,
Like mists together rolled,—

Henry LongfellowFata Morgana” (1873)

I took the photograph above in Kolyuchin Inlet, in the Russian Far East, one evening in September 2016. The curious “objects” on the horizon are not clouds, as you might at first guess, but are an example of a kind of mirage called the Fata Morgana. Here’s a time-lapse film of the same phenomenon at Lake Michigan:

In both cases, we’re seeing distant land distorted by a complex mirage effect. The ice floes in the foreground of the video give a clue to the atmospheric conditions required for the Fata Morgana—there must be a layer of air at the surface that is considerably colder than the air aloft. This situation, the reverse of the usual condition in which the air becomes progressively cooler as one climbs higher, is called a temperature inversion. In Kolyuchin Inlet and Lake Michigan, the surface air was close to freezing point, but that’s not necessary for a Fata Morgana to appear. The important thing is the change in temperature with altitude, not the absolute temperature. Indeed, the Fata Morgana got its name in the Strait of Messina, the narrow channel that separates the island of Sicily from the “toe” of the Italian mainland. It’s not an area known for its ice-floes, but it is prone to temperature inversions. The proximity of two warm landmasses separated by a narrow channel of relatively cool water means that Sicilians can often observe this mirage effect distorting the mainland hills, while the residents of Reggio Calabria can watch the same thing happen to the Sicilian coastline.

La Fata Morgana is the Italian name of Morgan le Fay, a mythical adversary of King Arthur. From her origin in the Arthurian legends of Britain and France, she arrived in southern Italy along with the Normans, who established a short-lived kingdom in the region during the twelfth century. In her new Mediterranean home, Morgan was said to inhabit a castle that floated in the air. And so her name became attached to an optical phenomenon that occasionally produces the appearance of towers and battlements where none exist.

Here’s an example from Antarctica that shows how the Fata Morgana can convert the appearance of rounded hills into flat-topped towers:

Everyone probably knows that mirages, of which the Fata Morgana is a particularly complex example, occur because light rays are following unusually curved paths through the atmosphere. It’s perhaps less well known that light generally follows a curved path through the atmosphere, induced by the drop in atmospheric pressure with height, which produces a corresponding change in the refractive index of air. This normal curvature of light rays is concave downwards, so it routinely brings into view objects that are actually below the level of the geometrical horizon. In particular, if we were to wait until the setting sun appeared to be resting exactly on the sea-level horizon, and then removed all the intervening air, we’d discover that the sun was already entirely below the “real” horizon. I’ve written a lot more about that in my post on the Shape Of The Low Sun. The curving rays also make visible distant landscape features that would be invisible below the horizon in the absence of the atmosphere.

A temperature inversion, in its simplest form, simply accentuates this natural concave curvature of light rays, as increasingly warmer air aloft further reduces the atmospheric pressure and the refractive index of the air. Such temperature inversions further delay the setting of the sun, and lift even more distant geographical features into view—for instance, the towers of Chicago occasionally pop into view from the opposite shore of Lake Michigan, a good 60 miles away. It’s been calculated that a rise in temperature of 0.11°C for every metre of altitude induces horizontal light rays to follow a path with a curvature equal to the curvature of the Earth. Under such conditions, the surface of the Earth appears flat, and an observer can see as far as atmospheric haze and intervening topography allow. There have long been speculations that this sort of “Arctic mirage” informed the early voyages of discovery in the North Atlantic—allowing the first inhabitants of the Faroes to glimpse Iceland, and the early Icelanders to occasionally discern Greenland.

But temperature gradients much steeper than the critical 0.11°C/m exist, locally and over a few tens of metres altitude, at times when the Fata Morgana is visible. Under these circumstances, light rays emitted upwards by an object on the surface of the Earth can follow an arching course that brings them down to meet the eye of an observer some kilometres away. Like this:

The vertical scale of my diagram is typically a few tens of metres; the horizontal a few tens of kilometres—so you must imagine these trajectories stretched out a thousandfold from side to side.

If the temperature increases in a roughly linear way with altitude, the trajectories of the emitted light rays are all roughly the same shape. As a result, only one ray (the red one in my diagram) connects the object to the observer’s eye. So this sort of mirage merely lifts the image of the distant object so that it appears to sit higher in the sky than usual. At the extreme, sailors traversing a cold sea with warm air aloft can get the visual impression that their ship is sitting at the bottom of a bowl. In the parlance of English-speaking mariners, this effect was aptly named looming. And it was also known to those archetypal cold-water sailors, the Vikings, who called it hillingar. (Cleasby and Vigfusson’s dictionary of Old Icelandic translates this word as “an upheaving”.) So the general phenomenon is sometimes referred to as the hillingar effect.

The Fata Morgana requires a rather more dramatic temperature gradient, however—one in which the temperature changes abruptly over a short change of altitude, at a junction called a thermocline. Like this:

The thermocline provides such a range of temperature gradients in a relatively short span of altitudes, it can deflect a correspondingly wide range of ascending light-rays downwards towards a ground-level observer, acting like a sort of mirror in the sky. Like this:

So now our observer sees two images of the same object. The lower ray (let’s call it the direct ray) may or may not produce a noticeable hillingar effect, but the upper ray (call it the reflected ray) certainly produces an out-of-place image floating above the first.

And the thermocline really does act like a mirror. Here’s a diagrammatic plot adding the trajectories of light rays originating from a higher position on our distant object (marked in red):

The direct rays maintain their relative positions on their way to the observer’s eye, so that the higher parts of a distant object appear above its lower parts, in the usual way. But the reflected rays cross each other, so that the image of the higher parts arrive at the observer’s eye from a direction lower than the image of its lower parts. So we see an inverted image hovering disconcertingly above an upright image, like this:

But there’s more. For objects at a specific distance and height above the horizon, the range of temperature gradients associated with the thermocline can deflect multiple rays from the same source into the observer’s eye. Like this:

That means that part of the direct image can appear to be smeared vertically:

This phenomenon has another name that comes to us from the Vikings—the hafgerdingar effect. The original Norse word is haf-gerðingar, variously translated as “sea-fences” or “sea-hedges*”, which certainly fits the mirage appearance. But the Vikings may not actually have been referring to the mirage that now has this name; it’s possible they were talking about some sort of real, dangerous oceanic wave. The phenomenon is only ever discussed in an Old Norse text called Konungs skuggsjá (“The King’s Mirror”), which is far from clear:

Now there is still another marvel in the seas of Greenland, the facts of which I do not know precisely. It is called “sea hedges,” and it has the appearance as if all the waves and tempests of the ocean have been collected into three heaps, out of which three billows are formed. These hedge in the entire sea, so that no opening can be seen anywhere; they are higher than lofty mountains and resemble steep, overhanging cliffs. In a few cases only have the men been known to escape who were upon the seas when such a thing occurred. But the stories of these happenings must have arisen from the fact that God has always preserved some of those who have been placed in these perils, and their accounts have afterwards spread abroad, passing from man to man. It may be that the tales are told as the first ones related them, or the stories may have grown larger or shrunk somewhat. Consequently, we have to speak cautiously about this matter, for of late we have met but very few who have escaped this peril and are able to give us tidings about it.

But whatever the original haf-gerðingar was, it’s the merging of the hafgerdingar effect with the inverted reflected image that produces the full Fata Morgana appearance, like this:

Added to this appearance of distant, fluted towers and battlements, there’s a degree of animation to the Fata Morgana, because the thermocline is never entirely still. If you watch the time-lapse video near the head of this post carefully, you’ll be able to see the occasional wave running along the top of the mirage, produced by real wind-driven waves in the thermocline itself. These produce a gentle billowing effect at the upper margin of the miraged image, which on occasion can look like banners wafting gently in the wind.

So that’s an “edited highlights” explanation of the appearance of the Fata Morgana. Proper detailed treatments are hard to come by, I find, and much of the early mathematical analysis was published in German. So my primary reference has been one of the documents published in English by the Norwegian Polar Institute in 1964, reporting the scientific findings of the Norwegian-British-Swedish Antarctic Expedition of 1949-1952. It’s entitled Refraction Phenomena In The Polar Atmosphere (Maudheim 71° 03´S, 10° 56´W), and written by G.H. Liljequist.

* I’m somewhat surprised that there’s an Old Norse word for “hedge”. It’s like discovering that the Vikings had words for “herbaceous border” and “ornamental water feature”.

A couple of months ago I received this lovely picture from Mick Shaw, which I use with his permission. The sun is reflecting off a thin layer of sea-water covering the sand-flats of Morecambe Bay, and producing a pair of reflected-light rainbows in tandem with the usual primary and secondary arcs.

Reflected-light rainbows were the subject of one of my first posts on this blog, back in 2015, when I posted my own photograph of the phenomenon:

In that photograph, you can appreciate how still the waters of the Tay estuary were—an essential condition for this sort of rainbow to form, because the water surface has to be flat enough to reflect a near-perfect image of the sun.

Once the water surface reflects a clear image, falling drops of rain are illuminated by two suns—one above the horizon, and one below. So two rainbows can form, each centred on an antisolar point—the point directly opposite the sun. Rainbows formed by direct sunlight are centred on a point as far below the horizon as the sun is above it—this angular distance between sun and horizon is called the solar altitude, in astronomical jargon, and that’s what I’ll call it from now on. But the reflected-light rainbow arcs are centred on an antisolar point that is at the same height in the sky as the sun. Like this:

Rainbows form a circle around the antisolar point, but the sections of their arc that fall below the horizon are generally invisible, unless the observer is in a high place looking down on clouds. The reflected-light rainbow forms an arc that is identical in shape and size to the portion of the normal rainbow that lies below the horizon, except it is flipped vertically to sit above the horizon. So the two arcs, from direct and reflected sunlight, converge exactly at the horizon, forming a prominent lopsided “V” shape, evident in the photographs above.

A little bit of spherical trigonometry lets me plot the divergence between our two intersecting rainbows—that is, the angle at the base of the “V”. Here it is for the primary rainbows:

With a solar altitude of zero, there is no divergence. The sun is sitting on the horizon, and the direct and reflected antisolar points have merged on the opposite horizon, so that the direct and reflected-light rainbows are precisely superimposed, forming a single semicircular arc. With the sun farther above the horizon, the “V” between the two rainbows opens up steadily, until suddenly shooting up towards 180 degrees as the sun approaches an altitude of 42 degrees. This critical angle of 42 degrees corresponds to the radius of the primary rainbow. With the antisolar points 42 degrees above and below the horizon, the two rainbows are circles that touch each other at the horizon, with the primary rainbow invisible below the horizon, and the reflected-light rainbow forming a complete circle in the sky, sitting on the horizon. Like this:

Have you ever seen such a thing? Me neither. So I began to wonder why that doesn’t seem to happen.

There are a couple of practical considerations that strongly limit our opportunities to see a circular reflected-light rainbow entirely above the horizon. One is that rainbows are huge. A circular primary rainbow would stretch from the horizon almost all the way to the zenith. We’re used to the upper part of even a normal rainbow fading out, as it extends from a region of sky near the horizon where we have long sight-lines through falling rain, producing bright rainbow “legs”, into a region where our sight-lines are blocked by the rain-clouds themselves, so that the upper part of the arc is faint or invisible, like this:

Compared to the sunset bow above, a full circular rainbow would extend twice as far upwards into the sky, taking it into regions that normally provide few raindrops along our line of sight.

There’s also the issue that a reflected-light rainbow needs a sizeable reflective surface. Areas simultaneously large enough and calm enough to produce a complete circular rainbow are probably fairly rare.

But there’s a more fundamental reason that precludes our seeing a perfect circular reflected-light rainbow, hanging in the sky. It has to do with the reflective properties of water. The amount of light a flat water surface reflects depends on the angle at which the incoming light hits the water surface. Here’s a graph of the amount of reflection of (unpolarized) sunlight according to solar altitude:

You can see that, by the time the sun is thirty degrees above the horizon, its reflected image is only a fraction of the brightness of the real sun—about 6%, if we do the sums. So the reflected-light rainbow will be comparably reduced in brightness.

Another, more minor, influence on the brightness of the reflected-light rainbow is that the light reflected from a water surface is usually quite strongly polarized:

Now, as I described in my post about rainbow rays, the repeated process of reflection and refraction inside a raindrop mean that the rainbow itself is polarized, in a fashion that follows the arc of the rainbow. The horizontal top of the rainbow arc is polarized in the same sense as the reflected light from the water surface; the vertical sides of the rainbow are polarized transversely relative to the reflected light. So the upper part of a reflected-light rainbow should be a little brighter than its “legs”, if all other things are equal (which they’re generally not).

Here’s what happens when I feed the reflected light from a flat water surface into the light-path of the primary “rainbow rays”:

The centre of the graph corresponds to the sunlight that reaches us after reflecting off the water surface and then passing through the horizontal top of the rainbow arc. The sides of the graph show the same information for the vertical “legs” on either side of the rainbow.

At the top of the graph is the line corresponding to a solar altitude of zero. With the sun on the horizon, the water surface reflects all the light that falls on it, so there’s no polarization—illumination from the reflected sun is the same as that from the real sun, and our reflected rainbow is as bright as a normal rainbow (though that means only about 4.5% of light rays striking the reflecting surface find their way back to our eyes in the rainbow).

But the rapid fall in reflection from the water surface with increasing solar altitude means that our reflected-light rainbow fades out quickly. The onset of polarization also means that the “legs” of the rainbow grow fainter faster than does the top of its curve. With the sun just 10 degrees above the horizon, the reflected-light rainbow is already fainter than a normal secondary rainbow, which we know is frequently faint or invisible.

So our best chance of seeing a reflected-light rainbow occurs when the sun is close to the horizon, because these rainbows fade into invisibility as the sun gets higher. We generally see them as they appear in the two photographs at the head of this post—as a fainter rainbow that echoes the “leg” of a normal rainbow, while converging with it at the horizon. The supposedly brighter top of the reflected-light arc is often invisible, for the same reason that top of a conventional rainbow is often invisible—because it extends higher than the region in which long sight-lines extend under the rain-clouds. So I’ve never seen the upper arc of a reflected bow, but I live in hope.

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