By Liaf
OK folks. To continue on from last month's basic sundial lesson to this month's correction factors that we have to add to sundial time to get your clock time. If you understand last month's lesson regarding the geometry of the sundial, and at least a general feel for the corrections one has to add to make the sundial tell the correct time in this lesson, you will indeed have a pretty good understanding of all sundials. The rest is just design variation of the sundial itself to make an interesting-looking timepiece.
I am talking about a correction factor called the equation of time. Indeed, this sounds very Einsteinish, but this is just a fancy name for the equation that explains why the sun does not move at a nice steady rate along the ecliptic during the year. There are two main factors that contribute to this equation. The first is the fact that the earth is tilted on its axis 23½º relative to the equator. Our clocks are what we call "mean solar time" as if the sun moves along an ecliptic that is on the equator. However, in actuality, the sun moves around the ecliptic inclined by that 23½º angle to the equator. This introduces cyclical various where the sun moves ahead, then back relative to our hour lines. More on that shortly. The second factor is that the earth moves along an elliptical orbit instead of a circular orbit. For those of you that know a little about Kepler's Law of Planetary motion, the earth has to speed up when it gets closer to the sun so that it does not fall into it (called the perihelion at its closest point) and at it's furthest point slows down. The solar day is 24 hours, but as I mentioned in earlier lessons on the stars (see the star index page), the earth has to turn about an extra degree per day to catch up with the sun because of the revolution around the sun. An actual rotation is 23 hours and 56 minutes, but the extra 4 minutes is when the earth spins slightly more to face the sun due to its offset in revolution. However, one can see that if the orbital speed is faster at perihelion, for example, in the course of one rotation the earth revolves a little more around the sun that day, making it necessary to turn just a tad more to catch up with the sun. In others words, and in simple terms, the day is longer. The opposite happens at the other slower end of the earth's orbit. Needless to say, this introduces variations in the mean solar time where again, the sun goes ahead, then behind the mean solar time. When we add these two effects together, we have a funny curve called the equation of time to add to the sundial time to get our clock time.
So, let's try to demonstrate these two effects by starting out with a simple earth. The earth has a perfectly circular orbit (meaning it orbits at a constant rate) and it is not tilted on its axis. The sun is always over the equator. I think most people can understand that if the earth moved this way, the sun would move along the equator once a year at a nice, even rate. The first day, the sun would be at an arbitrary 0º as a starting point. The next day, it would be at 4º about, and the next day about 8º and so forth. Since clocks tell solar time. The sun would be due south noon sharp like "clockwork" very day (since clocks account for this daily offset by telling time relative to the constant-moving sun). Likewise, sundials would agree nicely with the clocks. That would be awfully boring. Here is a picture of such an earth and what I mean.
We see that there are stars (very far away) in the background and as the earth revolves in its nice, circular orbit without a tilt, that the extra turning needed daily to face the sun (red arrow) would be constant. The sun would still gradually move relative to the background set of stars the way it moves through the zodiac today once per year. Again, the sun would move at a nice even rate like a steady car in cruise control relative to the background.
Here comes the hand of God. He takes his Godly forefinger and now tilts the earth 23½º. However, He left the earth's orbit still circular. If we look at the next illustration, we see that the sun will move along a circle of the earth tilted relative to the equator. The problem is that hours are measured relative to the equator, because that is the primary motion of rotation (the meridians or longitude lines). So, 10º moved along the tilted circle does not equate to 10º along the equator because we now introduced some vertical component to the sun's motion. They will agree at the solstices and equinoxes, making this cycle complete twice per year. At other times, the sun will race ahead then move behind the hour lines.
Another way one can visualize what I am saying about the motion not being uniform is to compare this to a linear example. Suppose you are in a car traveling on a highway on cruise control. Along side of you is a drunk whose speed is the same horizontal speed as yours, but he has an added component. He swerves left and right in a cyclical motion. I think you can see that when he is swerving either to the left or right, the horizontal component is less and he falls back behind you. When his car is straight and facing the same direction as yours, he gains speed and goes in front of you. The idea is that the more left-right component is added, the less horizontal and he loses speed. The opposite is true when he straightens out the car.
When we graph the cyclical component of the sun's movement due to tilt (remembering that the both circles agree at the solstice and equinox), we get a behavior that looks something like that below:
The vertical axis represents minutes, and we can see that the waves max out at about 10 minutes. In other words, an earth tilted 23½º but having a circular orbit will result in a sun moving ahead and behind our clocks about 10 minutes maximum. The heavy vertical line of the graph represents the zero point of the horizontal axis, and takes place January 1. Each division represents a month. In the early months, we see that the curve is downward meaning the sun is behind the clock time. This makes the days longer, hence the sun drops behind.
OK---- So now we demonstrated the cyclical motion of the sun when only a tilt is involved. Now comes the hand of God and He straightens out the earth's tilt but now makes it's orbit elliptical instead. If we superimpose an ellipse to a circle, the ellipse's major axis (its longest part) will agree with the circle but in between times, the speed will vary so again the sun moves ahead and behind the hypothetical earth on the circular orbit. This motion is easier to understand than the previous tilt motion. Again we use the drunk on the highway analogy. You are traveling at a constant 60 mph speed, but the drunk in the lane next to you does not swerve, but this time simply speeds up to 62 mph, then backs off to 58 mph then back up to 62 mph and so forth in a cyclical fashion. I think most people can see that the drunk will at some times race ahead, then other times fall behind your car. Likewise, if we have an earth on an elliptical orbit compared to one on a circular orbit, the elliptical earth will speed up during the winter season, but slow up during the summer. They will agree at the closest and farthest points, but that is all. So, from winter to summer the earth speeds ahead, and from summer back to winter it falls behind introducing a cyclical pattern once per year shown below:
The scale is the same as the previous graph, and we can see here that calculations show that this effect accounts for about 7 minutes (not quite as marked as the tilt). We can also see the graph crosses early in January, and again in early July using each division along the horizontal axis as a month. These times are when the earth is closest and farthest from the sun respectively.
Finally, here comes the Hand of God one last time to reintroduce the tilt of the earth but He keeps the elliptical orbit in place. What will be the result? Maybe you guessed it. The result will be the sum of the two effects, and will result in a graph of the sun's variability throughout the year looking something like the graph below:
We see that the graph crosses the heavy horizontal line 4 times in a year, which are the 4 times the sundial agrees with the clock. They are at about Christmas time, about mid April, mid June and early September.
Likewise, we can study the significant peaks. Around mid February, we see the sun falls behind the clocks by about 14 minutes. In early-mid May it peaks out by only about 3 minutes ahead of the clocks, then slows behind them again to bottom out about 6 minutes behind the clocks around late July or August. This swings rapidly the other way until it peaks out at a whopping 16 minutes ahead of the clocks around Halloween time. One may even notice sunsets getting earlier rapidly once autumn starts.
These curves were the results of my calculations entered into a graphical calculator program. We can see the truth of this when the resulting curve is extremely close to an official equation of time graph from a site specializing in sundials. Read this article and view their equation of time graph and see that indeed it looks the same as Liaf's here.
ONE LAST THING. Now that I enlightened my readers as to how to correct the sundial to obtain clock time, there are actually TWO other corrections to consider. Ugggg! Why did you not tell us? The clock time I was referencing is called the old fashioned "local mean time". Each town had its own until train travel became prominent and we had to consider time zones instead. Before time zones were introduced, the town next to you may have been ½º longitudinal difference from your own. Therefore, their clocks differed by a mere (but annoying) two minutes. Consequently, precise schedules had to be adjusted for one's location. However, time zones allowed everyone within a 15º longitude band (one hour) to be on the same time. Therefore if you live, say, 80º West longitude, but are on Eastern Standard time (which is for the 75º west), then you have a difference of 5º or 20 minutes (remembering that one degree is four minutes from our previous lesson). In this case, your sun is "slow" by 20 minutes because when the sun is over the 75º meridian, yours is still west of the sun (making the sun a little eastward) at the same time since your clock tells the same time as your 75º longitude neighbor. THE GOOD NEWS is that this correction is due to location, and is CONSTANT throughout the entire year. This will effectively shift your Equation of Time graph DOWN 20 minutes. For example, at November 1st when the sun is 16 minutes fast according to the graph, we take 16-20 and for your location is actually 4 minutes slow. In mid February, the graph says -14, so for your location the sun is -14-20=-34 minutes. The sundial time will be over a half hour slow that day. Do you see what happened? If that is the case, the sun will always be behind your clock time. It is just behind and "behinder" all the time. But consider yourself lucky. You have later sunsets (because the sun is "slow")at the same time and can enjoy your outdoors later than your neighbors at using the same time.
One last correction is similar. Do not forget to change your sundial time by one hour for those living in areas where daylight savings is practiced (which are most nations today). Much like the longitudinal adjustment, this is also constant. This makes the sun even "slower" by an hour, hence the later sunsets (which is the reason for daylight savings). With all these corrections, we can see why the uniformed sundial observer often quips, "This does not tell the right time." In all likelihood, if the sundial is set up properly, the time is the correct solar time, but mankind has to have mean clocks and wants everyone to live in the same time zone, so this corrupts the meaning of the solar position somewhat.
Let's summarize sundials and how to tell your clock time from it:
Using lesson one, make sure you have an accurate sundial, then use this formula:
SUNDIAL TIME + EQUATION OF TIME CORRECTION + YOUR LOCAL LONGITUDE CORRECTION + DAYLIGHT SAVINGS TIME CORRECTION= YOUR CLOCK TIME.
From the above formula, we can see how someone living in the western part of their time zone on daylight savings can have a sundial that differs by over an hour from clock time. When I look at my armillary, I can tell the time within five minutes and oftentimes on the button using these corrections. If my sundial gets off by more than 5 minutes, it usually means I have to adjust its position since the treated lumber stand that it is on is subject to change from weather conditions. A better route to go would be to have a concrete stand that has settled well.
Now that we see the corrections that go into sundials, I hope some of our readers will appreciate these fine timepieces in their next garden tour somewhere. And you are now educated enough not to say, "Why doesn't it tell the right time?"