By Liaf
No, folks, I am not crazy. Astronomy is something I like to dabble in a little, and one needs to know a little something about it to appreciate God's Word in the stars (for signs and seasons). Sundials were a past interest of mine, and lately I resurrected my interest in them. I think it's time to take a breather from the controversial topics and esoteric teachings to come "down to earth" for awhile (we'll get back to the old stuff for the next messages). For many that know me personally, I am not known at all the same way I portray myself on this web site. I'm actually a fun person to be with. Bring the hamburgers, chips and beer for a summer outing, and I will be quite happy. And if it's at my sunny house, sometimes I give my guests a simple lecture on sundials. The way I explain them, even those who have no initial understanding will have a good intuitive feel as to why they work.
Simply put, sundials tell solar time by the position of the shadow on some plate with marks. I emphasized solar because although clock time is based on it, this varies slightly due to irregularities in the earth's orbit. In addition, "clockwise" motion is based on the direction the sundial's shadow moves in the Northern hemisphere since that is where clocks were invented. I will touch on that in part two of this very short series. To be even more precise, sundials are really meridian markers. Since the earth turns 360º in about 24 hours, the sun moves 15º per hour. "Now what in the name of God are meridians?" you may ask. Glad you asked. Remember the globe in 5th grade geography class where you learned latitude and longitude lines? The longitudes are the lines that converge at both the North Pole and South Pole that are shaped like orange wedges. Latitude lines run across the globe in circles: the biggest one being the Equator that separates the top half from the bottom half, and then the latitude circles get smaller as they near the poles. Anyway, meridians are the longitude lines. So, when we say the "sun" moves 15º per hour, what is that respect to? That is respect to the axis of the earth. Let's illustrate this with a picture.
The red line at the top (pole) of the globe represents your gnomon (pointer) of a sundial on the North Pole. Notice that it is parallel and indeed an extension of the earth's axis. See where the longitude lines all converge? Looking from the "top down" they would appear like spokes coming from the center of a wheel. I think all but the most geometrically obtuse individual can see that. If that wheel spins at 15º per hour, I think most people can see that the shadow of the sun would align with the longitude line opposite the direction that the sun was shining from. The shadow would just keep going around and around at a fairly constant rate (the differences being due to the slight orbital irregularities) just like a clock hand. Pretty boring, isn't it? Since the sun does not rise or set during the summer months at the pole, we could designate any meridian we wanted to represent midday and then the shadow would move around that pole onto the black surface every hour. In this illustrative case, the faceplate with the markers is represented by the perpendicular black line. One other thing. Notice that there is a green line superimposed over the black. That is also the horizon at the North Pole. This means a sundial at the North Pole is a vertical stick. I think this makes sense. Just shine a light onto a spinning globe with some sort of extended axis and see how the shadow goes around in a circular motion relative to the pole on that globe. If this is difficult to understand for anyone up to this point, I'd say don't waste your time and log off now, because this is pretty straightforward in my opinion.
Now, look at the globe illustration again and see that I have two other sundials on it. The sun is very far away, and for all practical purposes the rays are parallel, in this case lets say from the right side of the picture. Therefore, if we take our polar sundial and move it to a middle latitude, say 40º North, or even to the equator, in both cases these sundials would work perfectly well as long as their gnomon (pointer) is parallel to the earth's axis. 15º rotation at the Equator would be the same as 15º at the pole, and the shadow would shift the same. But notice one thing. The green line (your horizon) in the middle latitude is tilted relative to the black dial plate and the red gnomon. On the left I drew the diagram relative to your horizon so that it's "flat". See what happened? The red gnomon is tipped at the angle of your latitude. This is why sundial pointers where most of us live are "crooked". But if the hour marker plate (the black line) is perpendicular to the gnomon, the hour angles are still graduated evenly like at the pole. Such a dial is called an equatorial sundial because the face plate is along the "equator" perpendicular to the pole. If the reader noticed something here, the sundial is sort of a model globe. When we get to the equator, the gnomon is actually laying on its side! In a case like that, another form of sundial works best there called a "polar" sundial. In this case, the shadow lengths mean more on a faceplate, although an equatorial dial on its side would work just fine.
On popular type of equatorial sundial is called the armillary. I have a picture of mine below. Notice the tilt of the center gnomon axis so that it is parallel to the earth's axis (I live at about 40.6º North latitude), and the hour markings are evenly spaced on a ring perpendicular to the pointer, just like in the globe illustration above. In this picture, you can see the hours 7, 8, and 9 marked. This was taken near noontime and my only gripe about this sundial is the polished finish on the hour circles which causes a lot of glare. Otherwise, you can see that a sundial is no more than a "skeletal" globe marking the meridians the sun is over.
We can see that if one wants to take an old globe and cut half of it away (along a longitude line) and put a string down the middle from north pole to south pole that the sun would move around it evenly and this would make a nice novelty sundial. Again, the pointer would have to be parallel to the earth's axis. And by the way, in case this has gone unnoticed, if the gnomon is parallel to the axis, it goes without saying that the sundial gnomon points north at the elevation of one's latitude.
Now that we all should have a good intuitive understanding that a sundial is nothing more than a globe and the hour lines are really longitude lines, let's change the shape of the sundial a little. No matter how weird the sundial looks and no matter how unevenly spaced the hour lines look, one thing is for certain: the gnomon is parallel to the earth's axis. The sun, no matter what season, always moves about a 15º per hour rate around that gnomon. Always. So, what does this mean? It means that we can imagine a "ray" emanating from the tip of the pointer (perpendicular) and where that ray falls onto the dial surface is where the hour marking will fall. One fellow told me you could stretch string---- but I like to calculate, and for those understanding basic high school trig, this is not difficult at all.
A horizontal dial is the most familiar. This is the flat hour marking plate with the pointer (gnomon) tilted. You most often see it in gardens. But, because the hour plate is parallel with your horizon instead of perpendicular to the gnomon, the hour lines are marked unevenly. This is due to the fact that at solar noon, the gnomon is closest to the dial plate and then as the sun moves around the gnomon at even rates, the shadow is cast onto the plate that is further away from the gnomon, and therefore the hour spacing is greater (stretched) at later hours (or early in the morning).
Here is a horizontal plate I made from wood minus the gnomon. The gnomon, if it were there, would face upwards in this picture with a tilt equal to the latitude. As a matter of fact, you can see the center line where the gnomon once was.
This is graduated in half-hour increments. Notice near noontime the markings at the top are pressed together whereas the earlier or later hours are further apart. And notice one other thing. Noon marking is ALWAYS facing north like the gnomon. And the 6 o'clock morning and evening marks are ALWAYS in line with the base of the sundial' gnomon. In other words, if you drew a line to connect the 6 o'clock markings, the base of the gnomon should touch there. NO EXCEPTIONS. If you want to buy a sundial and it does not have those properties, then it is most likely bogus and decorative only, not functional. The exception may be a custom made sundial but that is not sold generically anyway. And if we understand already that the sun moves 15º per hour, we can see why the 6 o'clock marks are where they are. 6 hours translates to 90º. Looking down at the edge of the gnomon at that angle (and for conventional sake say that 0º marks noon), we see that at the base the shadow would face due east or due west depending on whether its morning or evening.
Now for the math. We can further demonstrate the truth of what I just stated mathematically and calculate the angles of each hour. Here's the illustration that will demonstrate this:
Red triangle VAB represents your (tilted) gnomon with VA pointing north and indeed represents the axis of the earth. Notice that this forms Lº angle based on your latitude. Also notice that this triangle by definition is perpendicular to the surface of the earth. The green triangle VBC forms the shadow casts from the gnomon and is level with your surface horizon. Angle A is the hour angle we are going to calculate soon. Line BA points south to your local meridian and when the sun passes this is defined as solar noon when the shadow is shortest and points north. For simplicity sake, we will use an equanoxial sun for our calculations which is why BA is perpendicular to VA. This makes perfect sense since VA is the axis and when the sun is over the equator, such as the first day of spring or fall, it is at a right angle to it. However, for those more advanced in math, it can be shown (but not here) that the calculations would be the same for other times of the year simply because the angles are relative to the axis.
So, for convention, noon is ZERO degrees and marks when the sun is south. Suppose we want to calculate angle A for one o'clock? We know this. The sun moves 15º per hour represented by angle HA. In other words, if you looked down axis VA like the site of a gun, we would rotate a line 15º (HAº) from the bottom. This would make another triangle ABC. Again, the red triangle is vertical and the green along your surface, so I think our readers ought to be able to see that BC is perpendicular to both AB and VB. I like right triangles because we can use sines, cosines and tangents to figure other angles. For convention's sake, I labeled VB as "one unit" in length. "But Liaf, what if it's 8.5 inches?" It does not matter. Let's then make up a new unit and call 8.5" one fgert. Since we are using trig here, we are interested in the ratios of the lengths to obtain the angle A. The rest is now straightforward using simple trig:
AB= sin(L)
BC/AB = tan(HA) or... since AB= sin(L) then BC/sin(L)=tan(HA) or....
BC=sin(L)tan(HA)
Furthermore, tan(A)=BC/1=BC
Therefore, tan(A)=sin(L)tan(HA).
Let's try this at Liaf's latitude. What is the angle A of the sundial at 10 o'clock in the morning? In case you haven't noticed it yet, the lines are symmetrical, so the angle at 10 o'clock is the same, but opposite of the 2 o'clock line, with the morning lines on the west side of the sundial.
L=40.6º; HA=30º (i.e. 2 hours times 15 degrees=30)
tan(A)=sin(40.6)tan(30)
tan(A)=(0.6508)(0.5774)=0.3757
tan-1(.3757)=20.6º
In other words, the 10 o'clock line would make a 20.6º angle with the north noon line pointing toward the west side of the dial. If you look at the above horizontal face plate I carved in wood, we can see that this is the case. You will also see that if we calculate for say, 7 o'clock in the evening (which only a summer sun will cast a shadow on), it will be a "mirror" image of the same angle as 5 o'clock. Think of the sun as shining from "underneath" the dial plate as it rotates past the surface on the axis and you can see why this makes sense. How about 6 o'clock? Let's try the formula.
L=40.6º; HA=90º
See where this is going? If we put HA=90 into the equation, we get infinity. The inverse tangent of that (for Aº) is still 90º. It matters not what your latitude is.
OK.... this concludes part 1. There is another part 2 coming later. This is where we convert sundial time to your local clock time. What we gave in this lesson is telling your local solar time. Before the invention of uniform-moving clocks, this was enough. The subtle differences in the length of the day is too little to be noticed by the human mind, but moderately appreciable to a precise timepiece. Think of time as this way: if we go 60 mph down the highway, we pass a mile marker (analogous to a meridian) each minute. When we passed the ten-mile marker, going at that speed, we know we've been going for ten minutes and therefore by the markers we can tell the time. However, the "car" called the earth does not go at a steady speed. It varies slightly, and hence the sun's movement is not uniform. This accounts for an error up to about 15 minutes from your local mean (clock) time (as opposed to a time zone). This is not severe, but enough to take a little notice with precise clocks.