This post is about making sure my new garden beds don’t end up in the shadow of my back porch, during the summer. Based on the length of the shadows today, in late winter. And, ultimately, based simply on the height of the porch roof.
To cut to the chase: If you use Excel, and the NOAA sun-angle calculator, you can accurately predict the length of a shadow, for any date and time, anywhere on earth, via this formula:
Shadow length = obstruction height * cotangent (solar elevation angle in degrees * π / 180)
The π / 180 is there because Excel wants to see angles expressed in radians. If you’re using a calculator that accepts angles in degrees, omit that.
As surely as the sun rises in the east
The length of the shadow cast by an object, at any time, on any day of the year, is perfectly predictable. That’s because the location of the sun, in the sky, is perfectly predictable. For any location on earth, for any reasonable date, you can simply look it up, as with this National Oceanographic and Atmospheric Administration (NOAA) calculator (reference).
Notes: On the NOAA site, pick a city, determine whether to check the box for Daylight Savings Time or not (I suggest not), fill in a date and time, and click “calculate”. You want the results in the box labeled solar elevation. Alternatively, instead of clicking a city, you can click “enter lat/long” and manually enter a latitude/longitude pair, but it’s tough to imagine why you’d need to do that. There’s also a map-based version of that, but I find it a lot easier to make mistakes with that version. Solar elevation shows up in the very last data box returned by the map-based version.
Once you know the height of the object, and you’ve looked up the angle of the sun above the horizon (alpha, above), the length of the shadow is the height of the obstruction, times the cotangent of alpha.
Don’t remember what a cotangent is? No problem! You’re going to have your spreadsheet or calculator give you that value anyway. All you need to know is which button to push/formula name to enter.
Almost. There’s one twist. Excel wants to see the angle expressed in radians, not degrees.
Don’t recall what a radian is? Still no problem! Excel gives you the formula. Radians = degrees*(pi/180). Just plug-and-chug. Your calculator probably gives you the option of using radians or degrees.
So, in Excel, the actual formula is:
- Shadow length = obstruction height * cotangent (sun angle in degrees * π / 180)
This is all by way of doing the last bit of calculation before locating my new garden beds. For best sun, I’d like to place them as close to the house as possible. But I need to keep them out of the shadow of the house/porch. But not the shadow that I see today, at the end of winter. I need to be sure that they won’t fall into the shadow of my covered back porch, during the summer.
It’s easy enough to measure where the shadow falls today. The problem is, I want to know where it will fall during the growing season.
The first step is to look up the sun angle today (reference), and see how the observed length compares with the theoretical length, based on the formula above.
Hey, geometry works. That is plenty good enough agreement for gardening. (That last value — 21 versus 22 — was measured in a raised bed, not at ground level. See my point on shaving the values a bit, below.)
What does geometry tell me about those shadow lengths throughout the growing season?
There is one further twist, in that this is the length of the shadow at ground level. But my plants will be growing at least a foot or two above ground level. So I can safely shave these numbers a bit, and still have the plants themselves in sunlight.
In any case, the location of the garden beds will depend on how much I value late-afternoon (4 PM) sunlight, and in particular, how much I value it late in the growing season.
All things considered, placing the raised beds 10′ from the back porch is probably adequate, 15′ is arguably a bit better. Surely 15 is better if I expect to have (e.g.) very-late-season greens growing there. So 15 it is.
Afterword on equatorial sundials.
Source: Ebay
Here’s a weird little fact about sundials. You will note that the hour lines on a sundial are quite unevenly spaced. They typically look like the black-and-quite illustration at the very top of this post. Similarly, the distance the shadow moves, in my table above, is nowhere near constant, hour-by-hour.
That irregularity is a bit odd, if you think about it. The earth rotates at a constant (360/24 = ) 15 degrees per hour. The hours tick away like, well, clockwork. And yet the lines on a sundial are anything but regular.
That’s entirely due to the fact that the earth is flat. By which I mean, we who live on the surface of the earth perceive level ground to be horizontal. Flat. And, in the typical case, we insist that our sundial face be horizontal as well. Or, perhaps in some eccentric situations, we insist that the sundial be vertical. Which, from our perspective, is perpendicular to what we perceive as horizontal.
Intellectually, we know that the little patch of ground we stand on is tilted, relative to the globe as a whole. But it sure does feel flat. Only at the poles is what we perceive to be horizontal ground really horizontal, with respect to the axis of the spinning globe.
The shadow of the gnomon has that odd, non-linear behavior because your horizontal sundial is out-of-kilter with the universe. Or, more exactly, it sits at an arbitrary angle, relative to the plane of the earth’s equator. That angle being determined by your latitude.
The grandparent of all sundials is the equatorial, or equatorial sundial. In that version, you tilt the face of the sundial so that it’s parallel to the plane of the equator. Once you do that — align your sundial with the earth, point your gnomon north (in the northern hemisphere) — then the lines are all evenly spaced. As shown, just above.
So it’s all a matter of perspective.
If you insist that your little patch of dirt is “flat”, then you have to go through some obscure calculations to get the lines right on your sundial (reference). And, in theory, the exactly placement of those lines depends on your latitude, and so on. Every latitude has its own, somewhat-unique, sundial.
But if you’re smart enough to realize you live on a globe, then one simple sundial — with hours marked in 15 degree segments — fits all. You just have to tilt it so that it’s parallel to the equatorial plane. That requires tilting it 90 degrees less your latitude. In my case, at 38 degrees north latitude, the sundial face would have to be tilted (90 – 38 = ) 52 degrees up from horizontal. Find the angle based on your own latitude, point the gnomon north (in the northern hemisphere), tilt the south edge up, and you have a one-size-fits-all sundial. (Or, in round angles (?), 45 degrees works pretty well for most of the contiguous U.S.)
The plate-type equatorial is hard to read, as the sun will sometimes shine on the back of the plate, so you have to have the hours marked on both sides (reference).
If you ever see a sundial with a scale tilted above the horizontal, and something opaque running down the middle, you’re looking at an equatorial (or armillary) sundial. It’s the elegant solution to the problem of telling time based on the angle of the sun.
Source: Instructables.com, sundial by JohnW539
If you’ve ever read instructions on how to find south, using an old-fashioned analog wristwatch, you’re looking at an equatorial sundial in reverse. If you had a 24-hour clock, and tilted it up so that its face was roughly parallel with the equator, and pointed 12 noon due South, the hour hand would point at the sun. And so, in reverse, if you tilt the watch face and point the hour hand at the sun, 12 noon would be due south. The only trick is that we use 12-hour clocks. Because of that, south is halfway between the hour hand and noon.
Finally, it is possible to create a sundial (of sorts) that doesn’t require that you know where north is. Novelty sundial finger rings provide the approximate time based solely on the elevation of the sun above the horizon, rather than its east-west location across the horizon. These do, however, require that you know your latitude and the approximate date, in order to get the vertical orientation of the ring correct, as the height of the sun above the horizon depends on both the date and your latitude (reference Amazon.com).
