Part 2 of this little series looks at a different phenomenon to do with the sun’s movement through the sky. Imagine yourself picking a time of day – let’s say 10:30 in the morning – and taking note of where the sun is in the sky. Do this at the same time every day of the year to build up a curve tracing the sun’s apparent movement. One way to do this would be to take a photo pointing at exactly the same angle at exactly this time, then overlay the photos on top of each other. Another way would be to put a stick in the ground as a rudimentary sundial, then mark out the end of the stick’s shadow each day. It’s an easy experiment in principle, but takes a lot of patience and accuracy to get right.

But suppose you’ve done that – what would you expect to see? We know that the sun goes up and down in the sky through the year – in winter it is lower and in summer higher. So i suspect that most people would expect to see a straight vertical line being plotted through the year as the sun cycles along its seasonal track. But actually what you get is not a straight line, but a figure eight shape. In the northern hemisphere the top loop of the 8 is smaller than the bottom, while in the southern hemisphere the loop nearer the horizon is the small one.

This curve is called the analemma, and has been known for a very long time – Greek and Latin authors wrote about it some two thousand years ago in the interest of designing a better sundial. My guess is that people observed this much longer ago, and that the creators of the great prehistoric stone observatory monuments tried to incorporate it in their designs.

We can describe this curve mathematically, and it is taught as a method of dead reckoning for those at sea. With a good watch to keep track of time, decent knowledge of the analemma shape, and some precise observations of the sun’s position in the sky, you can pinpoint your position down to around 100 nautical miles. Not bad if you’re lost at sea with no GPS!

The root cause of this is a combination of two factors in the Earth’s movement. The first is that the polar axis, around which the Earth spins to give day and night, is not at right angles to the plane of the Earth’s orbit. This offset angle, a little over 23 degrees, is what gives us seasons. The second factor is that the Earth’s orbit around the sun is not perfectly circular, but a slightly squashed oval. Moreover the sun is not at the centre of the oval, but offset to one side at one of the two focal points – we are about 5 million km closer to the sun in early January than we are in early July. The Earth does not move at a constant speed around this oval. We speed up at closest approach to the sun, and then slow down as we move further away. Those who can remember school physics might have come across this as Kepler’s 1st and 2nd laws of planetary motion, originally formulated in the early 1600s.

Now, for convenience we split our year into equal length days, which means that for one part of the year, a day according to our clocks gets ahead of its allotted portion of the orbit, and for another part it falls behind. By the end of the year it all comes out even. Also, the offset of the polar axis changes the degree to which these shifts make a real difference against the sky. The combination of these two factors is what generates the figure 8 shape of the analemma.

Let’s think back to our ancient ancestors and the stone monuments they built. We know that the positions of the stones encode astronomical information. The monument builders were aware of not just the annual cycle of the sun, but also of more subtle patterns, such as the 28 year cycle that the moon makes in its own path against the sky. Since the analemma can be mapped out with nothing more complicated than a stick to make a shadow, it seems to me quite improbable that they did not know it. Having said that, I don’t know of any specific stone patterns that can be linked directly to the analemma. Once people started making sundials, they soon found that there was no single division of hour markers that works consistently. The figure 8 shape ensures that your sundial sometimes runs fast and sometimes slow.

Moving into the future, every planet has its own variation of the analemma. The exact shape depends on interplay between the angle of the polar axis and the extent to which the orbit deviates from a pure circle. Our Earth has these two factors in approximate balance. So does Pluto, which therefore has a figure 8 shape like Earth, though in this case the top and bottom loops are almost the same size. But for other planets one factor or the other dominates. As a result, Jupiter has a simple oval shape, while Mars has a tear-drop. However, actually making the observations (as opposed to calculating them) might be tricky as you move out through the solar system. On Earth, you only have to wait 365 days. But a Jupiter year is almost 12 of our years, and Pluto takes nearly 250 years to circle the sun once. You would need extreme patience to plot out a full analemma cycle in both these places!

I guess pretty much all of us know that December 21st this year marked the winter solstice, and so – in the northern hemisphere – the shortest day and longest night of the year. But comparatively few people seem to know that this day is not the one when the sun rises latest and sets earliest. The exact dates of those events are, at the latitude of London, just over a week different from the solstice. Specifically, the latest sunrise this year is not until January 1st 2017, and the earliest sunset was on December 12th.

It turns out that the times when the sun rises and sets are governed by a moderately complicated algorithm called the Equation of Time. This obviously varies with your latitude and longitude, but also takes into account the small differences between the solar day and the sidereal day (the day length as measured against the distant and essentially fixed stars), seasonal variations in the earth’s distance from the sun, the apparent size of the solar disk, and a host of other relevant pieces of information. Strictly speaking, one’s height above sea level, and the details of the surrounding terrain also make a difference, but not in a way that’s easy to quantify here. Finally, there are several different definitions of what angle counts as the zenith line, and I have taken the civil definition as opposed to nautical or astronomical.

Once upon a time the calculations would have taken a very long time and lots of paper, but nowadays we can throw the calculation steps into Excel and find out the information for anywhere we want, and for a reasonably long span of time into the past or future. For the curious, a step by step description can be found at this link.

Out of curiosity, I plotted the changes for a series of latitudes from that of Reykjavik in Iceland (just over 64 degrees north) via Orkney, Penrith and London in the UK, through Rome and the Tropic of Cancer to the Equator. For simplicity I just took everything on the zero longitude line (through Greenwich) since I was only interested in changes in latitude. If you wanted to do this for yourself then you would need to adjust for your actual longitude east or west from Greenwich, and your official time zone.

Here’s the corresponding chart for sunset.

A few things stand out at a quick glance. First, the time of sunrise varies considerably at some times of the year even between London and the north of Scotland. Secondly, you don’t have to go all that far north to get to the ‘land of the midnight sun‘. Thirdly, the total range of variation of sunrise is very small at the equator – about 1/2 an hour, as compared with London’s 4 1/2 hours, or Iceland’s 8 1/2 hours. The places where all these lines cross over is at the spring and autumn equinoxes, where night and day are each 12 hours long across the whole globe.

Going back to where we started, and looking carefully at the early part of the year, you can see that the day of latest sunrise happens after the solstice. The further north you go, the closer the two days are together. So in Reykjavik the latest sunrise is on December 26th. Come down to Orkney and it’s the 28th. In London you have to wait until January 1st. In Rome, January 5th. If you lived on the Tropic of Cancer (say in parts of the Sahara, roughly on a level with Kolkata, India) you’d be waiting for the 8th.

If you live right on the Equator something else comes into play. You get not just a simple days-get-longer then days-get-shorter cycle. Instead there is a more complex curve. Something similar happens in the whole belt of the tropics. This is because there are times when the sun at noon is to the south (as always happens in the northern hemisphere north of the Tropic of Cancer(, but then times when the noonday sun passes overhead and is, for a while, to the north. As it swings over and past you, the day length lengthens and then shortens again – as you can see in the graph.

OK, that’s enough of the Equation of Time for this week. Next time – another oddity about solar movements through the year, together with some thoughts about what this all means for us humans as we have observed the sun through the years. I am convinced that our remote ancestors knew about these patterns (though probably didn’t dress them up in the sines and cosines used by modern maths) and incorporated this knowledge into their monuments and observatories. But more of that next time…

Today’s topic is – once again – inspired by an experience on the Scilly Isles. It has to do with how things look different from one point of view than they do from another. If you were going to navigate between the islands – say from Bryher round to St Martin’s – then the chances are you’d pull out a map. If you were feeling particularly cautious, or you were going in a larger boat which drew more water, maybe you’d go up a grade and get yourself a chart with underwater depths plotted. Coupled with some knowledge of the tide, you could then plot out a course.

Now if you also had a GPS navigation system, in principle you could then hand over the course to that, sit back, and enjoy the journey with no more effort. But if you didn’t have that, and you were reliant on personally converting all that map work into movements of the tiller, you’d find that it is rather more tricky than it looks.

A map is a top-down view of the world – the sort of thing you would see if you were flying. But out on the water you get a completely different perspective. You are looking along the (reasonably) flat surface of the water, seeing the sides of islands and rocks as they project up from that surface. Some things – perhaps your destination – are hidden behind other ones. Some things which look close together are actually far apart, having been brought together by visual accident. A passage between two rocks might appear wide and easy on your chart, but narrow and problematic on the water, since it calls for careful wiggling at key moments.

Now, of course, this was the norm for navigation throughout our history until very recently, and goes a long way to explaining why the profession of pilot was so important. The pilot knew his waters, and how to navigate ships of different sizes through them at different states of the tide – or indeed not to make the attempt until the tide turned.

Now, maps and charts to help navigation have been around for quite a long time – several centuries, at least. But a pilot in action was not so interested in the top down view given by a chart. What he wanted – needed – to know was the direction to steer in at any point. And this was done, by and large, by means of lines of sight. Certainly pilots necessarily built up a vast store of information of local conditions in all kinds of weather and tide. But the key to their navigation was the index of knowledge which knew that by lining up particular landmarks one behind the other, and trusting these remote guidelines over local phenomena, a difficult voyage could be broken down into a series of simple legs.

Each sight line gives you a stage in your journey: each journey consists of a chain of such guidelines. Arthur Ransome used this idea in Swallows and Amazons, where a navigation line was set up by visible marks in daytime, or lanterns after dark. John says: “This [stump marked with white cross] is one of the marks, and the other is that tree with a fork in it… come into the harbour without bothering about the rocks by keeping those two in line”. In this way they could easily get in and put of the harbour on Wildcat Island. The pilot’s job was basically the same, but upscaled to a much larger region and a hugely more complex set of navigation tasks.

But it is not just navigation on the water which has this difficulty. When we look up at the night sky, at stars and planets, we are looking at objects scattered through a vast three-dimensional space, as though they were arranged on the surface of a sphere. Extremely distant objects are pressed together, and small nearby ones can hide much larger further-away ones – as in this solar eclipse. And although changing tides are not a problem in space, things are moving at different speeds, and if you are trying to navigate – say – from one of Jupiter’s moons down to Mars, you have to take into account the change of location which will happen in the meantime. Human pilots still ply their trade in many parts of the world, and although they are usually supported by electronic backup, it still calls for in-depth familiarity with their coast, and day-by-day assessments of change. I wonder if the same will be true if and when we move into space?