If you live in the northern hemisphere like me then this Friday is hibernal or winter solstice. For those of you in the southern hemisphere it will be the summer solstice. Either way happy solstice to all! For those of us in midwinter the Earth’s north pole is at its maximum tilt away from the sun. This means that the apparent height of the sun is at its lowest point in the sky on the winter solstice.
Believe it or not a bunch of people — your friends and neighbors — have no idea what I’m talking about right now. But I suspect you think astronomy is cool or you wouldn’t be reading this right now. So I’m here to help you build an awesome solar calendar in your front yard both to mark the changing of the seasons and to help visualize the sun’s apparent motion across the sky. Your neighbor kids will love it trust me. And it will be fun for you to design and build. After the jump is a photo of the finished product as it looks in the parking strip in front of my house.
The solar calendar consists of a post (gnomen) along with three marker stones aligned on a north-south axis. Mine is installed on a narrow strip between the street and the sidewalk in front of my house. The post is at the south end as in this photo:
Use a straight kiln-dried 10 foot cedar 4 x 4 post. Don’t cheap out with fir or pine. It has to stand straight and not warp or twist over time. I put on three clear coats of Vermont Natural Coatings, which you can find at any any decent hardware supply. To protect the top of the post (and to raise the awesomeness level) I used a post cap from Madison Iron and Wood. Attach the post cap to the post with a good commercial-grade silicon adhesive. And use a sealant around the base of the post where it meets the concrete. All of this will run you about $150 or so but you’ll have a good, solid, and straight post that will weather the elements over time.
As solar noon approaches the post will cast a shadow, the length of which marks the time of the year. The three markers represent the summer solstice (closest to the post), the two equinoxes (middle stone), and the winter solstice (furthest stone). Here the shadow is approaching the fall equinox:
So now it becomes a question of where to place the markers. And to do that we have to know how long the shadow will be on certain days of the year as well as the height of the post. In short, we have to do some math!
Let’s start with determining the sun’s declination. Declination is a measure of the height of a celestial object in the sky. If the object is visible overhead it will be between 0° (horizon) and 90° (zenith). With the exception of our Sun, every star has a fixed declination. You can use this number, along with your latitude, to determine how high a star will be as it crosses your meridian. The meridian is a line that extends from your horizon due South and passes directly overhead to the horizon due North. This line cuts the sky into two halves: before midday (ante meridiem) and after midday (post meridiem). This is where we get the abbreviations A.M. and P.M.
The Sun’s declination changes every day. You’ve seen that it’s higher in the sky in the summer and lower in the winter. On the summer solstice, the Sun will reach its highest point in the sky as it crosses the meridian — at my latitude here in Portland, Oregon it is 68° above my horizon as I face due South at solar noon. On the winter solstice at my latitude, the sun never gets higher than 21° in the southern sky. And at the spring and fall equinoxes, when the Sun rises due East and sets due West, the sun will reach 44.5° at its height. Here is a sketch of the sun’s declination on the summer solstice:
How do you calculate the sun’s maximum height in the sky at your latitude on each of these four days? Here’s the formula:
A = d + 90° – φ
Where d is the sun’s declination and φ (phi) is the latitude in degrees of your location. If you want to seriously geek out then you can calculate latitude by hand. Or just go to latlong.net and plug in your place name.
The value for d will always be 23.5° (on the summer solstice), -23.5° (on the winter solstice), or 0° (equinoxes) because this represents the fixed angle of the Earth’s tilt on its axis. Here in Portland my latitude is +45.5° so the sun’s declination for me is:
| Summer Solstice | A = 23.5 + 90 – 45.5 | 68° |
| Winter Solstice | A = -23.5 + 90 – 45.5 | 21° |
| Equinoxes | A = 90 – 45.5 | 44.5° |
So now you know the sun’s declination on those four days for your location.
Next we need to determine two things: (1) the post height; and (2) the shadow length. Get out a tape measure to determine the post height. Mine is 269 cm. The shadow length is a bit more involved. To measure that we need to know the properties that describe the ratios of the sides of a right triangle. For this exercise we’re interested only in sine and tangent. Given this right triangle:
The trigonometric functions are sin A = a/c and tan A = a/b. Line segment a is the post height. So for me a is 269 cm. You will have to measure yours. The interior angle A is the Sun’s declination. Suppose it’s the summer solstice (68°) then we just have to solve for shadow length b. The first method we can use is the Pythagorean Theorem. First solve for the hypotenuse c:
sin A = a/c
c = a/sin A
c = 269/sin 68
c = 269/0.927
c = 290 cm (rounding down)
Then:
a² + b² = c²
269² + b² = 290²
72,361 + b² = 84,100
b² = 84,100 – 72,361
b² = 11,739
b =√11,739
b = 108 cm (rounding down)
The second method takes the tangent and is much shorter:
tan A = a/b
b = a/tan A
b = 269 / tan 68
b = 269 / 2.475
b = 108 cm (rounding down)
Therefore the shadow cast by a 269 cm tall post on the summer solstice will be about 108 cm in length (42 inches). You’ll have to do the math for the other three change of seasons as well. The fall and spring equinoxes (aequus nox is Latin for “equal night”) will cast shadows of identical length so this is why you use three stone markers for the four seasons.
If you want to go all Indiana Jones you could carve a spiral pattern in the marker stones so that the tip of the shadow passes right through the center. Or paint a design on the stone. If you have the space for it you could put stones to the left and right of the central marker denoting 11:00 A.M. and 1:00 P.M. respectively. This combines an analemmatic sundial feature with a solar calendar. But I’ll leave that to you. Have fun!