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Where is the sun in the sky?
Data, spreadsheet, calculations

Sub-page associated with discussion of
"How do we measure sun's position in the sky?"

This page is a sub-page from Where is the sun in the sky?. This page goes into some of the details of how we use the raw readings from our sun position measuring instrument, and turn them into more useful measures of where the sun is.

Before I go further, I have to explain that this page still has some pretty dreadful rough edges. I hope you will find that it has enough Good Stuff to repay you your time.. but it is incomplete and in need of some editing. That said... onward...

In the page this is subordinate to, I described an instrument, much like a horizontal sundial. The dial wasn't marked in hours, though. We merely used the "sundial" to measure the length of the shadow cast by a vertical gnomen, and the angle (in the horizontal plane) of that shadow from an arbitrary "90". Our "90" was where the shadow would be, roughly, at noon.

The worksheet

I've prepared a worksheet in connection with the observations of 23 June 2014. ("Worksheet": The document processed by a spreadsheet program. I use OpenOffice's "Calc".) At the heart of it...

-

Please find, just above the main block of numbers, "C1", "C2", "C3", etc. These stand for "column 1", "column 2", and are for coordinating this text with the worksheet.

In C2, C3 and C4 are the basic measurements.

In this page, I am going to try to show you how these numbers can be turned into measures of how high (as an angle) the sun is above the horizon, and the angle of the sun from true north.

For data point 1, we see 11:02:19, 361.0, -235.0. (Yes... all through this, I need my knuckles rapped in various places in respect of inappropriate precision.)

In detail: Those figures say that when my watch was reading 11:02:19, I placed an "X" on the instrument, putting it were the tip of the shadow was at that time. I subsequently measured the horizontal distance from vertically below the tip of the gnomen to the "X", and found that to be 361.0mm (+- 0.5mm). We'll come to the "chord" measurement (-235.0mm) in a moment.

In the page this one is subordinate to, we have already seen how the elevation of the sun above the horizon can be determined from the length of the shadow and the height of the gnomen. There are elements in the worksheet to take you directly to the elevation figure (54.3 degrees (in C5)). The value for the height of the gnomen is entered in a cell set up for that, and it is referenced as needed for calculating the elevations. (Yes, the worksheet with all its formulae will be available for download.)

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So! That's one of the two numbers we need to say "where the sun is/ was" dealt with.

Rotation/ direction

Now we need to deal with the number which says what direction we need to face, to be looking directly towards the sun. (Don't look at it, remember!)

That number could be obtained with a protractor... but not as accurately, or, in practice, as easily as the method much of the rest of this page is about. It is easy to use, if not, perhaps, immediately easy to grasp... or indeed describe!

Recall there were arcs on the "face" of the instrument, the arcs to facilitate measuring the shadow's length. Think in particular of the one with a 30cm radius. And recall that we marked a point on it (arbitrarily), and called that "90". If a shadow lay in that direction, it was, relative to the instrument, "at 90 degrees".

While some shadows may not be long enough to cross the 30cm arc, any short shadow's line could be extended to the 30cm arc.

We then measure, for each shadow, the distance from where "90" lies on the 30cm arc to where the shadow crosses it. Points to the left of 90 are recorded as negative numbers.

Those readings are what appear in the worksheet column I've labeled "C4". Easy to measure, anyway. Now let's see how those numbers can be turned into angles....

The joys of trigonometry

The conversion rests on the precepts of trigonometry... but, as before, you don't need to know any. I will tell you the bits you need.

-

The diagram above is what you would see looking down on the sun-position instrument, with the sun at your back.

"C" is the point directly below the tip of the gnomen.

A shadow is lying along the line from C to S, i.e. "line CS".

At "B" begins the 30cm arc which we drew on the face of the instrument to facilitate measuring shadow lengths.

En route to finding the angle which specifies the direction to the sun, we will measure from "T" (Top) to "S" (Shadow)... from where the "90" line crosses the 30cm arc to where the shadow crosses it.

For the first datapoint, in the table above, in the column labeled "C4", we see -235... which says that line TS, in the case of the first datapoint, was to the left of "T", and 235 mm long.

That's the only measurement we need to make. We do the rest with mathematics.

Trigonometry is less hassle if you are dealing with triangles with a 90 degree angle in one corner. Hence the line CR, which splits the angle SCT neatly in half.

We know the length of RT (235/2 for the first data point) and the length of CT (30cm).

With the relevant Neat Trick from trigonometry, arcsine, we can determine the size of the green angle: RCT. Twice that gives us the size of SCT. If we take THAT away from 90, we get the size of BCS... which is what we really wanted, from the start. That angle is the number we report as the direction of the sun, relative to the instrument. In the case of the shadow shown in the figure above, it would be about 30 degrees, or "030", if you had a nautical upbringing.

To make our reading more generally useful, we correct the "relative to the instrument" reading and turn it into a "relative to true north" reading. All it takes to do this is adding (or subtracting.. unlikely...) a constant number to all of the "relative to instrument" readings.

You'll be pleased to know that the worksheet will take care of all the arithmetic (and trig) for you. But you should try to understand why when you put -235mm into C4, you get 47.2 degrees in C6. The 130 degrees in C7 comes from combining the 47.2 with a constant entered into a part of the worksheet you can't see in the bit displayed above.

The worksheet: It is available for download, in the OpenOffice format. If your spreadsheet cannot open OpenOffice spreadsheets, I suggest you install it. It is free. It opens the worksheets of many spreadsheets you have to pay for. You might want to think about why your spreadsheet can't "play nicely" with worksheets from a spreadsheet which will "play" with worksheets from your spreadsheet...

Related material

If you visited the sub-page with the spreadsheet and calculations, and read about the alternate way of measuring angles, and want to read more on that, you might be interested in another of my pages.





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