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Scottish Hill Lists: The Corbett Revisions

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Cover of 1953 edition of Munro's Tables

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.

Height-prominence chart of Munros, Corbetts and Donalds
Click to enlarge

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 And although the idea of freezing these tables into historical documents has been discussed, particularly in the early days of Hugh Munro’s table of 3000-footers, there was also a countervailing idea that the compilers themselves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:

Height-Prominence plot of original Corbett list, with revisions
Click to enlarge

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:

One-inch "Popular" map of Sgurr nan Eugallt c.1950
Click to enlarge

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.

OS six-inch map of Sgurr nan Eugallt, 1902
Click to enlarge
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Phenomena

Finding Apollo Trajectory Data

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Apollo 11's departure orbit relative to Van Allen Belts (2)
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Prepared using Celestia

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.

State vectors for Apollo 11 TLI
Click to enlarge
(Edited for compactness from Apollo/Saturn V Postflight Trajectory AS-506)

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 thatthe 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.†

Julian Day of Apollo 11 launch
(Source)

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‡.

Apollo 11 Translunar Injection conditions
Click to enlarge
(Source: Apollo/Saturn V Postflight Trajectory AS-506)

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:

Apollo 11 entry trajectory parameters
Click to enlarge
(Source: 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:

Diagram of geocentric and geodetic latitude of a spacecraft
Click to enlarge

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 (a–b)/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”.

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Phenomena, Walking

Scottish Hill Lists: The Classics

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Cover of 1953 edition of Munro's Tables

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:

Geographical distribution of Munros, Corbetts and Donalds
Click to enlarge

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:

Height-prominence chart of Munros, Corbetts and Donalds
Click to enlarge

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”.

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Phenomena, Reading

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

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Dropped toast

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

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Keplerian Orbital Elements

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Portrait of Johannes Kepler
(Source)

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 Copernicus’ heliocentric 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.

The metric orbital elements, a and e
Click to enlarge

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.

The angular orbital elements
Click to enlarge

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

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Fata Morgana in Kolyuchin Inlet, Russian Far East
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As the weary traveller sees
In desert or prairie vast,
Blue lakes, overhung with trees,
That a pleasant shadow cast;

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

Henry Longfellow “Fata 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:

Fata Morgana, Black Island
Click to enlarge
Photo credit: United States Antarctic Program

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:

Simple superior mirage
Click to enlarge

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:

Relevance of thermocline to Fata Morgana formation
Click to enlarge

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:

Fata Morgana formation (1)
Click to enlarge

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):

Fata Morgana formation (3)
Click to enlarge

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:

Diagram of Fata Morgana mirage (1)

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:

Fata Morgana formation (2)
Click to enlarge

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

Diagram of Fata Morgana mirage (3)

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:

Diagram of Fata Morgana mirage (2)

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”.

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More About Converging Rainbows

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Reflected-light rainbows over Morecambe Bay, by Mick Shaw
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Photo credit: Mick Shaw

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:

Double rainbow from reflected sun
Click to enlarge

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:

How double rainbows form

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:

Divergence of directed and reflected-light rainbows at horizon, by solar altitude

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:

Direct and reflected light primary rainbows at solar altitude of 42 degrees

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:

Rainbow at sunset, Kylerhea
Click to enlarge
Rainbow at sunset, Kylerhea, © 2021, The Oikofuge

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:

Proportion of sunlight reflected by water surface, by 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:

Degree of polarization of sunlight reflected from water

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”:

Brightness of reflected-light rainbow arcs, by solar altitude

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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Same Sun, Other Skies

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Cover of Russian edition of Asimov's "Lucky Starr and the Big Sun of Mercury"
Russian edition of Isaac Asimov’s “Lucky Starr And The Big Sun Of Mercury”
(Chosen because, of all the editions of this novel, it does the best job of delivering exactly what the title says.)

A section of the horizon was etched sharply against a pearly region of the sky. Every pointed irregularity of that part of the horizon was in keen focus. Above it, the sky was in a soft glow (fading with height) a third of the way to the zenith. The glow consisted of bright, curving streamers of pale light.
“That’s the corona, Mr. Jones,” said Mindes.
Even in his astonishment Bigman was not forgetful of his own conception of proprieties. He growled, “Call me Bigman.” Then he said, “You mean the corona around the Sun? I didn’t think it was that big.”
“It’s a million miles deep or more,” said Mindes, “and we’re on Mercury, the planet closest to the Sun. We’re only thirty million miles from the Sun right now. You’re from Mars, aren’t you?”
“Born and bred,” said Bigman.
“Well, if you could see the Sun right now, you’d find it was thirty-six times as big as it is when seen from Mars, and so’s the corona. And thirty-six times as bright too.”
Lucky nodded. Sun and corona would be nine times as large as seen from Earth.

Isaac Asimov, Lucky Starr And The Big Sun Of Mercury (1956)

That’s Isaac Asimov (writing under the pseudonym “Paul French”), being very Asimov about things, in one of his “Lucky Starr” science fiction juveniles. In Asimov stories, characters explain things to each other quite often; in his “Lucky Starr” stories, doubly so. This particular passage introduces my theme for this post—the Sun as seen from other planets of the Solar System.

First, there’s some basic geometry to deal with. The farther a planet is from the Sun, the smaller the Sun will appear in the planet’s sky—there’s a simple inverse relationship between planetary distance and the apparent width of the solar disc as seen from that planet. But it’s the apparent area of the solar disc that determines how much light and heat the planet receives from the Sun—that bears an inverse-square relationship to the distance between planet and Sun.

We can distil that down into a simple table if we list the average distance at which each planet orbits the Sun, giving that figure in Astronomical Units, one AU being the Earth’s orbital radius. In the table below, the second column of numbers, indicating the apparent width of the solar disc, is simply the inverse of the first column; the third column, showing the apparent area of the disc, is the square of the second. (I’ve rounded the numbers, so the relationship between the tabulated figures isn’t exact.)

PlanetSun dist. (x Earth)Sun width (x Earth)Sun area (x Earth)
Mercury0.38712.586.67
Venus0.72331.381.91
Earth1.00001.001.00
Mars1.52370.660.43
Jupiter5.20290.190.037
Saturn9.53670.100.011
Uranus19.18920.0520.0027
Neptune30.06990.0330.0011

That’s not the full story, however, because the planets have elliptical, rather than circular, orbits. So the apparent width of the solar disc will vary somewhat around the mean values listed above, getting larger and smaller during the course of each planet’s year. For most planets, the change is very slight. From the Earth, for instance, the solar disc has an average apparent width of 32 minutes of arc, which increases by about half a minute of arc in January, when the Earth makes its closest approach to the Sun (its perihelion), and correspondingly decreases by about half a minute in July, when the Earth is at the farthest point in its orbit (the aphelion). It’s not a particularly noticeable change. But two planets, Mercury and Mars, have significantly elliptical orbits, and I can improve my table by listing maximum and minimum values for their solar discs.

PlanetSun dist. (x Earth)Sun width (x Earth)Sun area (x Earth)
Mercury0.38712.14 – 3.254.59 – 10.58
Venus0.72331.381.91
Earth1.00001.001.00
Mars1.52370.60 – 0.720.36 – 0.52
Jupiter5.20290.190.037
Saturn9.53670.100.011
Uranus19.18920.0520.0027
Neptune30.06990.0330.0011

The area of the solar disc (and therefore the light and heat from the Sun) varies more than two-fold during the course of a Mercurian year! Mars undergoes a more modest change, but the Sun gets 40% larger in the Martian sky as the planet moves from the farthest to the closest point in its orbit.

Another thing we can tell from my table above is that Asimov got his numbers wrong. (Now, there’s a phrase I thought I’d never write.) We can defend his characters’ claim that the solar disc appears nine times larger on Mercury than on Earth, if we assume they’re talking about its apparent area at a time when Mercury was close to perihelion. But there are no circumstances under which the Mercurian solar disc can appear even thirty times larger than that seen on Mars, let alone thirty-six times.

Before I go through the list of planets in more detail, I need to give a couple of definitions. The apparent surface brightness of the solar disc is called its luminance, and (perhaps counterintuitively) it doesn’t change with distance within the planetary system. The reason the solar disc sheds less light on more distant planets is because it is smaller, as detailed in my table above, not because it’s dimmer, area for area. The amount of light a planet receives from the Sun is called illuminance, and it’s what gives us the sense of whether our surroundings are dimly or brightly lit, and what determines the settings our cameras need to use to get a properly exposed picture. The SI unit of illuminance is the lux (say “looks”, not “lucks”), and sunlight on a clear day on Earth provides about 100,000 lux.

Now, let’s go through my list of planets, one at a time:

Mercury

Sunlight on Mercury is going to be brighter than anything we experience on Earth—five to ten times brighter. But, although science-fiction illustrators tend to depict the Mercurian sun as huge in the sky, it wouldn’t actually be that large. You can easily cover the solar disc, seen from Earth, with just the tip of your little finger. The Mercurian sun could be obscured with a couple of finger-tips, even at its largest.

Mercury’s slow rotation has an interesting effect on its daylight. It’s locked into what’s called a spin-orbit resonance. One Mercurian “year” lasts 87.97 Earth days; one Mercurian rotation takes 58.65 Earth days, meaning that the planet rotates exactly three times on its axis for every two orbits around the Sun. This means that, for any point in Mercury’s orbit, the planet returns to that point one orbit later with an extra half-rotation. So if some point on the surface experiences noon at a particular orbital location, it’ll experience midnight when it returns to that orbital location after one Mercurian year, and then noon again the next year, and so on. If it’s noon at a particular location when Mercury is passing through perihelion, it will be noon during perihelion every two Mercurian years (and midnight on the alternate years). So there are two points on the Mercurian equator, on opposite sides of the planet, that experience the brunt of the solar heating during Mercury’s closest approaches to the Sun. One of these hot points, on the equator at 180 degrees longitude, lies some distance to the southeast of a huge Mercurian impact basin, which has accordingly been named Caloris Planitia, or “Plain of Heat”.

There’s more fun to be had with Mercury’s spin-orbit coupling and eccentric orbit, but it strays away from the chosen topic of this post—I’ll come back to it another time, perhaps.

Venus

Venus, at about three-quarters of Earth’s distance from the Sun, sees the solar disc about a third larger than it appears from Earth. So light levels in orbit around Venus are close to double what we’d experience while in orbit around the Earth, and about two-and-a-half times the illuminance at the surface of the Earth on a clear summer’s day. That difference occurs because our atmosphere, even on a clear day, absorbs and reflects some sunlight before it arrives at the surface—but not nearly as much as the atmosphere of Venus, the surface of which lies under a perpetual dense overcast.

The light level at the surface of Venus, under all that cloud, was measured by the Venera series of Soviet-era landers. Venera-9, which made the first successful landing in a location with the Sun high in the Venusian sky, reported an illuminance of 14,000 lux, which is about what you receive if you stand in the shade under a bright, blue sky on Earth. Results from Venera-13 and Venera-14 were a bit lower, if the numbers quoted in a slightly batty paper entitled “Hypothetic Flora of Venus” can be considered reliable. Again with the Sun fairly high in the sky, the Venusian light level reached a value of 3,500 lux, about a thirtieth of a sunny day on Earth, and representing just a seventieth of what arrives at Venus’s cloud-tops. Even that low figure can be considered bright, by Earthly standards. 3,500 lux is the equivalent of the illuminance provided by the lights positioned above surgical operating tables, sufficient to carry out extremely fine work. (Our eyes adapt readily to a range of lighting conditions, and most of us barely notice that the normal level of illuminance indoors is usually at least a hundred times lower than that outdoors.)

Not that we’d be doing much fine work on the surface of Venus, given that the massive greenhouse effect from its dense atmosphere pushes the surface temperature up over 450ºC. (It’s traditional at this point to say “hot enough to melt lead”. Consider it said.) But if we were standing on the surface, the illumination would be equivalent to that of a diffusely lit but extremely bright room. A day-night cycle would last 177 Earth days (Venus rotates very slowly), and the sun would rise in the west and set in the east (Venus has retrograde rotation). But we wouldn’t be able to determine the exact moment of sunrise or sunset, since the location of the Sun would be no more than a brighter patch in the sky—“a smear of light”, says NASA, though I haven’t been able to track down a simulation of Venus’s atmosphere that might provide more detail on that.

Mars

Because of Mars’s elliptical orbit, the solar disc varies in apparent size with the seasons, ranging between about 60% and 70% of its width seen from Earth. This means the illuminance at the surface of Mars on a clear day can vary between 47,000 and 68,000 lux—to the adaptable human eye, indistinguishable from daylight on Earth. The solar disc appears largest when Mars is at perihelion during its southern hemisphere summer, and smallest during southern hemisphere winter. The southern hemisphere therefore experiences more extreme seasonal temperature changes than the north.

The Martian day is not much different in duration from an Earth day—about 40 minutes longer—and planetary scientists who study Mars refer to the Martian day as a “sol” (from the Latin for “sun”) to avoid confusion. There’s no confusing a Martian sunset with an Earth sunset, however:

Martian sunset, Spirit rover, Gusev crater
Source

The dominant colours are reversed, with a blue sun in a red sky. The effect is due to the particular size of fine dust particles suspended in Mars’s atmosphere, which produces a diffraction pattern that preferentially reinforces the forward-scattering of blue light. (A similar effect is sometimes produced by smoke aerosols on Earth.) You can find the detailed optical explanation here.

Jupiter

The next planet out from the Sun is Jupiter. While there’s nowhere for an observer to stand on a gas giant planet, it has a retinue of moons that might provide convenient locations from which to observe the Sun.

Such hypothetical observers would see a solar disc shrunk to about a fifth of its width as seen from Earth, providing only about a twenty-fifth of the light. That is, however, still about 5,000 lux—a not-too-overcast day on Earth, and easily sufficient illumination for the finest of work. And still comparable to the surface of Venus.

Saturn

By the time we reach Saturn, the solar disc has a tenth of its Earthly width, and sheds about hundredth of the light—1,400 lux, which is the equivalent of a solidly overcast day on Earth, or the sort of lamp one uses for fine work indoors. Still not particularly dim, then.

Again, our hypothetical observer would need to station themselves on one of Saturn’s moons in order to have a clear view of the Sun. Any moon would do, with the exception of Titan, which is swathed in a thick atmosphere. And we have some pretty detailed calculations of what the Sun would look like from the surface of Titan. I quote from the linked paper:

At visible wavelengths, the sky appears as nearly featureless orange soup most of the time, with little if any increased brightness toward the Sun’s azimuth.

My linked paper also provides some helpful graphs, suggesting that the overall reduction in illuminance when the Sun is overhead on Titan is somewhere between five-fold and ten-fold—so down to around 150-300 lux, in round numbers. That’s what we get under the absolute densest of massive cumulonimbus storm clouds on Earth, and the range of lighting used in corridors and stairwells indoors. So some of us would need to reach for our reading glasses on even the brightest day on Titan. The illuminance is cut a hundred-fold by the time the Sun has reached the horizon on Titan (say 15 lux, getting down to the limit for reading newsprint), and the setting sun would be absolutely invisible. But Titan’s atmosphere scatters light so well that the sky would continue to illuminate the landscape with full-moon brightness (about 0.2 lux) even when it was thirty degrees below the horizon.

Uranus

At Uranus, the solar disc has a twentieth of its width on Earth, and provides only around 350 lux—a little brighter than the surface of Titan, but still a profoundly overcast day on Earth.

Neptune

By the time we reach the outermost planet of the Solar System, the solar disc is down to a thirtieth of the width we see on Earth, and providing just 140 lux. Still, that’s the equivalent of 700 full moons, and you’d have no more difficult finding your way about in the vicinity of Neptune than you’d have finding your way down a stairwell on Earth.

We reach a significant threshold at Neptune, however. The solar disc is now just one minute of arc across, which is the limit of resolution of the human eye. Looking at the sky from one of Neptune’s moons, the Sun would appear as an eye-wateringly intense point of light, rather than a clear disc. But it would have the same surface brightness as the Sun seen from Earth—you could still damage your retina by staring at it with the naked eye (though the naked eye would of course not be an option out at the edge of the Solar System).

If we move farther from the Sun than Neptune, the solar disc can never appear any smaller to our eyes—it will always appear as a little point of light, smeared by the physical limitations of our eyes into a tiny spot one minute of arc across. At first a searingly bright star, and then progressively dimmer as we move farther and farther away.

But that’s a topic for another post.

Comparison of sizes of solar disc seen from various planets
Click to enlarge
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Why Does The Illuminated Side Of The Moon Sometimes Not Point At The Sun?

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Illuminated part of moon apparently not pointing at sun
Click to enlarge

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

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

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

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

Gibbous moon at sunset
Click to enlarge

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

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

Railway line perspective, Carnoustie
Click to enlarge

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

Vanishing points in two directions
Click to enlarge

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

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

Perspective between two vanishing points
Click to enlarge

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

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

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

Crepuscular Rays © 2016 Marion McMurdo
Click to enlarge
© 2016 The Boon Companion

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

Anticrepuscular rays
Source
Click to enlarge

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

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

Sun rays in perspective
Click to enlarge

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

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

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

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

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

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

Here’s the result:

Illuminate part of moon actually does point at sun
NowClick to enlarge

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

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We Are Stardust (Supplement)

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The cosmic origins of the chemical elements that make up the human body
Click to enlarge

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

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

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

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

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

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

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

The cosmic origins of the atoms that make up the human body
Click to enlarge

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

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

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