Wednesday, December 21, 2011

Darkness falls across the land

Happy solstice festivals!  This year (as every year, but more so for some reason), a number of people have been asking this question: Why isn’t the earliest sunset on the date of the solstice? Indeed, the solstice this year is at 05:30 UT (12:30 am EST) on December 22, while the earliest sundown in many places (as many noticed, particularly Sabbath observers) was around two weeks earlier. (This depends on location, but here in the DC area, the earliest sundown was at 4:45 pm on December 7, while sundown on December 22 will be 4:49 pm.)

The answer to this question depends on what you mean by “why”, and I’ll try to answer both interpretations of the question.

(I apologize to readers in the southern hemisphere that this post is written from a borealocentric perspective.  To apply it to the southern half of the globe, either change "December" to "June" (and switch around some of the specifics), or change "shortest" to "longest" and swap "earliest" with "latest".)

1) Phenomenological:  I thought the solstice was the shortest day! If the earliest sundown was on December 7, why don’t we call that the solstice?

The winter solstice is indeed the day with the shortest amount of daylight (i.e. the shortest time from sunrise to sunset).  But sunrise and sunset times don't move symmetrically.  During the fall, sunset times are getting earlier and sunrise times are getting later.  But after we hit the earliest sunset (sometime last fortnight), sunset starts getting later again (albeit slowly*), while sunrise continues getting later (somewhat faster).  Thus, the day is still getting shorter.  This continues until the solstice.  After the solstice, sunrise is still getting later but has slowed down, while sundown is also getting later and has sped up, so the day is getting longer.  Finally, sometime in early January, we get to the latest sunrise, after which sunrise starts getting earlier.

This is all easy to overlook since many of us, for a variety of reasons, are more attuned to sundown times than to sunrise times.

[* For those familiar with calculus, this makes sense if sunset time is a smooth function:  on the date of the earliest sunset time, its derivative is zero; therefore, near that date, its derivative must be small.]

For a concrete example:  Here in DC, sunrise on December 7 was 7:12 am (so the time from sunrise to sunset was 9 hours 33 minutes), and on December 22 it will be 7:23 am (so sunrise to sunset will be 9 hours 26 minutes, which is shorter!).  The latest sunrise in DC won't be until January 6, when it is 7:26 am.  (On that day, sunset will be as late as 5:00 pm, so sunrise to sunset will be 9 hours 34 minutes, longer than on the solstice.)

(Ok, technically the latest sunrise was actually 7:39 am the first weekend in November just before we "fell back", because this country is addicted to Daylight Saving Time and stays on it much longer than it should, but that's just a clock trick that's neither here nor there.)

2) Mechanistic: Ok, but why don’t sunrise and sunset move symmetrically?

Yeah, why?  It doesn't seem like they should be independent, because the same mechanism is responsible for the changes in both sunrise and sunset times, right?  The earth's tilt means that you get different amounts of light and darkness each day depending on where you're located in the annual orbit (the standard explanation for the cause of seasons).  And there's no reason this should affect sunrise and sunset differently (after all, we would expect the earth-sun system to look basically the same if we ran time backwards).

The answer is that, separate from the seasonal variation in day length, there is also variation in the time of solar noon (the time halfway between sunrise and sunset, when the sun is at its highest point in the sky).  This variation is known as the equation of time, and is the same everywhere on the planet (unlike day length, which depends on latitude).  That Wikipedia link explains it in more detail than you ever wanted, but here are the basics:

The equation of time is the result of two factors: a) the eccentricity of the earth's orbit, b) the earth's tilt.  So if the earth's axis were not tilted and the earth's orbit were perfectly circular, then there would be no variation in solar noon, and the earliest sunset (and latest sunrise) would indeed be on the date of the solstice.

a) Eccentricity: The earth's orbit is an ellipse, not a circle.  It's close to a circle (the eccentricity is only 0.0167), so the earth-sun distance doesn't vary substantially over the course of the year; contrary to a popular belief, this is not the cause of seasons.  However, the eccentricity has another effect:  As described by Kepler's second law (and explained by conservation of angular momentum), Earth's speed as it moves around the sun is not uniform:  it moves faster when it is closer to the sun, and it is closest to the sun in January.
Not every solar day is 24 hours: that's just the average over the entire year.  The solar day (time from one noon to the next) is equal to the sidereal day (the time it takes for Earth to rotate once on its axis, relative to faraway points such as other stars:  about 23 hours 56 minutes), plus the extra amount of time it has to rotate so that the sun is at its highest point again, to account for the fact that the earth moved a little bit during that day.  The earth moves through about 1/365 of its orbit every day, so this extra rotation should be about 1/365 of a circle, and should take about 1/365 of a day:  about 4 minutes.

But at times of the year when the earth is moving faster (e.g. close to January), it moves through more than 1/365 of its orbit each day, so this extra rotation is more than 4 minutes, and the time from one solar noon to the next is more than 24 hours.  Solar noon gets later each day, which is exactly what we see above, with the sunrise and sunset data.  At times of the year when the earth is farther from the sun and moving slower (e.g. close to July), it's the opposite.

b) Tilt:  Around the two solstices, part a is basically the whole story.  But around the equinoxes, Earth's rotation is at an angle (up to 23.5°) relative to its motion around the sun.  This means that the Earth's motion around the sun in a single day corresponds to less rotation around the axis, so it has the same effect as the earth moving slower: solar noon gets earlier each day.

Putting the two parts together, the combined effect is greater around the December solstice, since the two parts act in the same direction: both the solstice and the nearby perihelion cause noon to drift later.  Around the June solstice, the effect still exists, but is less pronounced, because the solstice and the nearby aphelion act in opposite directions.  (In DC in 2012, the summer solstice is on June 20, the latest sundown is on June 27, and the earliest sunrise is on June 13 or 14.  So there's still a difference, but it's not as big.)


Advanced section:

In thinking about this question, it occurs to me that sunrise time and sunset time, as time-dependent variables, can (like any other pair of functions) be decomposed into a symmetric and an antisymmetric part (or, if you like, a differential and a common-mode signal). The antisymmetric (differential) component is the length of the day (call it L), and the symmetric (common-mode) component is the time of local noon (call it N), relative to the average solar noon in that location. (There’s also a “DC offset” (call it D) representing the time of average solar noon, which is by definition constant throughout the year and depends only on longitude – essentially where you are within your time zone – but that doesn't tell us anything interesting.) I think this is a more natural choice of basis to understand what’s going on.

Sunrise time can be expressed as D + NL/2, and sunset time can be expressed as D + N + L/2. (This analysis requires the approximation that these variables change slowly enough that fluctuations on the scale of less than a day are negligible, so that on a given day, N and L have the same value at sunrise as they do at sunset.)

Then we can look at each variable separately to see what affects it.

L has two components:  L0 is constant, equal to 12 hours plus a few extra minutes to account for the refraction of light in the atmosphere (so in a vacuum, it would be exactly 12 hours).  L1 is a periodic function with a period of one year.  The amplitude of this function depends on latitude, while the period and phase are the same everywhere.  At the equator, the amplitude is zero (so the day length is 12 hours and change for the entire year), and the amplitude increases as you go up in latitude.  In the southern hemisphere, the amplitude is negative (or you could call it positive and call the northern hemisphere negative; you'd just have to shift the phase by 180°).  Inside the polar circles, it starts to break down, since there are parts of the year when the sun never sets/rises, so L isn't well-defined.  The minimum and maximum of L are on the solstices.

N is the equation of time (but with the reverse sign because of the convention of how the equation of time is defined).  As shown in the Wikipedia article (and its graphs), it has two Fourier components: N1 (for eccentricity) with a period of a year, and N2 (for tilt) with a period of half a year.  This function is the same everywhere on earth.  The two zeroes of N1 are at perihelion and aphelion, so the maximum and minimum are about halfway in between.  The four zeros of N2 are at the solstices and equinoxes, so the maxima and minima are about halfway in between those.

Putting this together, it becomes clear why the date of the earliest/latest sunrise/sunset depends on latitude:  you're combining functions with two different periods, and the extrema of the combined function will depend on the amplitudes of the components (set the derivative to zero and solve!), and the amplitude of L1 varies with latitude.

1 comment:

  1. So it's not a function of astrology!
    I've been noticing the apparent anomaly. Thanks for explaining.