Neptune Completes First Orbit Since Discovery: 11th July 2011 (at 21:48 U.T.±15min)

Happy anniversary, Neptune!

According to those tables of numbers you get in books about the Solar System, the planet Neptune takes 164.79 years to travel once around the Sun. And Neptune was discovered 164.77 years ago as I write this post (1st July 2011).

This means our blue ice giant has still not made even one full journey around the Sun since being spotted and recognised for the first time by humans.

At some point this month, that 'first' orbit will be completed. The inhabitants of Neptune will be wryly noting the first anniversary of the inhabitants of Earth first realising we were looking at their planet.

Here on Earth, it's an excuse to celebrate a big round icy blue thing in space! Which is cool.

So I was wondering: when exactly is this anniversary? How do we know, and how accurately do we know? There are plenty of blogs and other sites claiming various dates, but few explain where their figures had come from, and none say how accurately they are known.

I thought this would be straightforward to find out, but it turns out it's not... and the quest was fascinating.

If you prefer raw facts to processes and explanations, here's the answer: the first orbit since discovery will be completed within about fifteen minutes of 21:48 U.T. on Monday 11th July 2011.

If you're wondering why this date is different to other dates you might find elsewhere, it's because I've thought it through in a lot more detail (I can get a bit carried away) and done it properly.

At least, that's what I think. Let me know if you disagree.

For those who'd like to read further, here is some suitable musical accompaniment:

To get a date for the completion of an orbit, a number of questions need to be asked:

When, exactly, was the planet "discovered"? (What does that even mean?)

Where was Neptune then? (What does that even mean?)

Where is Neptune now, and how accurately can we track it?

What is a "complete orbit"? Is the "first orbit" any different to any other orbit?

When was Neptune "discovered"?

Neptune is the first planet to be discovered that is definitely not visible with the naked eye. The first instruments capable of rendering Neptune visible were the telescopes made by Galileo in 1609.

Astonishingly – and this is 169 years before even the discovery of Uranus – Galileo himself observed Neptune on 28th December 1612, and recorded it as a star.

He observed it at least once more during the subsequent month. The motion of Neptune across the sky in one month is fairly small (about a sixth of a degree), but there has been speculation that Galileo was quite aware of it having apparently moved between sightings.

It's practically impossible that anyone could have seen Neptune before Galileo. But although he may have suspected something, he didn't consider it important enough to publish or investigate further, so it's fairly clear that he didn't believe it to be a planet.

[Edit 7/7: it is suggested that stars as faint as magnitude 8.0 may be visible under perfect conditions; perhaps even fainter for some individuals. Neptune can reach a peak magnitude of 7.78, which is 20% brighter than a magnitude 8.0 star; so it is theoretically possible that it could have been seen before the invention of telescopes. As there are many tens of thousands of stars with a very similar brightness, it would be unimaginably difficult to pick out even if you knew where to look. I'm not aware of anyone ever seriously claiming to have seen Neptune with the naked eye.]

Many further sightings were made of Neptune, always noting it as one faint star among many. The subsequent discovery story is full of intrigue and controversy. For me, what is fascinating is the new significance of mathematics: in fact, some have gone so far as to say that Neptune was discovered by mathematics before it was seen with a telescope.

Using Newton's Laws of motion and gravitation, astronomers were able to calculate the path Uranus should follow under the gravitational influence of the Sun and the other known planets. But Uranus gradually drifted away from this path. In 1845, John Adams and Urbain le Verrier both hypothesised that it was being pulled by something else. Independently of each other, they used the calculations to determine where this something else was, and suggested astronomers should look for a planet there.

In September 1846, Johann Galle, with his student Heinrich d'Arrest, made the first definitive sighting from the Berlin Observatory, locating Neptune less than a degree from where Le Verrier said it would be.

I disagree with those who say Neptune was "discovered" by mathematics. A hypothesis was made, and known laws of physics were employed to mathematically infer the most likely position of a new planet, assuming the hypothesis and the known laws of physics were valid for what was being observed. In this case, of course, they were absolutely valid, and the prediction led immediately (for Galle) to the discovery.

Johann Encke, who collaborated with Galle on later observations, said in 1846: "this is by far the most brilliant of all the planet-discoveries, since it is the result of pure theoretical researches, and in no respect due to accident."

Mathematics had established itself as a powerful tool for exploring the Universe.

Although Neptune had been seen many times before (including by people who were looking for it), and it had also been located using indirect observation and persuasive mathematical reasoning, it's clear that the first person to definitively observe it and know he was observing it was Johann Galle.

The date was the night of 23rd September 1846, and we can even pinpoint the time. It's in the Monthly Notices of the Royal Astronomical Society, Vol. 7, November 1846, page 155:

The planet's position was first recorded at a time of 12 hours, 0 minutes and 14.6 seconds, Berlin Mean Time. (It does seem strange to record the time to one tenth of a second! I imagine them taking all their readings diligently with a stopwatch.)

There were no national time zones in 1846, let alone any idea of a Coordinated Universal Time. Astronomers naturally used their own local mean time. The Berlin Observatory is located at 13º23'39"E, which means their local mean time would be UTC + 53 minutes 34.6 seconds.

(How accurately this was measured, or how accurately their stopwatch was synchronised to it, I do not know, but I would imagine the uncertainty could be reduced to a matter of seconds.)

So Neptune was first recorded at 23:06:40.0 U.T. on 23/09/1846.

Is that the "discovery" time? It must have been seen by Galle before that time... but was it recognised as being the planet they were looking for before or after the stopwatch reading was taken? Does it matter? Either way, it seems sensible to take this as the discovery time, to remember that the act of discovery is rather more fuzzy than the act of pushing of a stopwatch button.

But if what you're after is a time to let off blue fireworks or Neptune-themed party-poppers... we have a starting point to work with!

Where was Neptune at the moment of discovery?

The table reproduced above also tells you exactly where Neptune was recorded to be when it was discovered, but to be honest I don't know exactly what system they were using to measure it, so it's not really any use as it is.

What we need is system that tells us where planets are at any given time: an ephemeris.

The obvious place to go for this is NASA. Their HORIZONS ephemeris is as good as it gets. If anyone knows where the planets are, HORIZONS does. It's provided by the Solar Systems Dynamics group of JPL/CalTech (the Jet Propulsion Laboratory in the California Institute of Technology), and it's open to everyone.

Now what is best way to tell exactly where a planet is in its orbit, and how will we know when it has returned to that place?

I have to briefly be a little technical and long-winded while I make the case for employing ecliptic coordinates around the solar system barycentre. If you'd rather take my word for it (or if you know it all already), you can skip it.

First of all, there's no point in using the R.A. and Dec values that astronomers use to locate planets relative to the Earth, because we're interested in Neptune's orbit, and it doesn't go around the Earth.

Secondly, for a very similar – but more subtle – reason, we shouldn't use R.A. and Dec values from the Sun (known as heliocentric coordinates), because it doesn't really go around the Sun either. What Neptune orbits, to the extent that it can be said to orbit anything, is the Solar System barycentre, which means the centre of mass of the entire Solar System.

In many-body systems like the Solar System, there are no true periodic orbits. The planets, all the smaller bodies, and the Sun itself, all move in the gravitational field of each other. If a system is dominated by a single massive body like our Sun (which makes up nearly 99.9% of the mass of the Solar System), the other bodies tend to arrange themselves over time into approximately periodic orbits.

Normally, approximately is good enough, and we can just say Neptune goes around the Sun every 164 and-a-bit years. But if we want to know what date an orbit is completed, we're asking for (at least) an accuracy of 1 day in 164-and-a-bit years. Crude approximations are not the way forward.

Neptune's System:

Neptune is part of a small gravitationally-bound system of its own, comprising the planet itself, a large moon called Triton, and a dozen or more smaller satellites. Neptune is 5000 times more massive than Triton, and Triton is 200 times more massive than all the other satellites put together. For the most part, this system consists of Neptune and Triton both orbiting their common centre of mass, plus some flotsam.

The bound system of Neptune and its moons can be thought of as a single rotating thing, making its way around the Sun. And it is the centre of mass – the barycentre – that most closely follows a smooth periodic orbit.

As it happens, in the case of Neptune, this barycentre is only 74km from the centre of the planet. The planet travels at around 5.4km/s around the Sun and in very nearly the same plane, so wherever the barycentre goes, the planet will never be more than 14 seconds ahead or behind it. 14 seconds is a tiny amount of time compared to 164 and-a-bit years, so this is not something that will affect our results. Nevertheless, it's the barycentre that most closely follows a periodic orbit, so it's the barycentre we'll be following.

What Pulls Neptune Around:

The Neptune system (which I'll just call Neptune from now on) is around 4.5 billion km from the Sun. It is pulled around by a collection of massive objects within its orbit, all of which tend, on average, to pull Neptune continually towards the 'centre' of the Solar System. From Neptune's perspective, the giant planets – Jupiter at a mere 0.78 billion km and Saturn at 1.4 billion km – are all pretty close to the Sun. They may pull a bit to the left or a bit to the right, but for the most part, they pull in.

Even Uranus, at 2.9 billion km tends to pull inward, although often at more of an angle than the others. (It may at times be closer to Neptune than Jupiter or Saturn, but is less massive, and always has less gravitational influence than either of the gas giants.)

So the 'centre' that Neptune is pulled in towards is not the core of the Sun, but the centre of mass of all those objects within Neptune's orbit plus Neptune itself.

There are vast numbers of objects outside of Neptune's orbit, but (a) they are pulling in all kinds of directions, and these pulls will tend, on average, to cancel each other out; (b) they are usually a very long way away; and (c) they are tiny – the total mass of all of them comes to barely a dozen times the mass of Neptune's moons.

There is also a strange collection of objects that actually inhabit Neptune's orbit, herded by Neptune's gravity into two little clusters, 4.5 billion km ahead of and behind Neptune. They're curious animals; but they're also very very tiny and very far away.

The Barycentre:

It makes sense to ignore all the other little things, and say that Neptune is, on average, attracted to the barycentre of the Solar System, and is in an approximately periodic orbit around the barycentre of the Solar System.

Below is a diagram showing the motion of the barycentre relative to the Sun over a period of 50 years. (Note that we might more properly think of the Sun as moving relative to the barycentre, but that would be harder to plot). The most prominent effect is the 12-year cycle as the Sun does its tango with Jupiter. But it is obviously being pulled around in other ways too. (source)

Because of this motion of the Sun, tracking an orbit of Neptune relative to the Sun will give rise to some unnecessarily complicated relative motion.

Below are two plots of the distance to Neptune, the first measured from the Sun (in red), and the second measured from the barycentre of the Solar System (in blue), over six centuries from 1700 to 2300 (click on images to enlarge).

The periodic variation in distance associated with any elliptical orbit is clear in both plots. A closer look, however, reveals that the path around the barycentre is much more smooth. (Data from HORIZONS. Thanks to W. Folkner for suggesting this comparison.)

The Motion of the Solar System Through the Cosmos:

We've considered the motion of Neptune and its moons relative to their barycentre, and the motion of the centre of that system relative to the barycentre of the Solar System. What about the motion of the barycentre of the Solar System relative to the rest of the Universe?

I might say more on this in a future post, because I like that kind of thing. For now, I'll just say that that it's completely irrelevant to the motion within the Solar System. The Solar System is in virtually perfect freefall through its stellar neighbourhood and, like any object in freefall, unless there are appreciable tidal effects, what goes on within the system is entirely isolated from the gravitational effect of anything beyond it and unaffected by the nature of the path it follows. There aren't any appreciable tidal effects from outside the Solar System because the distances involved are far too large.

Some people like to think of the planets corkscrewing their way through space. If that's your thing, go right ahead, but it doesn't mean anything in physical terms. If there were an absolute frame of reference relative to which the Solar System could be considered to be moving, that might be useful in some objective way. But there are no absolute frames of reference in space. The choice of frame of reference is ours to make. For what we're interested in, the frame of rest of the barycentre of the Solar System is by far the best one we've got.

How to Measure Position Relative to the Barycentre:

Now we can come back to using the JPL ephemeris, HORIZONS, to establish a location for Neptune relative to the barycentre of the Solar System.

The settings I used are shown below:

These settings select a coordinate system centred on the centre of the Solar System, and use those coordinates to tell us where Neptune is. I've entered the time of the moment Neptune was first recorded (I've entered 23:06 and 23:07 on that date, as there's no scope for entering seconds; but we can always interpolate).

The full output can be seen here (or you can do it yourself). The figures that matter are:

X=25.74504003 A.U., Y=-15.4126128 A.U. and Z=-0.27570192 A.U.

These are the x-, y- and z-coordinates of Neptune. They tell us that if we want to get to Neptune from the centre of the Solar System (on the day it was found), we should go 25 and-a-bit times the Earth-Sun distance in the direction of the ascending node of instantaneous plane of the Earth's orbit and the Earth's mean equator at the reference epoch, then, with the North Star above us, turn right into the plane of the Earth's orbit and go 15 and-a-bit times the Earth-Sun distance that way, then turn down out of that plane and go a bit more than a quarter of an Earth-Sun distance, and there it is.

(The coordinate system is somewhat awkwardly defined, being based on the orbit of the Earth, but its axes are fixed relative to the distant stars. They are the axes of a frame of reference that is at rest with respect to the barycentre of the Solar System and would be inertial in the region of Solar System if the Solar System were not there. And that is all we need.)

So that's where Neptune was then.

When does it return to that point in its orbit?

All we need to do now is to find out when it returns to that point in its orbit, 164 and-a-bit years later.

Of course it will never return exactly to the same point... so we will have to settle for the next best thing, which is to find out when it returns to the same longitude. From the Earth, we can measure the celestial longitude of a planet, which is just how many degrees it has moved along the ecliptic, relative to the position of the Sun at the spring equinox. (The ecliptic is the path of the Sun across the sky.)

The HORIZONS coordinates will give us a celestial longitude for Neptune easily, because X and Y are based on the ecliptic. Using the figures quoted above, the longitude is the inverse tan of Y/X in the range 0º to 360º, which is 329º 5' 33.3"

From the centre of the Solar System, however, a longitude based on the ecliptic is not very useful. From the barycentre, the ecliptic is the current path of the Earth across the sky, but we're interested in the orbit of Neptune. The best general purpose longitude for the Solar System is the angle around the invariable plane.

Unlike the ecliptic or any other orbital plane in the Solar System, the invariable plane is absolutely constant. The paths of orbits of all the planets oscillate slowly about this plane, over tens or hundreds of thousands of years, but the law of conservation of angular momentum ensures that the invariable plane can never be changed by any of the complex dynamics of the Solar System. All orbits are ultimately paths around the invariable plane, with some additional movement above and below it.

The diagram below shows the relationship between the ecliptic (path of the Earth) and the invariable plane. The angle between them is exaggerated for clarity. I've marked the position of Neptune at the time of its discovery:

The longitude I want to use is the angle θ shown on the diagram. θ is the angular position of Neptune relative to the line of intersection of the planes.

The calculations are not worth reproducing in their entirety, but here's a little vector wizardry that looks pretty if you don't know what it means, but is enough to explain what I was doing if you do:

which gives θ = 41º 30' 35.86".

As you can see, this is so close to the angle of 41º 30' 35.6" on the diagram in red (on the ecliptic) that it's safe to say it isn't going to make much difference to the final outcome.

The next job is to use the ephemeris to locate Neptune at various times in July 2011, and find out when it returns to precisely this angle around the invariable plane. The result is:

11th July 2011, at 21:48 and 24.6 seconds U.T.

For the record, I also carried out this process using the ecliptic and using the current orbital plane of Neptune. The results are:

11th July 2011, at 21:50 and 27.7 seconds (ecliptic)

11th July 2011, at 21:47 and 31.7 seconds (orbital plane of Neptune)

These are so close, it clearly doesn't matter at all which definition of longitude you think is best. Agreement within a minute or two after 164.79 years is pretty good.

The output from HORIZONS for the relevant times is summarised below, so you can play around with it if you wish.

The reference to "Coordinate Time" in there tells us that it's the time when Neptune actually was at those coordinates, not the time when the light from Neptune reaches us. Light takes 0.1733 days (a little over 4 hours) to reach us from Neptune, as you can see from the output. If the distance has changed appreciably between two sightings, this can make a difference in terms of when we would actually see an orbit being completed. In our case, comparing the positions of Neptune in 1846 and in 2011 as measured from Earth, it's slightly more distant in 2011, but the difference is less than 30 seconds.

How close will it get to where it was when it was discovered?

At that moment, Neptune will pass within 1.5 arcseconds of its 1846 location relative to the barycentre. As shown in the image at the top of this post, this is less than the diameter of the planetary disk, so it will overlap its original place in the sky.

This corresponds to a distance of 32,460 km in the direction perpendicular to the invariable plane, or any of the other planes if you prefer. This is not the same as the change in the raw Z-coordinate from the ephemeris: the reason for this is that Neptune has moved in a little towards the Sun in 2011, and that alters the Z-coordinate for the same point in the sky. To be specific, it will be 347,750 km closer (0.0077% closer) to the Solar System barycentre than it was in 1846.

Combining these figures, we find that its closest approach to its discovery position is 349,260 km. These values are subject to uncertainty of the order of ±1000 km, as we'll see below.

How accurate are the data that I've used for this date and time?

It's probably fairly clear that giving these events to a fraction of a second is a bit silly. But it would be good to know how accurate we can expect them to be. For example, is it definitely on the Monday (11th)? Could it be out by a couple of hours?

Also, it's just good practice to establish a realistic degree of uncertainty.

The sources of uncertainty here are:

1. Our knowledge of the "time of discovery" of Neptune

2. A degree of arbitrariness regarding the choice of longitude

3. The degree of uncertainty in the HORIZONS ephemeris data we are using

The first one reflects the fuzzy interpretation of the word discovery, as discussed earlier. I don't know how to quantify the fuzziness, but given we have the exact time that its position was first noted, I'd be comfortable with ±10 minutes.

The second one has just been addressed, and we can see that it could introduce a vagueness (to add to our fuzziness) of a minute or two.

To find the uncertainty involved in the ephemeris data, I tracked down this report relating to the "JPL Planetary and Lunar Ephemerides DE405", which is the source of the data for the HORIZONS online ephemeris. The data was collected in its present form in 1997, and the report is dated 1998.

The report compares the positions of various bodies as given by DE405 with the positions from an older ephemeris, DE403. Assuming these are independent, the difference between them gives some indication of their level of accuracy. The newer DE405 was created with far more accurate information on the inner planets than its predecessor due to the various spacecraft that have taken precision equipment there, but the outer planets remain a little blurry.

Figure 8 at the very end of the report (reproduced below) shows that the differences between the two ephemerides for the longitudinal position of Neptune in the period 1846 to 2011 are around 0.1 arcseconds (or 0.1").

An arcsecond is 1/3600 of a degree. At the distance of Neptune, 0.1" corresponds to a little over 2000 km, and Neptune will cover that distance in around 400 seconds (nearly 7 minutes).

I think it's fair to assume that the majority of this difference will be due to inaccuracies in DE403 rather than DE405. But it's not unfeasible that a straight comparison could hide systematic errors common to both. Nevertheless I suggest that an uncertainty of ±5 minutes is reasonable.

Altogether, I believe we have around 15 minutes of uncertainty in the time of completion of the first orbit of Neptune.

With that, I'll give the final result (again):

The first orbit of Neptune since discovery will be completed within about fifteen minutes of 21:48 U.T on Monday 11th July 2011.

Other claims for this date:

12th July has been quoted widely for this event, including on Wikipedia which I cannot change as they do not permit "original research" (if this can be called that) or regard blogs as a notable source of information, and rightly so. Some have even gone so far as to specify a time on that day. This comes from heliocentric ecliptic longitudes as would be observed from the location of the Sun, and don't take into account the motion of the Sun during that period.

[Edit (7/7): The date of the 12th has also been quoted by a few informal NASA sources.]

10th July has also been quoted – the reasoning behind this is much more straightforward. The length of a year for Neptune is often quoted as 60190 days or 164.79 years (sometimes as 60190.03 days and 164.79132 Julian years). So some have simply use Excel or Wolfram Alpha to add 60190 days to the discovery date. If you do this, you'll get the 10th.

I've also seen the 8th July quoted, which is what you get if you use Wolfram Alpha to add 164.79132 years to the discovery date, without realising that this figure refers to Julian years (it is 164.79485 tropical years).

I wanted to see who was right and why, and it turns out that none of them were. (But then, who in their right mind would go to this amount of trouble anyway?)

Is the duration of this first orbit different from any other orbit?

I took a look at a few more orbits using HORIZONS ephemeris. I was pretty surprised at the amount of variation from period to period. The plot below shows how much the period changes over 26 orbits:

Some Neptunian years in this sample are over 50 days longer than others. There seems to be a periodic variation on a scale of several thousand years. And the period of this variation also appears to be increasing – the downward slope on this plot is steeper than the upward – which suggests the presence of more than one periodic driving force for the changes in year length.

As we know, Neptune is being pulled around by the other three giant planets in quite dramatic ways. As a result its orbit is subject to far more variation than that of the Earth.

The idea that "each year on Neptune lasts 164.79 times longer than a year on Earth" that we read in our favourite Solar System text books... it's not the whole truth, really, is it...

So now what?

Having done all that, the question now is: what shall we do on the 11th?

Any thoughts?

Here's a nicer link to this post: http://bit.ly/neptuneorbit

Please quote this link if referring to results from this post.

Arne Naess and the Call of the Mountain

In the last twelve months, this blog strayed somewhat from its tagline and became a vehicle for exposing the pseudoscience of some Hawaiian fruitloop with a cult following. It's been kind of fun, but I'm a bit tired of arguing with people now. I figured it was time for a change.

Today I was reminded by a friend about a man called Arne Naess, a visionary philosopher who was central to the foundation of the deep ecology movement. His writings were a massive inspiration for me when I studied ecology in the '90s, starting with this excellent little book. After having spent several years studying particle physics, which could be seen as an extreme form of reductionism and of abstracting oneself from the world (it needn't! but it can seem that way sometimes), this approach to investigating reality was a real breath of fresh air.

He spent nearly 25 years living in a hut high on a Norwegian mountain, and wrote An Example of a Place as a celebration of it. He saw the mountain in many ways, including as 'a great father'. Naess considered all of these relationships to be as genuine as any material reality, and saw them as calling out to be deeply experienced. He referred to them as being the key to "the establishment of a place as a Place."



I have a great deal of respect for someone who can exemplify and articulate his own radical philosophy with such brilliance. His approach could be described as being deeply spiritual (it certainly is by him), but it doesn't compromise on anything that science has revealed to us about our world.

We seem to be creatures evolved to encompass only our immediate environment and tribe. It is deeply challenging for us to take on board the reality the global reach of our interdependence with each other, with other species, with ecosystems and landscapes and the climate. The scientific facts themselves convey very little of the reality they describe. The reality cannot but be transformative for anyone ascribing to any kind of deeply-considered and heartfelt value system. It doesn't matter how much hyperbole is used, or how much melodrama and over the top CGI they are presented with, or how loudly or how often they are repeated, the facts themselves cannot give us that.

In addition, we're asked to rely on increasingly complex scientific inquiry to hand the current picture down to us, which puts us at an even further remove from it. It shouldn't surprise us if people prefer to turn away altogether from consciously putting their trust in science and devote themselves to the safe haven of simplistic opinion.

One of the primary motivations of deep ecology, as Bill Devall says elsewhere in the documentary (see link at bottom), is "the search for meaning in a world of facts."

We need to build our own philosophy, as an active participant, to find our own personal way of seeking that meaning. The aim is "self-realisation", a way of being in the world that embraces our interdependence with nature, using imagination, deep reflection, appreciation of wildness, fullness of experience and, above all, action.

It stands in contrast to the continual stream of denial that modern life twists our arm to accept on a daily basis. It's so easy to find ourselves falling into the trap of believing that the less attention we pay to the source of everything we eat, drink, breathe and travel through, the better. For some of us – at least some of the time – a kind of wild awareness that this is no way to live becomes a thing to be cherished. For Naess it was far more: experiencing our interdependence as fully as possible lies at the heart of inquiry, and living in accordance with that inquiry lies at the heart of the true Self.

This might look like fluffy subjective ecopolitics, at least at first glance, to someone of a materialist disposition. For me personally, this man's vision stands at the very heart of what science is all about: the attempt to transform the way we see our world, and live in it, in accordance with What Is.

Arne Naess died in 2009 at the age of 96, two years ago as I write this. I feel sad that I found out only today.

The full 51 minute documentary can be watched here, on Daily Motion.

British Birdlists

Today I finished off two neat little bird lists. If you click on either the images below, you can grab yourself a copy.

The first is a straightforward list of all the native birds of Great Britain. If you print it and fold into quarters, you can slip it in your bird guide on a day out and make a note of what you've seen as you go.
I've included all species for which, on average, more than 5 pairs breed or more than 50 non-breeding individuals visit each year. There are 247. I've wanted a list like this for ages, to encourage me to record what I see while I'm out. I'm sure there are similar lists out there, but I couldn't find one exactly how I wanted it... so I've done it myself.

The second list consists of the same 247 species of birds, but with a note of how many of each species there are, because sometimes that's a very useful thing to know...
The note alongside each bird on the list indicates: the number of breeding pairs; whether they're predominantly resident (r) or migratory (m) breeders; how many individuals are present in winter (w) (or a note such as "w+" if winter numbers are 2-5 times higher than summer; a double ++ implies 5 to 20 times); and a note of how many passage migrants (p) there are, where these are significantly higher than both summer and winter populations. Where no note for winter is given, the number of wintering resident adults is similar to the population in summer – roughly 3x the number of resident breeding pairs – and the number of wintering migrants is zero or extremely low.

Most of the information is simplified from the 2006 report by the Avian Population Estimates Panel (APEP).

(Abbreviated notes are included right-hand side of the sheet.)

Adding up the numbers, it seems that there are around 66 million breeding pairs of birds in Great Britain – still a little more than one pair for every human being.

Long may it stay that way.

Rugbinatorics

The Six Nations Championship started today. I tried to watch the first game, but got distracted by the pretty numbers.

The scoring possibilities at any point in the game are 3 (drop goal or penalty goal, g), 5 (unconverted try, u) or 7 (converted try, c). Some scores (e.g. 4) are not possible at all. Some (e.g. 10 = cg or uu) are possible in more than one way. What the blazes is going on? There must be a formula.

It's best to start by ignoring goals and look at what's possible using tries alone. The table below is the set of scores uniquely possible from tries alone – converted or unconverted. The word uniquely implies that we exclude scores like 35 which could be generated in two distinct ways (u×7 or c×5). There are 35 such uniquely generated scores, and each takes the form 5a+7b, with 0≤a<7 and 0≤b<5.

Let's call this set A.

Adding 35 to any of these numbers will give a score that can be made in two ways by tries alone (simply because the 35 can be made in two ways, and the rest can be made in one). Adding 70 will give a score that can be made in three ways... and so on.

To introduce drop goals, we add multiples of 3 to the numbers above. For example, 19 can be scored in 3 ways: it is 10 (in the table) plus 3 goals; or 7 (in the table) plus 4 goals; it's also in the table in its own right (two converted tries and one unconverted) plus no goals.

It's helpful to arrange the 35 scores of set A into modulo 3 subsets – that is, to divide them into those divisible by 3 without remainder, those with remainder 1 and those with remainder 2.

This gives us three subsets, which we can call set 0, set 1 and set 2.

Taking our example of how 19 may be score, we see that it is present in set 1, and that there are two smaller numbers in the set. We know that these numbers – 7, 10 and 19 – can be scored by tries alone, and by adding the required number of drop goals, each can be made up to our score of 19. The three ways of scoring 19 are more readily visible using this table.

Thus, for scores less than 35, the number of ways C(n) of reaching a score n is given by the number of members of the set n mod 3 (i.e. set 0, 1 or 2, shown left) which are less than or equal to n.

If n ≥ 35, we must supplement the result from the partial formula above by considering the two ways in which 35 may be scored by tries (u×7 or c×5).

For example a score of 50 (which divides by three with remainder 2) can be made using any of the unique combinations of tries in set 2 (there are 11 of these) together with the required number of drop goals. In addition, we can score 35 by tries – in either of the two ways – plus 15. The above partial formula gives us 3 ways of scoring when n = 15. With two ways of scoring 35 in each case, we can therefore add 6 further ways to reach 50. Together with the 11 found earlier, we have a total of 17 ways of scoring 50 points.

For a general approach, we can proceed as follows. Obtain a starting result using the partial formula above. Then subtract 35, and obtain a second result, and multiply this by 2. If n ≥ 70, subtract a further 35, obtain a third result, and multiply this by 3. Continue in this way until no further 35 can be subtracted, and this is the final result.

The number of ways C(n) of reaching a given score n in Rugby Union is thus:

where A{n mod 3, ≤ n} is defined as the number of members of n's modulo 3 subset, 0, 1 or 2, of set A (which is the set of integers 5a+7b: 0≤a<7, 0≤b<5; as tabulated above left) which are ≤n. The sum is from r=0 to the integer quotient of n/35.

Example: a score of n=100...

n/35 is 2 and a bit, so we sum from r=0 to r=2.

r=0: 100 mod 3 = 1; there are 12 members of set 1, all ≤ 100. 1 × 12 = 12.

r=1: 65 mod 3 = 2; there are 11 members of set 2, all ≤ 65. 2 × 11 = 22.

r=2: 30 mod 3 = 0; there are 7 members of set 0 which are ≤ 30. 3 × 7 = 21

The total number of ways of scoring 100 is 55.

It would be straightforward to list all 55 should we wish. For example, there are 21 entries in the third (r=2) part of the total above: three for each member of set 0 up to and including 30. If we chose, say, the second member, which is 12 (corresponding to uc in the notation set up at the beginning), we can pinpoint the three ways of scoring 100 associated with it. The 12 from set A in this case represents scoring 30 by uc g×6 . The remaining 70 may be either u×14 or u×7 c×5 or c×10 – the three combinations represented in the sum. The precise contribution of any member of each set included in the steps of the sum can be identified and listed by this process.

The result follows extraordinarily closely to a quadratic function:
Using this approximation for C(n), and rounding to the nearest whole number, gives the correct answer in over 80% of cases, and I've yet to find a value of n for which it is out by more than 1. (For n=100, it gives a neat 55.)

I can see where the 1/210 comes from. Where n is a multiple of 35, summing (r+1) from r=0 to n/35 - 1 gives (n/35 )(n/35 +1)/2, then multiply this by the average full set of A, which is 35/3, and you have your 1/210 coefficient of n². But I can't for the life of me figure out why there's a consistent coefficient of 1/14 for n. So there's a puzzle for you.

So now you see how hours of fun may be had at a rugby union match without any need to understand of the rules of the game. One might be tempted to conclude that the game hardly need be played at all. But if it is, then a team being 4 points behind towards the end of a match would do well to be aware that, by my calculations, were they to score one more try, they'd be winning.

The Upxkcd Number

Following on from the previous post, it seemed fitting to launch this blog's own stupidly big number.

Introducing the Up Function, Up(n): a Conway arrow chain (a very elegant, one could even say simple, way of producing very large numbers) containing n copies of any number n:

Up(1) = 1
Up(2) = 2→2
Up(3) = 3→3→3
Up(4) = 4→4→4→4
etc.

They look innocent enough, but the values of Conway chains grow real fast: Up(4) is already far bigger than Graham's Number.

What's more, there's nothing to stop us using the function more than once, thus:

Up⁽²⁾(4) = Up(Up(4))

The brilliant xkcd webcomic created their own xkcd number by sticking Graham's number in the wonderfully explosive Ackermann function:

xkcd = A(g₆₄,g₆₄)

Up likes the xkcd number, but – remarkably – even this is much smaller than Up(4). Indeed, in Conway notation, xkcd is barely distinguishable from g₆₄. Therefore, naturally, we're proposing to make use of our nice new function to turbo-charge it up a bit.

Cue the Upxkcd Number, U_x:

U_x = Up^(xkcd)(xkcd)

I think that's enough to be getting on with!

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Postscript

...however, if you do want more, bigger, and better, and you want it now, here are the best pages I've found...

Scott Aaronson's excellent article on the game of "who can name the bigger number"
Robert Munafo's encyclopaedic website on large numbers. It starts on the ground, and it goes all the way to the top.
"I have this vision of hoards of shadowy numbers lurking out there in the dark, beyond the small sphere of light cast by the candle of reason. They are whsipering to each other; plotting who knows what. Perhaps they don't like us very much for capturing their smaller brethren with our minds. Or perhaps they just live uniquely numberish lifestyles, out there beyond our ken." – Douglas Reay

Cube 2

Once upon a time, there was a man called Graham, and he was wondering about cubes. By the time he'd finished wondering, he'd come up with the largest numerical answer to a reasonable question* in the history of the world.

Here's the game. Draw a cube. Draw lines joining each corner to every other corner. Colour each line either blue or red, as you wish. Your cube might look like this:


Now look and see if there are any red or blue ☒s in your cube.

Technically, we're looking for a single-coloured, planar K₄ graph. Which means a ☒.

In the picture above, in amongst the tangle of blue and red, there is one red ☒ and no blue ones. The red one looks like this:


(There will be twelve ☒s in a single-coloured 3D cube, so twelve possible places to find a red or blue ☒ in a bi-coloured cube.)

Now, can you make this cube into one that has no ☒? In this case, it's easy. Change any one of the red lines of the ☒ above to a blue line, and the cube will have no ☒s anywhere in it.

So it's easy to construct a 3D cube with no ☒s. Things get trickier when you look at cubes in more than three dimensions (sometimes called hypercubes). Ronald Graham and Bruce Rothschild, in 1971, found cubes in four and five dimensions having no ☒s. Six dimensions was too tricky at that time. However – and this gives an indication of the complexity of the problem – with modern computing, this has more recently been extended to ten-dimensional cubes... but no further.

What Graham and Rothschild wanted to know was this: if you make your cubes in ever higher dimensions, is it always possible to make a cube with no ☒s?

How they answered this, I can confidently state I will never know. Mathematics is a strange and beautiful world, full of many lands I'll never have either the will or the means to travel very far in. This particular land is known to devotees as Ramsey Theory, belonging to a branch of mathematics called Extremal Combinatorics, and I have come across some neat little problems to explore there. But if we're talking travelling, I never made it out of the airport. All I can say is that in 1971, the year of my birth, our explorers proved that the answer is no, it isn't. At some point, once you get to a certain number of dimensions, there will have to be a ☒somewhere in the cube: there is no way of making a bi-coloured cube in so many dimensions without it containing at least one ☒.

So what's the maximum number of dimensions for a cube with no ☒s? As yet, we don't know. Graham and Rothschild – as noted earlier – showed that it was at least six.

A few years later, Graham went on to prove that it's certainly no more than the number we now know as Graham's Number.

So it's got to be somewhere between those two.

What is Graham's Number? Well, it's big – rather too big to describe succinctly. The only way to get there is by a set of stages, and by introducing arrow notation (↑), so that is what I'll try to do here.

Graham's number is built up using lots and lots of the number 3.

One arrow: ↑ (exponentiation)
3↑3 is simply 3³ – that is, raise 3 to the power of 3. This is "3 times 3 times 3", which is 27.

Two arrows: ↑↑ (tetration)
3↑↑3 is "3 to the power of (3 to the power of 3)", or 3 to the power of 27, which is 7,625,597,484,987.
3↑↑4 is 3 to the power of 7,625,597,484,987 (already greater than the number of random monkey keystrokes required to reproduce the Complete Works of Shakespeare – or, for that matter, the entire contents of the Bodelian Library – in one burst), and 3↑↑5 is 3 to the power of that.
And so on.
(seven trillion steps later...)

Three arrows: ↑↑↑
3↑↑↑3 is 3↑↑(3↑↑3), which is: "3 to the power of (3 to the power of (3 to the power of (3 to the power of ... (3 to the power of 3)...)))". The dots signify the omission of repeated elements such that, if written fully, the number 3 would appear 7,625,597,484,987 times here.

3↑↑↑4 is 3↑↑(3↑↑(3↑↑3)) – as above, but the number 3 would appear 3↑↑↑3 times instead.

3↑↑↑5 is 3↑↑(3↑↑(3↑↑(3↑↑3))) – as above, with the number 3 appearing 3↑↑↑4 times.

Continuing in this way we get 3↑↑↑6, 3↑↑↑7... and eventually, after 3↑↑↑3 such steps, we get to 3↑↑↑(3↑↑↑3).

Four arrows: ↑↑↑↑
3↑↑↑↑3 is 3↑↑↑(3↑↑↑3)
And this, 3↑↑↑↑3, is the starting point for Graham's Number. We call this g₁
g₁ = 3↑↑↑↑3.

We could plough on like this to get to five arrows – obviously this would be an even more vast set of steps up for our number. But we have to move a lot faster than this, otherwise we'll never get there.

So imagine not five, not six, but g₁ arrows...

3↑↑↑↑3 arrows
g₂ = 3↑↑↑...{g₁ arrows}...↑↑↑3.

Even more arrows!
If we use g₂ arrows, we get g₃.
If we use g₃ arrows, we get g₄.

And so on, another sixty times... until we arrive at:

g₆₄= 3↑↑↑...{g₆₃ arrows}...↑↑↑3.

And that's Graham's number.

A cube in this number of dimensions would have 2^g₆₄ corners. And each one joined to every other one, so square that and divide by two, and you have a lot of blue and red lines. Graham proved that there would definitely be a ☒ in there somewhere.

So, returning to the question: what's the maximum number of dimensions for a cube with no ☒s? The answer was "well, it's at least six, but not more than Graham's Number". This must be the least precise answer in the history of mathematics. And the 21st Century update – using computers to narrow of the range to "well, it's at least eleven, but not more than Graham's Number" – doesn't, on the face of it, make it a great deal better.

The thing is, there's something magical about the unsolved problems in mathematics and the quest for their solutions – especially the ones that are on the edge of solubility like this one. The efforts of those attempting to solve the problem thus far might seem fruitless. Indeed, it might be hard to imagine what the point would be even if the answer to our question were ever found. But for anyone who's ever allowed themselves to be fascinated by these things, they soon quite naturally appear as ends in themselves. Their point is in their own poetry, their own mystery, which the application of precision, logic and reason only magnifies.

This particular question can be re-stated in terms of sets of people joining committees, rather than bi-coloured hypercubes. It might also be related to computing (as I alluded to here). Who knows what use it might turn out to have. I'm not sure I could care less about its uses. The purpose is in the thing itself, on its own terms and in its own world. In the sensing, by whatever peculiar human faculty allows it, of the fantastic nature of these worlds – and in the marvelling at the minds of the peculiarly eccentric beings who explore them.

*by a 'reasonable question', what I really mean is a question that makes sense without being framed in terms of incomprehensibly large numbers. So, no, "what's Graham's number plus one?" doesn't count...

The Wrath of Bob

I went on a long, long journey to the end of the world.

Camped at Wrynose Pass, Lake District, with my little car.


Road to Ben Nevis from Fort William. Ben Nevis is on the left. The road runs along Glen Nevis, an astoundingly beautiful region and somewhere I'm very keen to return to explore properly before long.


The empty, dream-like A838 in the Far North


Gorse-lit view from the Borgie Forest, near Tongue


View of Strathmore River from Ben Hope. I scampered from a layby by the river at 9m altitude to the summit at 927m in less than 90 minutes, and loved it (this would be a snail's pace for a fell-runner, but very exciting for a Bob in fell-running shoes). There is mileage in this scampering business. Also it's a Munro in the bag! One down, 283 to go.


Bay of Keisgaig, on the 28-mile trek from Blairmore to Cape Wrath and back, via the idyllic and remote beach of Sandwood Bay.

Unless you take the passenger ferry and tourist shuttle bus (which gives you a fine half hour at the lighthouse before whisking you back again), I can report that Cape Wrath is an Absolute Bugger to get to.

Strangely, and some might say suspiciously, I managed to delete all but three of the photos I took on the Cape Wrath walk, and therefore have no evidence at all to offer of having made it there at all. I remember impossibly high cliffs, strange, straggly creatures, and sleeping in a bivvy bag in the clouds, a lighthouse beam sweeping above me.

So maybe it was a dream after all.
And the strange, straggly creature was me.

In relation to my quest for the UK's darkest skies, as related here, I can say that (a) it's mostly cloudy in NW Scotland anyway, and (b) it's close enough to the Arctic Circle that, when I chose to go at the beginning of June, it doesn't actually ever get fully dark at all. Which I would have realised if I'd thought about it. So, in addition to accidentally deleting my best photos, it was also a complete waste of time, and I feel very silly.

Luckily, it was also an incredible adventure and a life-changing journey. Paid my respects to the extreme NE, N, NW, W and Up points of my home island, added the dastardly Bonxie and the snowy Ptarmigan to my life list of birdies, did things I really didn't believe I could do, and came back all sprightly and nourished and Full of the Bigness of Stuff.

Down

I went to the highest unbroken waterfall in England. Higher than Wales' best, higher even than Ireland's. But all that is visible to a normal human being is a very disconcerting hole in a Yorkshire hillside.

The sheerest cliff tops have nothing on this place. I didn't kneel to peer over - I couldn't stay near the edge for longer than a few seconds.

Planck Monkeys

Give a monkey a typewriter, and if you wait long enough it will type out the Complete Works of William Shakespeare.

This is the infinite monkey theorem.

I've been thinking about how much monkey typing is needed, and decided to do some tests. I acquired a small gedanken monkey that can type randomly at 48wpm (4 keystrokes per second) without stopping for food or rest. I gave my monkey a small typewriter with 26 letter keys (capitals only) and a space bar five times as big as the others, and (forgiving soul that I am) excused it from any punctuation. I wanted to see if he would type the word MONKEY at some point over the next 5 years.

In fact, he did, and it was hidden away like this: "... SJKBV SDG FMMONKEYP SRGH DKAFJI ..." near the bottom of page 230 of volume 798 of a thousand volumes of monkeyprint. I have built a library to hold these volumes for the benefit of future generations.

To clarify my "it is likely": the probability of the word appearing in 5 years is about one half. So I was lucky: it's 50:50 whether the word MONKEY would be found at all.

I decide to raise the bar to MONKEY WANT BANANA. (If my monkey had typed this, I would have fed it. But it didn't, did it.)

Monkeys have been around for about 50 million years. What if I had acquired all the monkeys in the world from the very start of monkeys, and had them type continuously at 48wpm for 50 million years? If anyone can help me with how many monkeys there are and have been over the last 50 million years, please let me know. I'm going to suggest 10 billion, on average. They would have produced a hundred thousand trillion tons of monkeyprint by now. Might they have managed a MONKEY WANT BANANA? Yes: the probability that one of them would have done it is an impressive 89.5%.

This is promising. Now, what if the universe was filled with tiny monkeys right back to the Big Bang, typing as fast as possible until now?

So, how tiny? One per atom? That would be good. And how fast can they type? A billion keystrokes per second? Now we’re talking! We have a 98% chance of one of them managing a SHALL I COMPARE THEE TO A SUMMERS DAY THOU ART MORE LOVELY AND MORE TEMPERATE. 98% is good! But add one more word, and the probability of one of them knocking that out drops to one in fifty million, which is not so good.

I have to make my monkeys smaller! One per atom is not very many, as most of the universe is intergalactic space with less than one atom of matter per cubic metre. There should be more monkeys than this. I shrink the monkey until it occupies the smallest possible space there is. Quantum gravity physics tells us that there is a smallest possible distance, known as the 'Planck length'. So of course I want to use this.

If you were to try to examine anything on a smaller scale than the Planck scale, the sheer effort required to do the examining would be so great that you’d create a tiny black hole there, bigger than the size of the thing you were trying to see. (The black hole would vanish very quickly as soon as you stopped looking.) The same thing would happen if you tried to chop anything into smaller pieces than the Planck length – you’d make a black hole and end up with bigger things than you started with. In fact, nothing can happen on a smaller scale than this – at least, not without radically altering the laws of physics so much that you would all but obliterate any meaning in the word 'smaller'. Experimentally, no-one has come anywhere near getting there, which is probably just as well.

As well as being the size limit on smallness, the Planck length is the fundamental unit of distance – the only one that is not made up with reference to anything else, as are metres and miles and the like. This elegance and purity makes it very important in physics.

I can't see it catching on though – it's way too silly. You wouldn't really want to measure your inside leg in Planck lengths. One Planck length is a hundred billion billion times smaller than the distance across a proton, which is a tiny speck of a thing that sits at the very centre of a hydrogen atom (the smallest atom).

Back to the monkeys.

I’m going to divide the universe into Planck-sized regions, and put a monkey in each one. You will ask what the monkey is made of, when nothing can be smaller than the Planck scale, and I will say that it is not made of anything – it is a single, fundamental monkey particle. One in every Planck sized region of space. These regions are very small – there will be nearly as many monkeys inside the space occupied by a single atom as there are atoms in the universe. And there will be monkeys in the spaces not occupied by atoms too.

And they will type faster. How fast can a thing happen? Just as there is a shortest possible distance, there is a shortest possible time, and it’s called the Planck time. The Planck time is how long it would take you to cover one Planck length if you travelled at the speed of light.

My monkeys will type at a rate of one keystroke per Planck time.

They will type so fast because the energy required to confine a monkey to such a small region will make the monkey extraordinarily hot.

You will ask what the typewriter is made of, and I will say it is not separate from the monkey: typing is what a monkey particle does. (I don’t know what happens to the letters that the monkeys type. There is no room for them or anything else, as the cosmos is jam-packed with hot monkey particles. But I’m not going to let this stop me.)

So, from the Big Bang, with a monkey in every last tiniest unit of space possible, typing at the fastest speed there is, for the entire history of the growing Universe, and do we have a deal?

Yes! The first four lines of the sonnet “SHALL I COMPARE THEE TO A SUMMERS DAY THOU ART MORE LOVELY AND MORE TEMPERATE ROUGH WINDS DO SHAKE THE DARLING BUDS OF MAY AND SUMMERS LEASE HATH ALL TOO SHORT A DATE” will be knocked out somewhere in the cosmos several times a second!

This is good! In fact, every few dozen thousand years, it’ll come together with the next word – SOMETIME – to boot. Will we ever get the next two words (SOMETIME TOO)? We might be lucky – there’s something like a one in three chance in the age of the universe.

So there we are. One in three. Ladies and gentlemen, I give you, from the Monkeys of the Cosmos, four lines and two words of a sonnet!

...FOESZH GIMCED GHN ASIO AKHPS WRSHALL I COMPARE THEE TO A SUMMERS DAY THOU ART MORE LOVELY AND MORE TEMPERATE ROUGH WINDS DO SHAKE THE DARLING BUDS OF MAY AND SUMMERS LEASE HATH ALL TOO SHORT A DATE SOMETIME TOOSFB L FPGPAAO XUN WVIKGXWS TX FSAOL PABK...

I don't know about you, but I think that's rather impressive.

If you want the Complete Works, as the theorem says, you'll have to wait.

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