Dark Energy - the basics

The Nobel Prize for Physics was awarded yesterday for the discovery of dark energy in 1998.

What is dark energy? Do we really need to just accept that it's complicated and freaky, unless we're boffins?

I don't think so.

It's an apparently constant 'energy' per unit volume of space, which causes space to expand.

In 1915, Einstein developed a theory of gravity, out of essentially nothing more than the assumption that the laws of physics in free-fall are the same as those without gravity.

One of his very clear conclusions was that dark energy – which he called a "cosmological constant" – could be a physical aspect of gravity. It emerges naturally from following through the logic from that one starting point.

The question of how to follow the logic is the tricky bit... but unless you're masochistic or deeply suspicious or fantastically curious and patient, it's ok to just think of it as something that's been accepted as a logical implication for nearly a century, and go with it.

So it's been there in the standard modern theory of gravity since the very beginning, although there was no evidence that it was anything other than zero until 1998. It's not a new thing - it's just a part of the nature of the force of gravity.

It's the part of gravity that causes space to expand so that very distant things accelerate away from each other.

I think calling it 'energy' and 'dark' makes it sound freaky and mysterious and new and unknown.

What's new is that it's been measured. Nobody expected it not to be zero, but it's not; and now we can't just pretend it's not there any more. And this is what the winners of the Prize – Perlmutter, Riess and Schmidt – with the help of many, many others, have given to the world.

It's not some kind of bolt-on to the laws of physics to explain something nobody understands – not in any way. It's a sophisticated measurement of something surprisingly simple, and what's more it's an interpretation that's been verified by many other independent observations of what's out there.

And yes, lots of research needs to be done to check that it's not this kind of field or that kind of modification or that it's doing this or that crazy thing, which is very important and great fun for the scientific community... but as it stands, there's no evidence for anything beyond good old gravity, doing its good old Einsteinian thing.

And if it turns out that it is as simple as that, then it means the fate of the universe is that clusters of galaxies will separate over time until they're no longer visible to each other.

Within clusters of galaxies, which is where we live, dark energy doesn't really do anything at all. (Apart from handily ensuring that the entire rest of the universe won't come falling in and crush us at some point in the future!)

I (try to) study this stuff, so it's fascinating to me. The details of the logic of the theories can be daunting, but I like to think that the real substance of ideas like this are accessible to anyone. But perhaps this belief just helps me feel less isolated from those not mad enough to dive into it all in detail.

If you kinda knew all that, and have been trying to come to terms with how it all fits together and the various questions and apparent paradoxes it throws up, the excellent Sean Carroll has provided a very clear and detailed FAQ.

Census of the Dead as the Living Reach 7 Billion

A question for Hallowe'en. For each person alive on Earth, how many ghosts are there? And what are they like?

If you want to skip the numerical details, click on Ghosts.

Contents:

The population of the living

The population of the dead

Ghosts

The Population of the Living

The human population of our planet has increased at a tremendous rate over the past few decades, and is poised to exceed 7 billion.

You may find people are telling you when this milestone be reached, but let's be honest: nobody can truly say they know what year it will happen, never mind what day. We may well already have passed it.

The world population at any time is probably known to an accuracy of little better than ±1%. Wikipedia's page of world population estimates gives a selection of independent estimates of the world population at different years, and the agreement for recent years is around ±0.5%, which means an uncertainty of ±35 million people.

The rate of increase is something like 217,000 people per day, so if we wanted the date when the population of the world exceeds 7 billion, realistically we're looking at an uncertainty of nearly six months either side.

Nevertheless, it's sometime about now, and it's a significant event, so it's good to have a date to focus our attention. The United Nations has chosen 31st October 2011 as the day on which world human population will nominally exceed 7 billion.

There's a lot to be said about the implications and consequences of this vast and still growing number of people. You'll hear plenty about carrying capacity, fertility rates, resource crises and environmental concerns over the coming weeks, along with the inevitable speculation about what drastic measures might be needed to deal with it.

So far as I know, the only large-scale socio-political force that is known to put the brakes on population growth is long-term investment in and commitment to the education and health of women. Which is hardly a drastic measure. Hopefully the news will focus on this, and how we can help; and hopefully at least some journalists will try to recognise other valid perspectives instead of just pressing the easy sensationalist hype button. Let's see.

That aside, what intrigued me today is something altogether different.

(The choice of date by the United Nations may have something to do with it.)

The Population of the Dead

If there are 7 billion people alive on Earth, how many people have ever lived?

This report by the Population Reference Bureau, which is discussed in this article in Scientific American, gives us an answer. (There may well be alternative studies which are more recent, more thorough or more objective in some other way, but I haven't seen them. I'd welcome comments from anyone who can put this in a bigger context.)

Their result is that 106.5 billion people were born between 50,000 BC and mid-2002.

Is this realistic?

The authors don't give an estimate of the degree of uncertainty in their figures, but they have modelled the population based on assumptions that appear to be reasonable.

The most significant assumptions are the very rough values they've chosen to use for birth rates, which are listed in the table on their report. To my eye, they seem to be on the high side, implying very high rates of infant mortality. If anyone has any clear ideas as to how world average annual birth rates per 1,000 of the population would have varied over history and pre-history, I'd be interested to hear them. I could re-run the authors' model with different birth rates if there are good reasons to do so.

Meanwhile, I'm taking the view that their figures are fine, though perhaps veering towards the high end of what's feasible, and presumably with an uncertainty of a few tens of billions.

One thing I will briefly look at – because it's interesting – is the date range of 50,000 BC to mid-2002 used in the report.

1. Adding in the period mid-2002 to 31st October 2011

This is fairly easy to do. The worldwide birth rate over that period has decreased steadily from around 20.8 to around 19.2 per year for every 1,000 of the population, so let's call it 20; the population has risen from 6.22 to 7.0 billion, so let's take an average of 6.6 billion over a period of 9.25 years. This gives a figure of 1.22 billion births, which we should add to the result.

2. Adding in the period before 50,000 BC

If we extend the period back to the origins of Homo sapiens around 250,000 years ago, or of the genus Homo 2.3 million years ago, or to our diversion from chimpanzee cousins around 5 million years ago, even with a tiny population, the total number of births during these enormous timescales would substantial.

The population is thought to have dropped to around 10,000 in 70,000 BC due to extreme climatic changes following a catastrophic volcanic event. Before this period, the number of humans is estimated to have been around 50,000.

Using the authors' suggestion of a birth rate of 80 per year for every 1,000 of the population, the number of births prior to 80,000 BC is in the region of 4 billion births - and deaths - per million years. If we went back five million years, we'd be adding something like 20% to the total. But that's pushing the limits of what we consider to be human further than we might wish: if we went any further than that we'd implicitly be including chimpanzees. It's a matter of taste where we choose to stop, but if we're talking about humans, it definitely won't be further back than that.

To summarise:

107.7 billion people (give or take a few tens of billions) were born between 80,000 BC and 31st October 2011.

Perhaps 20 billion homininans were born in the 5 million years prior to that, of whom around 0.8 billion were Homo sapiens going back to 250,000 BC. If we stretch our human period back to around a million years to start including some close relatives, we bring the total to 111.5 billion.

Of the 111.5 billion who have been born in the last million years, 7 billion are alive, and 104.5 billion are not. Which means...

For every human being alive today, there have been about 15 who have died.

Ghosts

So each of us can be allocated 15 ghosts of humans past. Assuming we are to share them equally, which seems only fair.

Who were they?

Using the table in the report, our ghosts were born as follows:

7.4 of them before 1 AD

3.8 of them between 1 and 1200 AD

1.8 of them between 1200 and 1650

1.0 of them between 1650 and 1850

1.0 of them after 1850

The last figure tells us that, of all people born since 1850, half are alive now. (The total number of births since 1850 comes to 13.92 billion.) Which means each of us can have only one ghost of a person born since 1850.

We each get one ghost of someone born in the period 1650-1850 too.

If we allocate a period to each ghost, the distribution would look something like this:

1 born after 1850

1 born between 1650 and 1850

2 born between 1150 and 1650

3 born between 200 AD and 1150

4 born between 1400 BC and 200 AD

3 born between 7000 BC and 1400 BC

1 born between 1,000,000 BC and 7000 BC

The last guy in the list may or may not have actually been Homo sapiens – it starts to get blurry here. And there may even be another couple of fairly closely-related hominids kicking around from earlier still.

(The splits at 1400 BC and 7000 BC follow from the assumption of steady exponential growth between 8000 BC and 1 AD, which the authors of the report used in their model.)

The other thing we can say, of our 15, is it's likely that:

5 or 6 survived to adulthood

3 or 4 died in childhood after the age of one

and 6 died before reaching the age of one

That's two baby boys and two baby girls from earlier than 200 AD

and one baby boy and one baby girl from after 200 AD.

And those are our ghosts.

As the Northern Hemisphere approaches the dark nights of winter, and the veils between the worlds of the living and the dead are at their thinnest, as our Pagan friends would say, we could take this opportunity to each bear in mind our 15 ghosts.

To consider the lives they had, the world they lived on, the world they've passed down to us, and what we might do now that we're in it. To consider how we might choose to leave our world for those who will later think back to us, wondering what we were doing and thinking as we watched the world pass 7 billion.

I'm not an artist – numbers and abstractions spark my imagination. But it occurred to me that some people might want to sketch their ghosts, in clothes or settings for their period; or else do some unpredictable creative thing to acknowledge them. If you do, send me a link.

And happy Hallowe'en – or however it is that you choose celebrate this time of the year.

Don't have nightmares. They were humans just like us, after all.

Be good to your ghosts and they'll see you right.

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.