Solar Derivative

Today is the first Earth-Solar event of 2021 — the Vernal Equinox. It happened early in the USA: 5:37 AM on the east coast, 2:37 AM on the west coast. Here in Minnesota, it happened at 4:37 AM. It marks the first official day of Spring — time to switch from winter coats to lighter jackets!

Have you ever thought the Solstices seem more static than the Equinoxes? The Winter Solstice particularly, awaiting the sun’s return, does it seem like the change in sunrise and sunset time seems stalled?

If you have, you’re not wrong. Here’s why…

The chart below shows the time of sunrise (red curve) and sunset (blue curve) throughout the year. The brown curve shows how many hours of daylight the combined sunrise-sunset times provide.

Figure 1. Sunrise, sunset, and daylight times for +45° latitude.

Note that the chart is for where I live, almost exactly halfway between the equator and the North Pole — at +45° degrees latitude.

The times differ depending on your latitude. For example, if one lives right on the equator (0° degrees latitude), sunrise and sunset never vary, nor does the length of the day:

Figure 2. Sunrise, sunset, and daylight times for 0° latitude.

Which seems boring to me. Twelve hours of daylight every day, sunrise and sunset at the same time. It’s like living in permanent Equinox. I like the change of seasons. One appreciates summer more when it’s temporary.

On the other hand, if one lives just below the Arctic Circle, at +66° degrees latitude, then the changes are more extreme:

Figure 3. Sunrise, sunset, and daylight times for +66° latitude.

Remember that above the Arctic Circle, during the winter the sun doesn’t rise at all, and in the summer it never sets. You can see that the curves are trending such that there is nearly 24 hours of daylight in mid-summer and almost no daylight mid-winter.

I can’t show you a daylight chart for latitudes above the Arctic Circle, because the simple formula I’m using blows up. For the curious, that formula is, for sunrise:

And for sunset:

See the Wiki page for Sunrise equation if you’re interested in the details. The advantage of this naïve formula is that it’s simple enough to calculate on any decent calculator or with almost any programming language.

What’s important here are the curves themselves (at least for those of us not living on the equator).

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Specifically, what’s important about those curves is that they are sine waves.

Which isn’t surprising given that the Earth is essentially round and rotating, and its orbit around the sun is essentially a circle. (In both cases, not exactly, but close enough for most calculations.)

Sine waves come from circular motion. (See Trig is Easy!) The electrical power that comes into our homes to power all our devices is called A.C. — alternating current. The voltage is a sine wave of 60 cycles per second (at least in the USA). More to the point, that power is created by generators that spin in circles.

Figure 4. Two cycles of a sine wave.

Above is a chart showing two cycles of a sine wave. Notice the blue line segments. Each of them is a straight line. They show the slope of the sine wave (at the center point of each blue line).

The four lines at the crests and troughs show the slope at the extremes of the curve, just before it changes direction. For a brief instant, as the slop changes from going up to down (or vice versa), the slope is flat — that is to say, the slope is zero.

However the three lines in the center have steep slopes. The two on the outside are sloping down (negative slope) while the center one slopes up (positive slope).

The slope of a line is simply the change on the Y-axis divided by the change on the X-axis. I’ll get more into that below. What’s important here is that the former set are flat compared to the steep slopes of the latter set.

Figure 5. Close up of the crest of a sine wave. It’s flat at the zero point.

This makes sense. If you’re going in one direction and then turn around, no matter how fast you turn around, for at least a brief instant you have to come to a complete stop.

These flat parts are the Solstices. The days have been getting shorter and shorter (or longer and longer), but at the Solstice that trend reverses direction. Looked at very closely, the rate of change slows down, comes to a stop, and then reverses.

That is why the daylight change at the Solstices seems to stall. Because that’s literally what happens.

On the other hand, the Equinoxes are at the highest rate of change during the year. The time of sunrise and sunset is changing most rapidly then.

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Let’s dig into this slope thing a bit more and learn some very basic calculus in the process. (It’ll be fun, trust me.)

Mathematically slope is exactly what we think of when we think about how steepness a hill or incline. If slope goes up we call it positive slope, whereas if the slope goes down it’s negative.

Figure 6. Basic examples of slope for straight lines.

With hills, up and down depend on where we’re standing. With graphs, it’s from left to right. More precisely, it’s from lesser numbers to greater numbers (which on nearly all graphs is from left to right).

As mentioned above, the slope of a line is the change on the Y-axis divided by the change in the X-axis. For a straight line we can simply use the end points of the line (or any points along that length). Straight lines have constant slope along their length.

In Figure 6, the top examples have slopes of +1.0 (left) and -1.0 (right), because the change in the Y-axis exactly matches the change on the X-axis. The bottom left example has a slope of zero because there is no change on the Y-axis. The example on the bottom right has undefined slope because there is no change on the X-axis, which results in dividing by zero.

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Lines on graphs are (almost always) functions — that is, there is some function (formula) that, if graphed, produces that particular line. In Figure 6, these functions are about as simple as can be.

For the positive slope of 1.0:

For the matching negative slope of 1.0:

And for the zero slope:

Let’s focus on that last one for a moment. It doesn’t have to be y=0. If, for example, y=4, then the slope is still zero:

Figure 7. Function f(x)=4 and its derivative g(x).

Notice the blue line along the zero axis. For some function f(x), there is a function, call it g(x), that is the derivative of f. That is, g is a function that tells us the slope of f at any point along the X-axis.

In Figure 7, since our function f produces a line with a slope of zero, function g is zero at all points.

On the other hand, if function f is a line that does have slope, then its derivative will be non-zero:

Figure 8. Two straight-line examples and their derivatives.

Figure 8 shows two examples. On the left, a line with a slope of +1.0 and its derivative, which is +1.0 at all points. On the right, a steeper line with a slope of +2.0 and its derivative (which is +2.0 at all points).

As you might imagine, as the red line becomes steeper and steeper, the blue line approaches infinity — another reason why a vertical line has undefined slope; its derivative is infinite. (And infinity is a concept, not a numerical value.)

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I realize we’re getting a bit deep in the weeds here, but bear with me a bit longer, we’re almost done. The takeaway so far is that when the slope is a straight line, then its derivative is a flat line. Straight lines have constant slope.

But what about functions that produce curved lines? Let’s consider the simplest possible example:

Which produces a parabola, a natural curve that appears widely in nature (anything ballistic from thrown balls to bullets to missiles follows a parabolic curve).

The graph of f(x) and its derivative looks like this:

Figure 9. A parabola and its derivative.

Now the derivative is a straight line with a slope! The key point here is that, on the left, where the red line is sloping downwards, the blue is negative. At the point x=0, where the red line turns around and slope upwards, the blue line goes positive.

Note also that, as the red line gets further away from from x=0 on both sides, its slope is more and more extreme (trending towards, but never reaching, the vertical). Therefore the blue one has larger and larger values as it gets further away from x=0.

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I was going to get into higher functions, their derivatives, how derivatives are calculated, and what their relation is to integrals, but I’ll spare you that (for now).

Here’s the punchline relevant for Solstices and Equinoxes:

Figure 10. The derivative of sine wave is another sine wave!

The derivative of a sine wave is another sine wave shifted 90° behind. Notice how the blue line crosses the X-axis (has a value of zero) at the crests and troughs of the red line. The slope of the red line is zero at those points.

Also notice that the blue line has its largest values (crests and troughs) when the red line is crossing the X-axis — the time of its steepest slope.

Finally, note that the blue line is below the X-axis (has a negative value) when the red line has a negative slope, and is above the X-axis (positive) when the red line has positive slope.

In fact:

The derivative of a sine function is the cosine function. The notation dy/dx is a common way of saying ‘derivative of’ — it’s a reference to slope being defined as the change (‘d’) on the Y-axis divided by the change on the X-axis.

[If you read this far, congratulations. You’re (mathematically) worthy!]

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All of which is to say that, yeah, the time change is happening fast during the Equinoxes, but does literally slow down and very briefly stop during the Solstices.

It’s not your imagination that, in the winter, it seems to take forever for the days to start getting longer.

The upside is that, in the summer, it also takes forever for the days to start getting shorter, so enjoy the daylight while it lasts. Summer does fade all too soon.

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Question for the day: Why is it Eastern Time and Pacific Time? Why not Atlantic Time and Pacific Time? Or Eastern Time and Western Time?

Is it because Francisco Pizarro was so impressed by the Pacific ocean that for us European pale faces Pacific just means west? (How very confusing for Asians.)

Stay pacific, my friends! Go forth and spread beauty and light.

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