# Physics Stop

## The equation of time strikes againMarcus WilsonJun 17

Some of us are rather looking forward to getting to 22 June. That's when the days get longer again. Yes, the reality is that no-one's really going to notice much difference for a while, but it's encouraging to think that the days will be getting lighter again, if only by a little bit. Don't confuse that with temperatures getting warmer – the coldest day (on average, of course) lags the darkest day quite considerably. Here it's around the end of July

But there's an interesting effect going on with sunrise and sunset. We've already had the darkest evening (hooray!) yet the darkest morning is still to come. Look at the sunrise and sunset times (for Hamilton) on the MetService website: today we've had sunrise at 7.32am and with cloudless skies the sun may stay out all the way to sunset at 5.07pm. But tomorrow sunset is recorded at 5.08pm (later!) and on Saturday sunrise has shifted to 7.33am (also later!). How can that be?

The point here is that the length of a day, meaning now the time between when the sun is at its highest to the next time the sun is at its highest, is only 24 hours on average (for some periods of the year its greater, for some periods is less), and isn't equal to the time it takes for the earth to spin once on its axis.

Let's take this last point first. It's solar midday, meaning that the sun is at its highest. Now, let the earth rotate exactly once on its axis. Do we get back to solar midday the next day? No. That's because, in the time taken for the earth to rotate once, it has also moved along its orbit (about 1/365th of the way around). That means it's got to spin a little bit more before the sun reaches its higherst point. The time to spin once (the siderial day) is about 23 hours 56 minutes – four minutes less than the mean solar day. Note that 4 minutes x 365 = 24 hours – which means one more revolution than you might expect  - the earth actually does 366 and a quarter revolutions each year.

However, the movement along the earth's orbit in a day is only on average 1/365th of the orbit. When the earth is closest to the sun (called perihelion – 3 January at present) it moves faster. That's Kepler's second law. When it's further away (at this time of year) it moves slower. That would mean that in January, we should perceive that solar midday gets later every day by our watch (since the earth needs extra time to spin that extra bit more), but that in July, the solar midday should be getting earlier. However, that's not what is observed. Our prediction for January is true, but for July it's the other way around – solar midday actually gets later as measured by our watches.

There's another effect going on.  This is because the earth is tilted on its axis. However, it's quite tricky to explain why that makes a difference.  Consider the transition from winter to summer, in the southern hemisphere. If we look at the position of the sun at sunrise and sunset, we see it move southward from one day to the next. What is significant is that at sunset the sun is further southward than for the previous sunrise. That gives us a shift in the measured time between solar midday and the next solar midday. A better explanation is given here.  This effect is 'zero' at the solstices and equinoxes, and does two cycles a year. Add this to the effect of Kepler's second law, and we get the odd-looking curve that is called 'the equation of time', and means that, at present, each solar day is slightly longer than 24 hours, giving both ligher evenings and darker mornings.

You can see a net result displayed on the ground under the sundial in Hamilton Gardens. The elongated figure-of-eight is called an 'analemma'. It will show you the position of the tip of shadow of the pole at different times at different times of year.

## When energy conservation doesn’t add up (or does it?)Marcus WilsonSep 23

In the last few weeks holes have been popping up all over Cambridge. They are being dug by 'ditch-witches'  - pieces of machinery designed for making small-diameter tunnels for cabling – as part of the installation of fibre-optic cables for the much vaunted ultra-fast broadband. A ditch-witch is about the ultimate in machinery-obsessed-toddler heaven. We've been avidly following their movement around the Cambridge streets, or at least the youngest member of our family has. They went down our street about seven weeks ago, and since then have been tracking southwards. I'm tempted to slip a GPS locator beacon on them and then write a ditch-witch locator app to help all those stressed parents cope with constant demands to find them.

So, Sunday saw Benjamin and I get on the bicycle and go on a ditch-witch hunt. (We're going on a ditch-witch hunt… We're going to catch a BIG one…we're not scared…). And, much to my relief, we found them, resting quietly on Thompson Street.

But this entry isn't about ditch-witches or diggers or cranes or other large pieces of machinery, it's about what we saw on the way. On the front lawn of one house, there was a teenage boy practising 'barrel walking'. He was standing on a barrel, and rolling it forward and backwards around the garden. He was obviously reasonably skilled at this since he had some pretty good control of where he was going.

An interesting observation is that to get the barrel to roll forwards, the rider has to walk backwards. That must feel a little disconcerting. To get the barrel (and you) moving forward at say 2 km/h, you have to walk backwards at 2 km/h . That's because the bottom of the barrel, in contact with the ground, is instaneously stationary, so if the centre is travelling at 2 km/h forwards, the top of the barrel must be going 4 km/h forwards relative to the ground. In order for you to go at the pace of the centre, 2 km/h forwards (and stay on top), you therefore need to go 2 km/h backwards with respect to the top of the barrel. In terms of mathematics: your speed relative to the ground = 2v – v = v, where the 2v is the speed of the top of the barrel, the  '-v' is the speed of you relative to the barrel, and the 'v' is the speed of the centre. Go it?

That kind of relationship crops up quite a bit in physics. I've talked about a case before – when a satellite in orbit loses energy because it hits air molecules, it speeds up. Uh! How does that work? It's because, as it loses energy, it drops to a lower orbit, one with less potential energy. But lower orbits have higher orbital speeds. It turns out that the loss in potential energy is exactly double the gain in kinetic energy. That is, if the satellite loses 100 J of energy, It's made up of a gain of 100 J of kinetic energy and a loss of 200 J of potential energy. It's another '2 – 1 = 1' sum.

There's also the neat but confusing case of a parallel plate capacitor at constant voltage. Let's say a capacitor consists of two large flat plates, a distance of 1 cm apart. The plates are maintained at constant voltage of say 12 V by a power-supply (e.g battery). This means that the plates have opposite charge, and so attract each other. (To hold them at constant distance, you have to fix them in place somehow). Now, consider pulling those plates apart. Since they attract each other, it is clear that you have to do work on the system to do this. One might therefore expect that the energy stored in the capacitor has gone up. But no. Do the calculation, and you'll see that the energy goes down. (Energy stored = capacitance times voltage squared, divided by two. The voltage stays the same, and since the capacitance is inversely proportional to plate separation, increasing the separation will decrease the stored energy.) Uh! Where does the energy go then? In this case, you have to consider the power supply. What happens is that you are putting energy back into the battery, by causing a current to flow backwards through it. It turns out in this case, that the work you need to do is exactly half the energy that goes to the power supply. The other half comes from the loss in energy stored in the capacitor. So, if we put in 10 J of energy, we lose 10 J of stored energy in the capacitor, and we gain 20 J of energy in the power supply. So, again, we have the '2 – 1 = 1' sum.

So, for every kilojoule of energy burned by the ditch-witch, doe the toddler also burns a kilojoule, thus meaning 2 kilojoules of heat end up in the air?  (As neat as it would be if that were true, I don't think the actual figures will come close).

## Seeing in the darkMarcus WilsonMay 21

No, nothing to do with carrots and vitamin A I'm afraid.

With dark evenings and mornings with us now :(, Benjamin's become interested in the dark. It's dark after he's finished tea, and he likes to be taken outside to see the dark, the moon, and stars, before his bath. "See dark" has become a predictable request after he's finished stuffing himself full of dinner. It's usually accompanied by a hopeful "Moon?"  (pronounced "Moo") to which Daddy has had to tell him that the moon is now a morning moon, and it will be way past his bedtime before it rises.

I haven't yet explained that his request is an oxymoron. How can one see the dark? Given dark is lack of light, what we are really doing is not seeing. But there's plenty of precedence for attributing lack of something to an entity itself, so 'seeing the dark' is quite a reasonable way of looking at it.

One can talk about cold, for example. "Feel how cold it is this morning". It is heat, a form of energy, that is the physical entity here. Cold is really the lack of heat, but we're happy to talk about it as if it were a thing in itself. Another example: Paul Dirac in 1928 interpreted the lack of electrons in the negative energy states that arise from his description of relativistic quantum mechanics as being anti-electrons, or positrons. In fact, this was a prediction of the existence of anti-matter – the discovery of the positron didn't come until latter.

In semiconductor physics, we have 'holes'. These are the lack of electrons in a valence band – a 'band' being a broad region of energy states where electrons can exist. If we take an electron out of the band we leave a 'hole'. This enables nearby electrons to move into the hole, leaving another hole. In this way holes can move through a material. It's rather like one of those slidy puzzles – move the pieces one space at a time to create the picture. Holes are a little bit tricky to teach to start with. Taking an electron out of a material leaves it charged, so we say a hole has a positive charge. That's a bit confusing – some students will usually start of thinking that holes are protons. Holes will accelerate if an electric field is applied (because they have positive charge) and so we can attribute a mass to the hole. That's another conceptual jump. How can the lack of something have a mass? Holes, because they are the lack of an electron, tend to move to the highest available energy states not the lowest energy states. Once the idea is grasped, we can start talking about holes as real things, and that is pretty well what solid-state physics textbooks will do. It works to treat them as positively charged particles. It's easy then to forget that we talking about things that are really the lack of something, rather than something in themselves.

A more recent example is being developed in relation to mechanics of materials as part of a Marsden-funded project by my colleage Ilanko. He's working with negative masses and stiffnesses on structures – as a way of facilitating the analysis of the vibrational states and resonances of a structure (e.g. a building). By treating the lack of something as a real thing, we often can find our physics comes just a bit easier to work through.

So seeing the dark is not such a silly request, after all.

## The earth’s magnetic field: much more complicated than you might thinkMarcus WilsonNov 13

At the recent NZ Institute of Physics conference, we were treated to a wonderful description of the earth's magnetic proceses, by Gillian Turner.  What makes up the earth's magnetic field? What effect does it have? How is it changing?

At first glance the magnetic field of the earth is pretty straightforward. There's a magnetic north pole and a magnetic south pole. In fact, the earth's magnetic field looks a lot like what you get from a simple bar magnet.

But look only a little more closely, and it's clear there's a lot more to it than that. For example, the magnetic south pole is not diametrically opposite to the magnetic north pole. In fact,  both are moving about quite substantially – like a lost polar bear / penguin, depending upon your hemisphere. At sixty-something degrees south, the magnetic south pole currently isn't all that far south at all!

Then there are the reversals in magnetic field, that occur from time to time. We're talking hundreds of thousands of years. But they're not regular, suggesting a great deal of randomness is going on.

It helps in interpreting what we see to remember that we are observing the field at the surface of the earth. It originates deeper within the earth – in the inner and outer cores. The distance from the centre of the earth is rather important. Why? Because as we move away from the centre, the smaller-scale variations get 'ironed-out' more quickly than the larger-scale variations. The effect is that the large-scale behaviour (i.e. the bar-magnet-like shape of the field) is emphasized at the earth's surface, whereas deeper down it is much less like a bar magnet.

For those more mathematically inclined, one can see this with multipole expansions. This is a mathematical way of breaking up the description of a shape of an object into different components. One starts with a monopole – how much like a sphere an object is. Then we move on to a dipole moment – this is describing how separation of material along an axis there is.  Next are quadrupoles, then octupoles – each describing finer variations in the shape. So, as an example, a perfect sphere has a monopole moment of 1, and no multipoles of any other type. A rugby ball is quite like a sphere, so it has a high monopole moment. While it's got a preferred direction (its axis) both ends of the axis are the same and so there is no dipole moment. The next term, the quadrupole moment, however isn't zero – it's this moment that describes the bulk of the distortion from a sphere.  The link above gives a nice example of a skittle – it has monopole, dipole and quadrupole moments.

Now, with the earth's magnetic field, there is exactly no monopole moment. Magnetic field isn't like electric charge – while one can have an isolated 'positive' charge, one can't have an isolated 'north' pole. The leading term for the earth's field is the dipole moment – there's an axis and a distinct split of field on the axis – at one end it points away from the centre, at the other towards the centre.  Now, the interesting thing is how the impact of the moments changes with distance away. The n-th order multipole has a strength that varies as the inverse of distance to the power n plus one. So the field due to a monopole varies as 1/r^2 (where r is distance away), the field due to a dipole varies as 1/r^3, that of a quadrupole as 1/r^4 and so on. As r gets large, the effect of the higher-order (higher n) moments diminishes quickly. Consequently, it doesn't matter what mish-mash of magnetic behaviour one has, at large enough distances away, the field from it  will look like that of a dipole.

At the boundary between the outer core and the mantle, there is such a mish-mash of magnetic behaviour. A picture from an impressive computer simulation of the field by Glatzmaier and others  is here. At the earth's surface, however, it is much smoother and we see it as approximately a dipole – with a clear north pole and south pole, (very) approximately diametrically opposite.

But the mish-mash of the field in the liquid outer core isn't the whole story. It's tempered by the solid inner core, which isn't going to change its magnetism so easily. It provides a large inertia against any changes, meaning that flipping the field of the inner core required some extreme behaviour in the outer core. It gets extreme enough just occasionally, and indeed the inner core can then be flipped, but it's not often. Our compasses are still likely to work tomorrow.

Glatzmaier, Gary A.; Roberts, Paul H. (1995). "A three-dimensional self-consistent computer simulation of a geomagnetic field reversal". Nature 377 (6546): 203–209.Bibcode:1995Natur.377..203Gdoi:10.1038/377203a0

## Precision Cosmology – Yeah, Right!Marcus WilsonSep 27

We've just had our first session at the NZ Institute of Physics Conference. The focus was on astrophysics, and we heard from Richard Easther about 'Precision Cosmology' – measuring things about the universe accurately enough to test theories and models of the universe. We ablso heard about binary stars and supernovae, and evidence for the existence of dark matter from observing high energy gamma rays.

Perhaps the most telling insight into cosmology was given in an off-the-cuff comment from one of our speakers, David Wiltshire. It went something like this. “In cosmology, if you have a model that fits all the experimental data then your model will be wrong, because you can guarantee that some of the data will be wrong.”

Testing models against experimental observation is a necessary step in their development. We call it validation. Take known experimental results for a situation and ask the model to reproduce them. If it can't (or can't get close enough) then the model is either wrong or it's missing some important factor.(s). Of course, this relies on your experimental observations being correct. And, if they're not, you're going to struggle to develop good models an good understanding about a situation.

The problem with astrophysics and cosmology is that experimental data is usually difficult and expensive to collect. There's not a lot of it – you don't tend to have twenty experiments sitting in orbit all measuring the same thing to offer you cross-checks of results – so if something goes wrong it might not be immediately apparent. And if you can't cross-check, you can't be terribly sure that your results are correct. It's a very standard idea across all of science – don't measure something just once, or just twice, (like so many of my students want to do), keep going until you are certain that you have agreement.

Little wonder why people have only very recently taken the words 'precision cosmology' at all seriously.

## Don’t miss the eclipse (hee hee)Marcus WilsonMay 08

Friday is the last opportunity to view a solar eclipse in New Zealand for a long time (till 2021 – or 2025 if you don’t count anything of a few percent or lower). I say ‘view’, but the reality is that such a smidgen of sun is going to be covered that you’re going to have to look carefully at the right time. And that’s only for us northerners – most  in the South Island are going to miss out. (Details for this eclipse are here).

For Hamilton, the eclipse hits its maximum coverage (a mere 5%) at 11:49 am.

But it’s not all bad news – an eclipse famine is followed by an eclipse bonanza – three total and three annular eclipses visible from New Zealand between 2028 and 2045. Worth looking forward to. I’ll be into my seventies for the last one of these. Ouch.

11:51am, Friday 10th May. Just caught a glimpse of the sun in a clear patch between the clouds. Can I detect any ‘nibble’ out of it. Nope. I thought 5% was a bit of an unlikely viewing situation.

## Pinhole cameras and eclipsesMarcus WilsonNov 15

Well, the eclipse yesterday was fun. There were enough patches of sky between the clouds to get some good views. I was pleased that the pinhole cameras I made out of miscellaneous cardboard tubes, tins, paper and tinfoil worked really well. Also, the trees around the front of the sciences building gave some nice natural pinholes as the sunlight worked it’s way through the gaps between the foliage – we could see lots of crescents projected onto the wall of the building. Not something you see everyday.

The trick with the pinhole camera is to get the combination of length between pinhole and screen and size of pinhole correct. (Basically – the f-number in photography-speak) A long length means a larger image – but also a fainter one. To increase the brightness, we need to let more light through (a bigger pinhole) but the drawback of this is that it blurs the image. It takes a bit of experimenting – best done well before the eclipse that you want to see.

On the subject of which…if you live in New Zealand…you don’t have a lot of opportunity for a while. We northerners get an iddy-biddy eclipse next May (10th) – sorry Mainlanders – you miss out – and then it’s nothing for ages before we get a few more feeble partials in the 2020s. BUT, as I said earlier, it’s then non-stop eclipse mayhem from 2028, with THREE total and THREE annular eclipses before 2045, for those of us who are still alive to see them. Details are all here courtesy of RASNZ.

There are a few videos up already from the Cairns region – here’s one. However, video does not do an eclipse justice, partly because of the difficulty in video capturing parts of the corona at different luminances simultaneously. If you want to see the fainter, whispy stuff at the far edge of the corona, you end up well overexposing the brighter area nearer the moon.  The naked eye does a far better job of capturing the totality phase than a camera.

I note a fair amount of pink on the video – this is the chromosphere – a thin, cooler area of the sun, between the photosphere (the bright yellow bit that we normally see) and the corona.

## Look out for the eclipse, 14 NovemberMarcus WilsonOct 30

There’s a great event coming to our neck of the woods soon (by neck-of-the-woods I mean Australasia and South Pacific) – a total solar eclipse, on 14 November (for those like NZ on the west of the international date line) or 13 November (for those on the eastern side – which won’t be many – save the odd ship). The NASA website above gives details in Universal Time (Greenwich Mean Time) and so reports it as 13 November – don’t get confused.

For those lucky enough to be in Cairns, there’s the full spectacle of a total eclipse. For us lesser mortals in NZ, it’s a pretty sizable partial eclipse, especially for those in the north of the North Island. Hamilton gets about 85% coverage at the maximum. (Note that anywhere in NZ will do – even Scott Base in Antarctica, I think, gets a few percent coverage, if you count that as NZ)

For Hamilton, the eclipse starts at about 9:20am, reaches its maximum at 10:30am, then is all done and dusted by 11:45am. Times for the rest of NZ are similar.

I thought hard about travelling over to Cairns for the event. The reason is simple – a large partial eclipse is nothing compared to the experience of a total eclipse. I was fortunate to be able to see the 1999 eclipse in Europe, from a small village in northern Bulgaria,  and, having experienced that, partial eclipses don’t have much interest. But, travel doesn’t come cheaply, and there’s a baby at home, so this time  I’m staying put. While it would be great to see another, I’m happy with one in a lifetime.

So what does a total eclipse give you that a partial one doesn’t. Here’s a list, that’s not at all exhaustive.

1. You get to look at the sun with your naked eye, quite safely.  DON’T do this at any other time.

2. The wispy corona comes into view.

3. If you’re lucky, so does the pink chromosphere (this was particularly prominent in the 1999 eclipse).

4. You get to experience the birds coming down to roost, and then taking off again.

7. Stars out during the day. Possibly a good view of Mercury, which is hard, though not impossible, to observe well otherwise, because it is so close to the sun.

8. The diamond-ring, as the bright photosphere bursts back into view.

And so forth. One of the things I remember from Bulgaria is just how quickly things went black in the final few seconds before totality. It was like standing in a well lit room and someone turning off a dimmer switch.

So, what do we get for 85% then? Well, not much, actually. You might not even notice that things have gone dim. The human eye is really good at adjusting to different light levels, and it’s really only when only a few percent of the sun remains that you’ll notice any obvious change in illumination. It’s fun to observe the crescent shape of the sun – but do so SAFELY – with decent eclipse glasses or solar projection. A fun thing to do is pinhole projection – put a tiny pinhole in a piece of card and project the sun’s image onto the ground or a sheet of paper.  In Bulgaria we had pinholes provided by way of the old tin roof on the cafe which our group occupied for the event – it was loaded with little tiny holes (not much good in the rain then) which gave some wonderful projections of the sun onto the tables below.

So, when’s the next total eclipse to hit NZ? There are actually a few coming ‘soon’ – ‘soon’ being used in an astronomical sense. 22 July 2028 sees most of Otago including Dunedin eclipsed totally. But it won’t be an easy eclipse to view, coming near sunset with the sun just 8 degrees above the horizon. The same eclipse, however, tracks right over Sydney (once again the Aussies get it – though there is far more of Australia for an eclipse to hit) so one might be better off heading westward.

But then, like buses, there’s a positive flurry of them. 10 March 2035 sees NZ get an annular eclipse (the moon doesn’t quite cover the whole sun – not as impressive but pretty spooky) – then 13 July 2037 and a total eclipse tracks over the central North Island, including Napier (Hamilton lies just to the north) and then 26 December 2038 we get another chance – this one over Golden Bay, Manawatu (including Palmerston North) and Wairarapa. (Wellington is just off to the south). That will add interest to the Boxing Day barbie on the beach. The really freaky thing is that there is a small slice of land near Waipukurau that will get a total eclipse in both 2037 and 2038.

## Distant galaxies and hobbitsMarcus WilsonOct 01

I haven’t read ALL of Tolkien’s work, but I suspect space-travelling hobbits don’t feature anywhere. However, what do feature are hole-dwelling hobbits, and I had the fun of seeing their holes in the countryside near Matamata yesterday. The original set for Lord of the Rings was mostly removed after filming, and rebuilt for the filming of the Hobbit trilogy.  (Trilogy? Since when was The Hobbit a trilogy? This is just milking money out of Tolkien fans, isn’t it?) But this time the set will remain, for all to see, for an appropriate fee of course. It certainly was fun to have a look around – what made it was the commentary provided by our excellent guide.

One of the fascinating things pointed out was the perspective tricks that were used. For The Hobbit, there are three different versions of some of the holes.  One, a ‘large’ version, appropriate for a normal-sized actor, dressed as a hobbit, to walk through. One, a smaller version, to make the dwarfs look bigger than the hobbits. And another, an even smaller version, to make Gandalf look bigger than the dwarfs. And the three had to be identical.

And then there are the perspective tricks. To make the view look like it is over a longer distance, the more distant holes are of smaller size than the nearer ones. On a 2d movie it works – your mind interprets what you see as being of equal-sized holes spread over a larger distance. But being there in 3d you see it more as it is.

That’s the problem that’s faced when determining the distance to distant stars and galaxies. Just how far are they away?  The moon, and anything further away, we perceive as 2 dimensional. We can’t get any 3-dimensional cues and so we have no idea, just by looking, of how far away they are.  So how can we measure distance to the stars?

One way, which works for the nearest stars, is parallax. The earth orbits the sun, and six months from now it will be about 300 million km away from where it is now. That gives a different viewpoint. The nearest stars, therefore, appear to move against the background of stars that are further away. We can therefore use a bit of simple trigonometry to work out the distance to the star. Indeed, one of the units of distance in astronomy is the parsec – one parsec being the distance over which the diameter of the earth’s orbit subtends a parallax angle of one arc-second.  Essentially, using parallax in this manner is like viewing the situation with two eyes – 300 million km apart.

Parallax, however, only works for our nearest stars, since the distances to our neighbours are so huge. To work out distances further away, there are other methods – such as looking at the intensity of Cephid Variable stars, and, for really long distances, the famous redshift. However, somewhat disappointingly, neither of these are exemplified by the Hobbiton movie set.