Category Archives: Phenomena

Blue Supermoon: Part 2

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

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

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

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

Moon distance, 2022-2024
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The short-period cycle is just the time it takes the Moon to move from one perigee to the next—this is called the anomalistic month*, and it lasts about 27.55 days. The longer-period cycle is that of the recurring apsidal alignments, which drive more extreme perigees and apogees. If the line of the apsides always pointed in the same direction, then that cycle would take half a year, repeating whenever either perigee or apogee points towards the Sun. But, just to make matters more complicated, the line of the Moon’s apsides is rotating slowly, completing one revolution every 8.85 years. So apsidal alignments actually happen at intervals of about 206 days.

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

Why a lunation is longer than an anomalistic month
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So now I’ll plot full moon dates on my previous diagram:

Moon distance and full moons, 2022-2024
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If you choose a full moon near perigee, and follow the progress of successive full moons across the diagram, you can see how they come progressively later, relative to perigee, with each anomalistic month that passes. But after 14 lunations and 15 anomalistic months, the full moon returns to approximately the same relationship with perigee—because 14×29.53 and 15×27.55 are both approximately 413 days. And, remarkably and (I think) coincidentally, that’s almost the same as two cycles of apsidal alignment, 412 days.

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

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

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

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

Moon distance and full moons, 2022-2024, with zero distance plotted
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That rather modest wobble means that a typical supermoon isn’t hugely different from an average moon:

Comparison of typical "supermoon" with the Moon seen at its average distance

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

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

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

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

Blue Supermoon: Part 1

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

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

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

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

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

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

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

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

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

I can put all this together in a chart:

Chart of all full moons, 2011 to 2030 (times are UTC)
Click to enlarge

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

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

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

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

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

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

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

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

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

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

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

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

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

Hill Lists: “On Top Of The World”

Map of Scotland's 20 "On Top Of The World" hills
Click to enlarge

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

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

Submission is defined in Xu’s paper as follows:

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

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

Diagram of "On Top Of The World" definition
Click to enlarge

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

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

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

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

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

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

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

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

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

Rockall image © Copyright Andy Strangeway, used under Creative Commons Attribution-ShareAlike 2.0 Generic licence

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

How Apollo Left Earth (And Returned)

Apollo 11 Translunar trajectory relative to Van Allen Belts 3
Click to enlarge
Prepared using Celestia

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

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

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

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

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

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

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

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

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

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

Diagram of velocity changes as Apollo spacecraft left Earth
Click to enlarge

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

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

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

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

Apollo 11 orbital angular velocity, relative to Earth, during first six hours of translunar trajectory
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We can see how Apollo moved faster than the Earth’s rotation for about an hour after TLI, but for the next hour was moving only a little faster or a little slower than the Earth, so that it would appear to hang in the sky over the Earth’s surface for a while.

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

Apollo 11 ground track after TLI, with hour markers and geomagnetic field
Click to enlarge
Base map prepared using Natural Earth data

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

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

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

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

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

Apollo 11 departure trajectory relative to VAB proton radiation
Click to enlarge

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

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

Apollo 11 ground track with VAB transit and Transposition Docking & Extraction manoeuvre marked
Click to enlarge
Base map prepared using Natural Earth data

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

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

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

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

Apollo 11 re-entry trajectory relative to inner VAB
Click to enlarge

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

Apollo 11 ground track with VAB transit and Entry interface marked
Click to enlarge
Base map prepared using Natural Earth data

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

Almost all the Apollo trajectories followed a very similar pattern—north over the VAB on the way out, south under the VAB on the way back. But there was one striking exception, the trajectory taken by Apollo 17. That’s what I’ll write about next time.

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

The Solar System’s Place In The Milky Way: Part 1

Location of our sun in Milky Way galaxy
Click to enlarge (with local arms labelled)
Base map: NASA / JPL-Caltech / R. Hurt (SSC-Caltech)

When I wrote recently about the pole stars of other planets, I was aware of one thing my sky maps didn’t show—the rotation poles of our galaxy. They weren’t really relevant to that discussion, but I’m now prompted to write a bit about the Milky Way galaxy, and our relationship to it, because I’ve just encountered a rather garbled, misleading and self-contradictory Wikipedia article on the topic—specifically the section entitled Sun’s location and neighborhood. (The last time I referred to a misleading Wikipedia page on this blog, the page was eventually fixed by an editor citing my article. This was gratifying, but it meant my link to the page no longer demonstrated the problem to which I’d objected. So this time I’ve linked to a Wayback Machine copy of the page dated 9 April 2023, which will at least give permanent context to my griping.)

The view of our galaxy at the head of this post is an artist’s impression, sketching in the major features as currently understood. Our sun, marked by the arrow, lies between two spiral arms. The Perseus Arm, which lies farther out than our location, is considered to be one of the Milky Way’s two major spiral arms; and inwards from our position is the more minor Sagittarius Arm. Spanning the gap between these two arms is a more diagonal structure, clearly visible in the painting, variously referred to as the Orion Spur, the Orion Arm, or the Local Arm, the status of which seems to be much debated. Our sun lies within the inner edge of that diagonal structure.*

Our distance from the centre of the galaxy isn’t known with great accuracy. From a couple of recent papers addressing the issue, the distance is somewhere in the region of 7.9 kiloparsecs (25,800 light-years) to 8.34 kiloparsecs (27,200 light-years), which you’ll see from the map above is very roughly halfway from the galactic centre to its rim. The two papers in my link are in much stronger agreement about the angular velocity of the sun in its orbit around the galaxy, which they place between 30.2 and 30.6 km/s/kpc. Call it 30.4, which translates to 9.85×10-16rad/s, or one orbit every 200 million years. But the sun is near perigalacticon at present (its closest approach to the galactic centre) and therefore moving faster than it normally does. Commonly quoted estimates for the duration of a true galactic year therefore lie between 225 and 250 million years.

So much for our radial location within the galaxy. How are we placed relative to the galactic plane? Estimates of our distance from the galactic mid-plane (which we can think of as the galactic equator) vary, but a recent measurement and review places it about 17 parsecs (55 light-years) “above” the plane. I’ll come back to what “above” means in this context later.

The orientation of the solar system relative to the plane of the galaxy turns out to be a little unexpected. In the view below, we’ve moved a short distance from the sun towards the galactic rim, and are looking back on the solar system framed against the nebulae of the galactic core, with the galactic equator running horizontally across the picture.

Orientation of the plane of the solar system relative to the galaxy
Click to enlarge
Prepared using Celestia and the “Milky Way” add-on by Guillermo Abramson

I’ve marked the constellations in the vicinity of the galactic core for orientation. The four visible orbits are those of the outer giant planets from Jupiter to Neptune.

So the plane of the solar system is tilted at an angle of about 60° relative to the plane of the galaxy. It also, as you see, comes close to aligning with the centre of the galaxy—the mismatch is only about 6°. Since the plane of the solar system remains fixed as it orbits the galaxy, a perfect alignment will occur twice in every galactic year—the most recent happened three or four million years ago.

Now I’m going to zoom in to look at the Earth’s orbit. The plane of the Earth’s orbit is called the ecliptic, and it’s pretty closely aligned with the overall plane of the solar system—close enough that I plan on using it as a reasonable proxy for the solar system’s invariable plane, later.

Orientation of Earth's orbit relative to the galaxy
Click to enlarge
Prepared using Celestia and the “Milky Way” add-on by Guillermo Abramson

I’ve included a cartoonishly large Earth to give you a visual cue to the orientation of the ecliptic along the axis perpendicular to your screen—the left side of the Earth’s orbit, as seen here, is farther away from our viewpoint than is the sun, while the right side brings the Earth closer to us. I’ve also marked four key locations in the Earth’s orbit—the solstices and equinoxes. And here’s a coincidence! The solstice points align pretty well with the galactic core. In 2023, for example, the sun will cross the galactic equator at about 11:00 on 22 December, the day of the solstice.

Now, we know that the Earth’s axis is tilted by about 23½° relative to the ecliptic. So how does the Earth’s tilt interact with the solar system’s tilt? There’s a hint in my previous picture, from the location of the solstices and equinoxes, but we can zoom in farther from the same vantage point to make the situation clearer.

Orientation of the Earth relative to the galaxy
Click to enlarge
Prepared using Celestia and the “Milky Way” add-on by Guillermo Abramson

The Earth is rotated in the plane of the image through the 60-degree tilt of the rest of the solar system, but the 23½°-tilt between its equator and its orbit is directed towards us, out of the computer screen. In other words, the Earth’s tilt relative to the ecliptic is almost at right angles to the ecliptic’s tilt relative to the galaxy. This is difficult to depict in a single diagram, like the one offered by Astronomy magazine recently (third image down on the linked page), in which it looks very much as if the Earth’s tilt is in the same plane as the ecliptic tilt. And it’s something that can catch out even the astronomers who wrote the entries for Swinburne University’s online Encyclopedia of Astronomy. For a few years their entry on the Galactic Plane claimed that:

The galactic plane is tilted at an angle of 63 degrees to the celestial equator. Since the ecliptic (the path of the Sun on the sky) is inclined at an angle of 23.5 degrees to the celestial equator, the galactic plane and the ecliptic are nearly at right angles (63 + 23.5 = 86.5 degrees), although this is purely coincidental.

Purely fantastical, more like. The entry was mercifully corrected in 2012; my link takes you to the original version embarrassingly preserved on the Wayback Machine.

But they were correct about the 63° tilt between the celestial equator (the extension of the Earth’s equator into the sky) and the galactic plane. The easiest way to see how all this fits together is not to try to diagram the intersecting planes of the Earth’s equator, the ecliptic and the galactic equator, but instead to plot the position of their north poles in the sky. The Earth has its celestial north pole, near the pole star, Polaris. The ecliptic has its own north pole, corresponding to the rotation axis implied by the movement of the planets in their orbits. And galactic north was defined by the International Astronomical Union, back in 1958, as being the extension of the rotation axis of the galaxy into the northern sky of the Earth. (Which, at last, allows me to say what is meant by the sun sitting about 55 light-years “above” the galactic plane—it lies north of the galactic plane.)

Position of celestial, ecliptic and galactic north poles
Click to enlarge

(As with my previous post about the pole stars of other planets, I’ve used a star map generated by

The north galactic pole lies in the obscure little constellation Coma Berenices. And you can immediately see how the angle between the ecliptic and celestial north poles is measured almost at right angles to the angle between the galactic and ecliptic north poles, so that the celestial and ecliptic poles end up at much the same angular distance from the galactic pole.

But there’s one little quirk to this trio of north poles, arising from the way the IAU defines north, which I addressed in more detail when I wrote about pole stars previously. If we looked down on Earth from the celestial north pole, we’d see it rotating anticlockwise; likewise, if we looked at the solar system from the ecliptic north pole, we’d see the planets moving anticlockwise around the sun. But if we were to look at the galaxy from the galactic north pole (the view in the picture at the head of this post), we’d observe it rotating clockwise. Like the planet Venus, then, the Milky Way galaxy turns out to be a retrograde rotator.

I can now go back to my original solar system diagram, and add some orientation arrows.

Solar System Orientation
Click to enlarge
Prepared using Celestia and the “Milky Way” add-on by Guillermo Abramson

The solar system, which participates in the galactic rotation, is therefore moving in a direction approximately aligned with my horizontal arrow. The extent to which it deviates from that alignment will be my topic next time I write on this topic—at which point I’ll also be able to explain my gripes about the Wikipedia article that started all this.

* For more detail on the layout of the spiral arms, and detailed maps of the solar neighbourhood, visit Kevin Jardine’s jaw-dropping Galaxy Map site, which is a thing of beauty and a labour of love.
This approximate alignment between the galactic core, the sun and the Earth on the day of the December solstice was of course one of the foundations of a fatuous end-of-the-world scenario predicted for 2012. (To be honest, though, I did quite enjoy Roland Emmerich’s accompanying disaster movie, 2012).
There’s a passage in Larry Niven’s classic science fiction novel, Ringworld (1970), in which he describes how the Puppeteer Fleet of Worlds abandons the Milky Way galaxy and sets out for the Small Magellanic Cloud. Elsewhere in the novel, Niven tells us that the Fleet of Worlds is “moving north along the galactic axis”. Unfortunately, the SMC actually lies south of the galactic equator. Oops. But this would make sense if Niven was using rotational north (that is, the direction from which a body appears to rotate anticlockwise), rather than the IAU’s parochial system based on the orientation of the Earth. (I once actually raised this possibility with Niven, but he responded that the Puppeteers were “taking the scenic route”.)

Converting Apollo State Vectors To Orbits

Apollo 11 Translunar trajectory relative to Van Allen Belts 3
Click to enlarge
Prepared using Celestia

OK, another mathematical one. This is the fourth in a series of posts about the orbits followed by the Apollo spacecraft as they travelled to and from the moon—something I suppose is getting a little more topical now that NASA has finally got underway with its planned return to the lunar surface.

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

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

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

In the first post I established that we need to know the following six orbital elements:

  • semimajor axis (a)
  • eccentricity (e)
  • inclination (i)
  • longitude of the ascending node (Ω)
  • argument of the periapsis (ω)
  • true anomaly (θ)

The first two give the orbit’s size and shape, the next three its orientation in three dimensions, and the last one tells us the spacecraft’s position in orbit—see my first post for more detail.

To go along with the true anomaly we need the time, called the epoch (t), at which our spacecraft is in that position. This is usually expressed in the form of a Julian Day, and there are online calculators that will convert a date and Greenwich Mean Time to a Julian Day—there’s one here.

And we need an orbital period (P). We can calculate this from the semimajor axis, if we know the mass (M) of the central body around which the orbiting spacecraft moves—in this case, the Earth. There’s a conversion factor, too, called the Universal Gravitational Constant (G), but since orbital calculations always involve multiplying M by G, and since M is constant for any given planet, the two can be lumped together into something called the Standard Gravitational Parameter (μ) for that planet. Currently accepted values for μ for the various planets are listed by the Jet Propulsion Laboratory.

In my second post I established that what we have available for Apollo are primarily state vectors—three-dimensional locations and velocities for given times. Here, for instance, is the Apollo 11 state vector for the time of Translunar Injection (TLI), which established the initial orbit on which the Apollo spacecraft departed from Earth.

TLI state vector for Apollo 11
Click to enlarge (Source: Apollo/Saturn V Postflight Trajectory AS-506)

I’ve marked the relevant data columns in red, and we have enough information here to produce all the required orbital elements and an epoch. We have, from left to right:

  • ground elapsed time (GET)
  • geocentric distance (r)
  • longitude (λ)
  • geocentric latitude (ψ)
  • heading (h)
  • flight-path angle (f)
  • space-fixed velocity (v)

I explained all these in my second post, and I’ll refer you back to that for more detail. GET measures the elapsed time since launch, and by adding that to the Julian Day calculated for the launch date and time, we get our epoch (t) for TLI—I walked through that process step by step in my second post. The next three parameters describe the spacecraft’s position in three-dimensional space, but relative to the rotating surface of the Earth, so there’s a little bit of work to do to convert these coordinates to the fixed celestial coordinate system we need for our orbit. And the next three numbers give the spacecraft’s velocity, again in three dimensions, and NASA have helpfully done the calculation to make this “space-fixed”—that is, they’ve taken the spacecraft’s velocity relative to the surface of the Earth (given in the “EF VEL” column), and added in the rotation speed of the Earth at that latitude, so that the quoted velocity is now relative to the fixed stars, rather than the rotating Earth.

In my second post, I worked out the epoch (in Julian Days) of Apollo 11’s Translunar Injection:

tUTC = 2440419.18209525 days

But, as I explained in my third post, this time is in Coordinated Universal Time (UTC), and to that figure we need to add the difference between UTC and TT on 16 July 1969, the date of the Apollo 11 Translunar Injection. In that previous post, I established that this difference, symbolized ΔT, was 39.746 seconds, and I’ll refer you back to that post for a description of the relevance and origin of that number.

So the Terrestrial Time epoch for our Apollo 11 orbit is:

tTT = tUTC +ΔT

tTT = 2440419.18255527 days

Now I need to convert the latitude and longitude coordinates to the equatorial coordinate system used in astronomy. The equivalent of latitude in this system is called declination (δ), and longitude translates into right ascension (α). The good news is that declination is equal to the geocentric latitude, so we can just set δ equal to ψ:

δ = 9.9204°

The conversion between λ and α is complicated by the Earth’s rotation. The zero meridian of longitude (the Greenwich meridian) only points at the zero meridian of right ascension (the vernal equinox) once per rotation. So to work out the right ascension corresponding to the longitude of TLI, we need to know in which direction the Greenwich meridian was pointing at time t, and this is a little complicated by the fact that the Earth doesn’t rotate at a constant rate. The angle between the vernal equinox and the Greenwich meridian is given by a quantity called Greenwich Mean Sidereal Time (GMST), which I introduced in my third post. Although this is commonly expressed in hours, minutes and seconds, it can be converted to an angular measurement in degrees, which is what we need. To find GMST we ideally need to know Universal Time in the form of UT1, which is approximated by the civil time, UTC. It was actually quite well approximated by UTC during the time of the earlier Apollo missions, and I looked up the difference between the two times on 16 July 1969 in my third post:

ΔUT1 = 0.0115 seconds

So to calculate GMST properly we need another version of the epoch:

tUT1 = tUTC + ΔUT1

tUT1 = 2440419.18209538 days

You’ll find all sorts of equations to convert between UT1 and GMST. The one I use below comes from Chapter 2 of a Jet Propulsion Laboratory publication entitled Explanatory Supplement to Metric Prediction Generation—see equation 2-15, but the whole chapter is a nice primer on the complexities of astronomical time measurements.

First we need to express our UT1 epoch in terms of Julian Days elapsed since midday on 1 January 2000:

d = tUT12451545 days

Then we plug this value d into a rather large equation it’s difficult to cram on to one line:

\tiny GMST = 280.460618375 + 360.9856473663\cdot d + 2.9079\times 10^{-13}\cdot d^{2}-5.3\times 10^{-22}\cdot d^{3}

This usually generates a big number, reflecting the many rotations that have taken place between 1 January 2000 and the epoch of interest, so we need to narrow it down to a value between 0 and 360 degrees by adding or subtracting whole multiples of 360. Most spreadsheets and programming languages let you do this using the modulo function, which gives the remainder after division. So mod(GMST,360), or something similar, will give the remainder after GMST is divided by 360, which is what we need.*

When I run the TLI epoch through this process, I get an answer of -180.1181 degrees—negative, because 1969 comes before the year 2000. This is the equivalent of 179.8819 degrees, so at the epoch of TLI, the Greenwich meridian was pointing towards a right ascension of 179.8819 degrees.

Our longitude (λ) from the NASA table is -164.8373, and the right ascension is just λ+GMST, giving us, at long last:

α = 15.0446°

So that’s the fiddly bit done, and we now have our state vector in the necessary space-fixed coordinates.

Many reference sources deal with the conversion from state vector to orbital parameters. Ulrich Walter’s magisterial Astronautics: The Physics of Space Flight is a great resource, but uses vector notation I’m going to avoid here. A useful on-line resource is Uger Guven’s slide collection, Satellite Orbit Dynamics.

For a given state vector, three of the orbital parameters depend on the planetary mass. To see why this is so, consider a spacecraft with exactly the same position and velocity as Apollo 11’s at TLI, but orbiting a planet with half the mass—it will go farther from the planet before falling back again under the influence of gravity. So it’s useful to define a constant, C, which I’ll call the “energy constant”, because it’s a measure of the ratio between the spacecraft’s gravitational potential energy and its kinetic energy:

C = \frac{r\cdot v^{2}}{\mu }

From the NASA data table, r=6711.964km, and v=10.8343km/s at TLI. But what about μ? My JPL link from earlier in this post gives a value of 398600.435507km3/s2. (In the Apollo era NASA used an estimate that was a little different, but only after about the sixth significant figure.)

So for Apollo 11’s TLI:

C = 1.9766

We can now calculate orbital parameters a, e and θ. The semimajor axis depends only on C:


a = 286545 km

The eccentricity and true anomaly also depend on the flight-path angle (f):

e=\sqrt{\left ( C-1 \right )^{2}\cdot cos^{2}\left ( f \right )+sin^{2}\left ( f \right )}

e = 0.976966

\theta =atan\left ( \frac{C\cdot cos\left ( f \right )\cdot sin\left ( f \right )}{C\cdot cos^{2}\left ( f \right )-1} \right )

θ = 14.909º

The other parameters merely require a bit of spherical trigonometry:

i=acos\left ( cos\left ( \delta  \right )\cdot sin\left ( h \right ) \right )

i = 31.383º

Then we need a couple of intermediate calculations before we can derive our final two orbital parameters:

\varphi =asin\left ( \frac{sin\left ( \delta  \right )}{sin\left ( i \right )} \right )

To get this value into the correct quadrant before proceeding, we need to check the heading angle h. This is always between 0 and 180º for eastward launches (that is, pretty much every spacecraft launch). If it’s less than 90º we let φ stand, but if it’s over 90º, we set φ=(180º-φ). Then:

\omega =\varphi-\theta

ω = 4.410º


\eta =acos\left ( \frac{cos\left ( \varphi  \right )}{cos\left ( \delta \right )} \right )

The longitude of the ascending node (Ω) depends on η, but also on which hemisphere we’re in, so we need to check δ before the final calculation. If δ<0º (ie, the southern hemisphere) then:

\Omega =\alpha +\eta 


\Omega =\alpha -\eta 

Ω = 358.383º

And those are all our orbital parameters. Finally, we need an orbital period, which is:

P=\sqrt{\frac{4\cdot \pi ^{2}\cdot a^{3}}{\mu }}

P = 17.6679 days

Bringing it all together, we have a complete description of the orbit followed by Apollo 11 on its departure from Earth:

t = 2440419.18255527 days
a = 286545 km
e = 0.976966
i = 31.383º
Ω = 358.383º
ω = 4.410º
θ = 14.909º
P = 17.6679 days

In my second post I described how to extract a state vector from the Apollo data for re-entry, and we can plug those data into the same sequence of equations used above, to derive the elements of Apollo’s return orbit.

I could stop here, and you’re very welcome to stop at this point if you’ve managed to get this far. What follows is a description of how I went about checking that my calculations above produced valid results, and then a short Appendix that deals with calculating some alternative orbital parameters that you may need to know: the mean anomaly (M) and the time of pericentre passage (T).

We can get a quick sanity check on the calculations by looking at the summary “Translunar Injection Conditions” table from the Apollo 11 postflight trajectory document:

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

The orbital inclination figure from my calculation matches NASA’s exactly; the eccentricity goes astray in the sixth decimal place, presumably because of the difference between the 1960s estimate of μ and the present-day value. But if we add 180º to the descending node in NASA’s table we get an ascending node of 301.847º, which is a long way from my calculated value. But the reason for this is straightforward—NASA didn’t measure the position of the node in the conventional astronomical coordinates I’ve used to produce these orbital elements. Instead, they used a coordinate system defined by the onboard inertial navigation system housed in the Instrument Unit of the Saturn V launch vehicle. This system was kept constantly updated from the ground until just before launch, and the moment at which the inertial navigation system “locked in” its own set of coordinates (called Guidance Reference Release) was marked by a phrase many of us can remember from the Apollo launch sequence: “Guidance is internal!” This happened at time marker “T minus 17”—that is, 17 seconds before launch, and that marked the zero point for longitude, as far as Apollo was concerned.

The Julian Day corresponding to Guidance Reference Release for Apollo 11 is 2440419.06369213 UTC. If we use that to calculate GMST (remembering to go through UT1), we find that the Greenwich meridian was pointing towards a right ascension of 137.140º at that time. The launch pad for Apollo 11 (Complex 39 Pad A) is at longitude -80.604133º, and so at that instant had a right ascension of 56.536º (this is called the launch pad’s Local Mean Sidereal Time). That’s the point from which NASA measured its descending node for Apollo 11, and we need to add that angle to NASA’s nodes to convert to standard right ascension. I’ve already calculated that the ascending node in NASA’s coordinates is at 301.847º, which implies a right ascension of 56.536º+301.847º=358.383º. Which (ta-da!) is exactly the figure I calculated earlier.

These orbital elements are going to be pretty accurate for the first couple of hours of the translunar trajectory, during which the Apollo astronauts performed the Transposition, Docking, and Extraction manoeuvre, linking up with the Lunar Module and removing it from its cradle in the S-IVB stage. Shortly after that, they fired their engine for three seconds to kick themselves clear of the spent stage, acquiring an additional velocity of six metres per second—a pretty slight modification to the very high velocity at which they were leaving Earth behind. A day later, they fired the engine for another three-second burn in a midcourse correction manoeuvre—finely tweaking their trajectory to achieve a very accurate arrival in lunar orbit. And the orbit is going to be increasingly inaccurate as Apollo approaches the moon, when the moon’s gravity will progressively shift the trajectory away from the original ellipse defined by the TLI state vector.

But we can make another check on the accuracy of these orbital parameters, by checking if they at least deliver Apollo 11 to the vicinity of the moon at the time we know it entered lunar orbit—called Lunar Orbit Insertion (LOI), this happened at 17:21:50 GMT on 19 July 1969. Here’s what happens when I plug the calculated elements into Celestia, and set the time to LOI:

Apollo 11 orbit defined by TLI, extrapolated to the moon at time of Lunar Orbit Insertion - no midcourse corrections, no lunar gravity
Click to enlarge

Without any midcourse corrections, or any effect from lunar gravity, my simulated Apollo 11 shoots straight past the leading edge of the moon and is a short distance beyond it at the time of LOI. In reality, the moon’s gravity bent the trajectory around the back of the moon, where Apollo 11 fired its engine to slow down and enter lunar orbit. But this close rendezvous in simulation is a pretty good test that I’ve got all my orbital elements correct.

With all that in place, next time I come back to this topic I’ll write a bit more about Apollo 11’s first few hours on its departure orbit, and last few hours on its return orbit.

Appendix: For completeness, I should mention that there are other commonly used measures that define a spacecraft’s position in orbit at a precise instant—that is, alternatives to the true anomaly and epoch I derived above. The first of these is the mean anomaly (M) at epoch t. This is an angle calculated on the assumption the spacecraft moves around its orbit at constant angular speed, rather than accelerating and decelerating as it moves closer to and farther from the Earth. We get to M through an intermediary calculation of something called the eccentric anomaly (E).

E=acos\left ( \frac{1}{e}-\frac{r}{a\cdot e} \right )
M=E-e\cdot sin\left ( E \right )

For the Apollo 11 TLI state vector, this gives us:

M = 0.0375º

This is specific to the epoch t in the same way as the true anomaly.

Another parameter that’s often used is the time of pericentre passage (T). This abandons the epoch to which the state vector applies, and asks what the epoch would have been at the time the spacecraft made its closest approach to the body around which it orbits. And it doesn’t matter whether that close approach ever actually happened—there’s a close approach implicit in the maths of every elliptical orbit.

To come up with this number, we first convert M to a value measured in revolutions (divide the value for M given in degrees above by 360º) and then multiply be the period (P) in days. This tells us how many days ago the spacecraft would have passed through its closest approach. Then we subtract that number from the epoch (t) of the state vector, and we have the epoch (T) of the pericentre passage. In this case:

TTT = 2440419.18071554 days

And in this situation we don’t need to quote any sort of anomaly, because the anomaly at time T is (by definition) zero.

* Be slightly wary of online GMST calculators—many of them calculate GMST for noon on the Julian Day given, rather than for a specific time on that date.
This way of defining the longitude of the nodes, based on the Local Mean Sidereal Time of the launch site at Guidance Reference Release, applies for Apollo missions 10 to 17. The descending nodes quoted for Apollo missions 8 and 9 are based on Greenwich Mean Sidereal Time at Guidance Release. The mission reports for earlier flights give no descending node data.

The Advent Of Atomic Time

Apollo 11 launch
Click to enlarge
NASA image S69-39961

This is, to a large extent, a companion piece to my post about leap seconds, in which I described how the irregular rotation of the Earth means that the time as measured by our atomic clocks would fall out of synchrony with the actual movement of the sun in the sky, were it not for the occasional addition of a leap second. In this post, I’m going to look back at how the various systems of time measurement we inherited from the nineteenth century were forced to adjust to the advent of extremely accurate atomic clocks in the 1950s.

But this is also, as you might have guessed from the head photograph, relevant to my continuing project to derive Project Apollo orbital data. NASA’s early space programme was conducted during a period when time-keeping standards were in a state of flux, as I’ll describe, and that has implications for how we accurately specify the timing of significant orbital events, like translunar injection and atmospheric re-entry.

But first, something about the various timescales we use.

In the nineteenth century, time was a pretty straightforward thing. A day was the length of time it took the Earth to rotate on its axis relative to the sun. Because that duration varies a little during the course of a year, clocks were set according to the average position of the sun—to a “mean time”. And the mean time measured at the Greenwich Observatory in the UK was Greenwich Mean Time (GMT), which was adopted as an international standard at the Meridian Conference in Washington, in 1884.

This type of day, measured relative to the position of the sun, is strictly called a solar day, and it’s the only kind of day relevant to most people. So GMT was the basis for civil time—the time displayed on public clocks.

Astronomers are also interested in another type of day, however—the time it takes the Earth to rotate once on its axis relative to the fixed stars. This is called a sidereal day, and it’s about four minutes shorter than the solar day. Like the solar day, the sidereal day is measured in hours, minutes and seconds, but each of these measures is just 99.7% as long as the ones we’re used to. There’s a Greenwich Mean Sidereal Time (GMST) that describes the Earth’s rotational position relative to the stars, and that’s what tells astronomers in which direction they need to point their telescopes.

But astronomers are also interested in the solar day, if only because they need to know when the sun rises and sets. And for slightly complicated reasons that I’ve fudged straight past in the preceding three paragraphs*, they took to referring to their own version of GMT as Universal Time (UT) during the 1920s.

It turned out there was a certain irony to the “universal” bit of that designation, because since the nineteenth century there had been a suspicion that the Earth’s rate of rotation was not constant, and that therefore any definition of time based on the Earth’s rotation would be similarly inconstant. And so it proved to be—by the 1920s, it was evident that the movement of the moon and planets ran to a more steady timescale than the rotation of the Earth. While astronomers still needed Universal Time and Mean Sidereal Time, they also needed a more precise timescale against which to measure the dynamics of the solar system.

And so was born the idea of Dynamical Time—time calibrated by the observation of solar system events, particularly the movement of the moon. This was formally adopted under the name Ephemeris Time (ET) in 1952.

It was immediately evident that a clock ticking out Universal Time would diverge steadily from one marking Ephemeris Time—UT seconds were longer than ET seconds, because the Earth was rotating progressively more slowly. From the way in which Ephemeris Time was formally defined, it turned out that UT and ET had been in perfect agreement at some moment in 1902 but that, by the 1950s, the slower ticking of a UT clock meant that Universal Time was about 30 seconds behind Ephemeris Time, and that difference has been increasing almost ever since.

So we embarked on the 1950s with a set of timescales based on the rotation of the Earth (GMT, GMST, UT), and one based on the movement of the moon and planets (ET). Universal Time soon separated into three flavours: UT1, which tracked the Earth’s rotation; UT0, specific to certain local observations and affected by polar motion; and UT2, which is UT1 with a small correction applied to reproduce the predictable seasonal changes in Earth’s rotation. UT0 is no longer used, and won’t bother us here. UT2 has likewise fallen into disuse, but at the time I’m discussing was considered the gold standard for civil time-keeping, so will feature prominently in what follows.

But the time-keeping game changed forever during the 1950s, with the invention of the caesium atomic clock, which soon proved to be a more precise time-keeper than anything that could be achieved by the most careful astronomical observations. Given the variability of the UT second, atomic clocks were calibrated to tick off the standard second defined by Ephemeris Time.

The Système International (SI) system of units was a little slow to catch up with the advantages of atomic time. Until 1960, it continued to define the second as being 1/86400 of a solar day. Then it shifted to the Ephemeris Time definition, which was based on the Earth’s orbital period. And then, only seven years later, it switched to a definition of the second based on the calibration of the caesium atomic clock, which it has stuck with ever since.

It wasn’t until 1970 that we’d see a standard definition of atomic time, International Atomic Time (TAI), but TAI was just the culmination of series of other atomic time-scales used during the ’50s and ’60s, and continues seamlessly from them. As a result, we find that TAI was effectively synchronized with Universal Time (specifically, UT2) back in 1958.

Trouble was, of course, that Universal Time (and GMT) immediately started to drift away from the time kept by the atomic clocks. So during the 1960s we saw a struggle to come up with a way of somehow applying the extremely regular output of atomic clocks to the slippery and evolving timescale of the rotating Earth. This hybrid of atomic time and UT was eventually named Coordinated Universal Time (UTC). It consisted of a set of instructions issued by the Bureau International de l’Heure (BIH), telling people how to modify the output of an atomic clock in order to produce a time-signal that closely matched UT2. From 1961 to 1972, this consisted of a frequency shift and the occasional step-change of a twentieth or a tenth of a second, but in 1972 the BIH shifted to one-second steps designed to keep UTC within 0.9 seconds of UT1—the “leap seconds” I’ve previously written about. The BIH was dissolved in 1987, handing over leap-second duties to the International Earth Rotation Service, but UTC continues as the civil time standard applied around the world.

So that’s how atomic time became the basis for civil time. It soon also took over the role of Ephemeris Time. In 1976 an atomic standard called Terrestrial Dynamic Time (TDT) was synchronized with ET, and later renamed to just plain Terrestrial Time (TT). This is the timescale we currently use when figuring out orbital motions in the vicinity of the Earth. There’s another one, Barycentric Dynamical Time (TDB), used for calculating high-precision orbits in the rest of the solar system—it exists because of General Relativity, and is always within a couple of milliseconds of TT, so can often be neglected.

Because TAI and TT tick at the same rate, they bear a constant relationship to each other: TT=TAI + 32.184 seconds. Where does that offset come from? It’s because Ephemeris Time (uniform with TT) was synchronized with UT back in 1902, whereas TAI was synchronized with UT in 1958. And during that time, UT had drifted away from ET by 32.184 seconds.

So nowadays our timescales look very different from the way they were in 1950. We have the Earth’s rotation, defining UT1, monitored by Very Long Baseline Interferometry, and reported by the International Earth Rotation Service. UT1 is interconvertible with GMST, so if we need to calculate GMST, we go through UT1. Civil times everywhere are based on UTC, which is an atomic timescale with added leap seconds to keep it close to the observed values of UT1. And the fine detail of solar system dynamics are calculated using TT and its associated atomic times.

If I draw a diagram of my discussion so far, you can see that the 1950s, ’60s and ’70s were a period of intense flux for time measurement.

The advent of atomic timekeeping
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This has relevance to my long, slow project of extracting orbital data from NASA’s original documentation, because NASA’s engineers would have been navigating to the moon using Ephemeris Time (which we can retrospectively call Terrestrial Time, because the one is a continuation of the other), but the mission documentation gives times according to GMT, or a time zone derived from it—either the time at the Florida launch site, or at Mission Control in Texas. So to correctly derive my Apollo orbits, I need to convert from GMT to TT—just at a time when the relationship between GMT and TT was at its most complicated.

To add to the complexity, the name “Greenwich Mean Time” was sometimes applied to UT1 when used for navigational purposes, so there’s a potential ambiguity to NASA’s usage of “GMT” in its Apollo documentation—did they mean UT1 or the evolving UTC standard? After wading through a lot of documents, I eventually turned up an answer to that question in the hefty Proceedings of the Apollo Unified S-Band Conference, which took place at the Goddard Space Center in July 1965. In the chapter entitled “Apollo Precision Frequency Source And Time Standard”, by R.L. Granata, we’re told that:

The method of obtaining time synchronization is to employ the WWV, HF signals.

WWV is the radio station used by the National Institute of Standards and Technology to broadcast time and frequency standards for the USA. These broadcasts were coordinated internationally by the BIH during the 1960s, and by 1965 had already been tied to the A3 atomic timescale, which was a direct precursor of TAI—so WWV was broadcasting the atomic-based civil time that would soon be known as UTC, and that is almost certainly what is meant by “GMT” in the Apollo documentation.

How do I get from UTC to TT? I need to turn to the Earth Orientation Center at the Paris Observatory, who maintain a dataset called the Earth Orientation Parameters, series C04. This provides daily values for UT1-TAI and UT1-UTC, stretching back to 1962. Bearing in mind the fixed relationship between TT and TAI, this is all I need to create a graph showing how UT1 and UTC were drifting away from TT during the course of the manned Apollo missions:

Time scales during the manned Apollo missions, 1968-72
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You can see how, by dint of frequent small step changes and adjustments in clock rate, the BIH kept UTC extremely close to UT1 and UT2 right up to the start of 1972, which was when the leap second was introduced—a small step change at the start of that year brought UTC to exactly 10 seconds away from TAI (42.184 seconds from TT), and the first leap second then occurred at the start of July.

Pulling up the data for 16 July 1969, the date on which Apollo 11 launched, the Paris Observatory tells me that:

UT1-TAI = -7.5505119 s
UT1-UTC = 0.0115221 s

And we know that

TT-TAI = 32.184 s


(TT-TAI)(UT1-TAI) + (UT1-UTC) = TT-UTC = 39.746 s

This value is sometimes symbolized ΔT, and it’s our route to converting NASA’s quoted GMT times to Ephemeris Time, or TT. And for strict accuracy, we also need to take note of the value for UT1-UTC, sometimes called ΔUT1. This is the conversion from UTC to UT1, and thence to GMST, which will be needed when I’m converting NASA’s state vectors to orbital elements in my next post on this topic.

So when NASA tells us Apollo 11 launched at 13:32:00 GMT, we can say that’s equivalent to 13:32:39.746 TT. It’s a significant difference—if we neglect it, it’s equivalent to a 16 kilometre displacement of the launch pad, and a 40 kilometre displacement of the moon. By comparison the conversion to 13:32:00.012 UT1 is trivial, and could easily be ignored, given the uncertainties in other data. More about that next time.

* Not like me, I know. If you want to know more about how the name “Greenwich Mean Time” came to be confusingly applied to several slightly different timescales, forcing the astronomers to give their own version a unique name, take a look at Dennis McCarthy’s “Evolution of Timescales from Astronomy to Physical Metrology”.
The progressive divergence of Universal Time stalled and reversed itself during 2020—solar days are getting shorter again, for reasons that are unclear.

Pole Stars Of Other Planets?

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

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

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

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

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

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

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

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

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

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

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

North celestial poles of various solar system bodies
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(You’ll notice that I’ve deviated from my habit of using Celestia to generate my astronomical and astronautical illustrations—all my plots this time are superimposed on star charts generated at, which are clearer for my purposes today.)

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

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

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

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

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

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

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

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

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

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

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

Which Way Does Space Station V Rotate?

Space Station V from 2001 A Space Odyssey

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

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

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

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

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

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

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

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

Rotation state of Space Station V, 2001: A Space Odyssey, Scene 1
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I’ve marked the picture above with an arrow to indicate the rotation direction, and have noted the timing of the shot, in minutes and seconds, on my old “Deluxe Collector Set” DVD, from which I’ve made these muddy screengrabs.*

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

Rotation state of Space Station V, 2001: A Space Odyssey, Scene 2
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Next shot, another hero shot of the space station, with the pursuing Orion III appearing later. This is the point at which, back in 1970, I said (rather loudly, I’m told), “Hang on a minute!” The station is now rotating anticlockwise.

Rotation state of Space Station V, 2001: A Space Odyssey, Scene 3
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Then a view from inside the cockpit of the Orion III. Still anticlockwise.

Rotation state of Space Station V, 2001: A Space Odyssey, Scene 4
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There follows a brief view of the instrument panel, in which a wire-frame model of the docking port continues the anticlockwise rotation, and then in the next shot we’re inside the station hub looking outwards. The stars outside are moving in clockwise circles, implying the station is rotating anticlockwise as viewed from inside, and therefore (dammit) clockwise when viewed from our stipulated vantage point.

Rotation state of Space Station V, 2001: A Space Odyssey, Scene 5
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Next shot, and we’re outside again, watching the spaceplane synchronize its rotation with the docking port. Anticlockwise again.

Rotation state of Space Station V, 2001: A Space Odyssey, Scene 6
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Then a view from the Orion III cockpit, now synchronized with the docking port ahead. The stars sweep past in clockwise circles beyond the station, so the station (and spaceplane) are rotating anticlockwise.

Rotation state of Space Station V, 2001: A Space Odyssey, Scene 7
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And this anticlockwise rotation is maintained in the final approach shot.

Rotation state of Space Station V, 2001: A Space Odyssey, Scene 8
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So that’s just a minor curiosity—there are many worse errors in many other films, after all.

Except …

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

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

The switching orientation of the Earth's illuminated limb in "2001: A Space Odyssey"
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(In the Moon’s southern hemisphere, where the action takes place, the correct orientation at lunar sunrise, when the action takes place, is to have the Earth’s illuminated portion to the right, facing east.)

Later in the film, from shot to shot, we see the character Frank Poole not only reverse the direction in which he’s running around the Discovery centrifuge (twice), but the entire centrifuge (and Poole) becomes mirror-reflected:

The bone thrown aloft by the man-ape Moon-Watcher, at the beginning of the film, reverses its rotation direction between the two shots that follow its trajectory. And there are multiple geometrical inconsistencies during the sequence set on the Aries 1B moon shuttle, relating to the orientation of the control cabin.

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

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

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

Scottish Hill Lists: The Donald Revisions

Cover of 1953 edition of Munro's Tables

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

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

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

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

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

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

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

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

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

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

Height-Prominence plot of original Donalds list, with revisions
Click to enlarge
Height-Prominence plot of original Donald Tops list, with revisions
Click to enlarge

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

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

Ordnance Survey one-inch mapping of Corserine and Carlin's Cairn (1920s)
Click to enlarge

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

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

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

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

Map of Donalds and Donald Tops
Click to enlarge

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

Ordnance Survey one-inch mapping of The Cheviot border region (1920s)
Click to enlarge

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

So Donald provided Auchope Cairn with a rather gnomic footnote:

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

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

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

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

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