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Three minutes of fame, for three weeks of pain

August 11, 2013

Sometimes I wish I could switch off and not think about things; but I can't; I do though find that thinking about chess or maths or other things is the best way to forget about work, so there is some merit in my madness. A few weeks ago, Jane made a chance remark that despite how home in Stockport being only 100 miles or so (googling suggests it is 148km, of which more later), there was a noticeable difference in the length of daylight between it and our cottage in Borrowdale, near Keswick. (cue for random picture of the Lakes; better view than in Stockport)

We were driving home at the time, a Sunday evening, and I had not long before watched a BBC4 science programme, Precision, from which I had learnt that the kilometre was defined as being the length such that there was 10,000km from the North Pole to the Equator. (In fact, the measurements made proved to be imprecise, and apparently the distance is 10,042km, at least according to a quick google. (Though see footnote 1)

So, and harking back to my recent post about my liking for approximation rather than precision, I did some mental maths (along the lines 100 miles is about 1/100, so the angle is about 2°, and 180° is 12hours, which is 360 minutes, so 2° is about 4 minutes, and announced the guess that the difference in sunrise times might be around 4 minutes…a few minutes later, daughter #2 announced that she had googled the result and it was…4 minutes…so, a mathematician's glow of pride. Alas, that feeling lasted for all of a few miles since she then announced that the difference in sensets was around 13 minutes (if I recall correctly): unexplainable, and the estimate of sunrise was seen to be no more than a lucky guess.

It was a long rest of the journey home.

Since then, I have being giving the subject occasional thought, and have realised what a task I have taken on. It would be easier to tidy teenage daughter #1’s bedroom than to get to a full understanding. (cue picture, of said bedroom, were it not truly shocking).

After some initial thinking, I did some googling and was swept into a world of complex mathematics with long formulae with equally long names. That wasn't for me. I wanted to be roughly right or, if this proved to be impossible, at least to have some understanding of the factors involved. I realised, for instance, that I had ignored the tilt of the Earth's axis.

Mistake #1 was thinking about the subject in the first place;

Mistake #2 was making a rough calculation and being smug when it chanced to be right for sunrise;

Mistake #3 was in mentioning the problem to a colleague, who I knew from some chance remarks I had made in a training talk, was interested in physics. Poor Paul, he got afflicted with the same bug, and he has written to me extensively as we try to grapple with it. In my blogs on this subject, which I am writing having to solve the problem (hopefully not making mistake #4, that in writing it, it will all become clear, and instead hoping that an apple (or, since I am writing this in Kas, Turkey, a pomegranate) will fall on my head, I will take freely and extensively from Paul's insights: but the errors, and I think there will be plenty including some howlers, will be mine.

Hopefully a reader will be able to improve what I write (when it is written, this post is just setting the scene)



The fateful journey which started it all

A tale of three cities

Before we can solve the Keswick-Stockport conjecture, and its extension, the Keswick-Kas konundrum, simplification is needed, and it so happens that three major cities in Britain are geographically well positioned

Cardiff and London are more or less on the same latitude, whilst, Edinburgh and Cardiff are at her enough the same longitude. This is very helpful, because there is a lot of data available for our major cities, and we can try to isolate the separate effects of the different coordinates.

Distances and times

The above table are the geographical coordinates and straight line distances between the various locations, and today's sunrise and sunset data. The figures are the best I can find, but may be erroneous. I was surprised to find that in many cases, the sunrise and sunset times for our majors cities varies, sometimes by a few minutes, between websites. Unfortunately, whilst the data for the three cities has come from the same source, so hopefully will hang together, the data for the 2Ks are from two different sources, and I do think there might be a risk of errors of a few minutes in the calculations due to divergent sources.

Estimation over a wider distance



I am trying to gain a better understanding of the differences in sunrises and sunsets between Keswick and Stockport.

With this understanding, then the aim is to estimate over a greater distance, from Keswick to Kas, Turkey, our holiday home. (cue for picture of sundown in Kas)


I will pause for now, and reflect on things another day, and hopefully post something with at least something of interest on another day. Now, it is Pimm's time.




Footnote 1: take care if doing your homework, children.

One site gives a vastly different distance:

Footnote 2: Pythagorus would be happy.

Using the data for the three cities, and with Edinburgh-London being the hypotenuse, the square root of the sum of the other two sides is 538km, less than 1% different from the stated 534 km.

Footnote 3: but I am not happy

Making approximations for the differences in latitude and longitude for Keswick and Stockport, with the Pole-Equator distance being 10,042km results in Pythagorus suggesting the distance between these towns should be 172km, not 148km: this is uncomfortably far out, and indicates an error somewhere. Help.

Footnote 4: shape of the Earth

The shape of the Earth approximates an oblate spheroid, a sphere flattened along the axis from pole to pole such that there is a bulge around the equator.[68] This bulge results from the rotation of the Earth, and causes the diameter at the equator to be 43 km (kilometer) larger than the pole-to-pole diameter.[69] For this reason the furthest point on the surface from the Earth's center of mass is the Chimborazo volcano in Ecuador.[70] The average diameter of the reference spheroid is about 12,742 km, which is approximately 40,000 km/π, as the meter was originally defined as 1/10,000,000 of the distance from the equator to the North Pole through Paris, France.[71]


From → Maths, Science, Travel

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