Why We Can't Predict Sunset Times Exactly

Introduction

So, if we can calculate astronomical refraction to a gnat's eyelash, and we know the position of the Sun to a fraction of a second of arc, we should be able to predict when it will set tonight, with an accuracy of a fraction of a second of time, right?

Wrong. (Although that's a reasonable question.)

What do we need to know? Sunset occurs when the apparent position of the Sun's upper limb reaches the apparent horizon; so we need to know the positions of both the upper limb (as a function of time) and the apparent horizon.

Well, we certainly do know the Sun's geometric position accurately. And we know its diameter, too; so we can compute the true zenith distance of its upper limb accurately at any time. That's the easy part (i.e., the astronomical part) of the problem.

However, the apparent positions of both the Sun's upper limb and the visible horizon are affected by atmospheric refraction — but in somewhat different ways. And, to calculate refraction accurately, we need to know the distribution of temperature and pressure in the atmosphere accurately. That's the hardest part (i.e., the meteorological part) of the problem.

Another difficult part has to do with just where the apparent horizon is. If it's a sea horizon, the apparent horizon depends not only on the observer's position, but also on the height of the waves. If it's a land horizon, irregularities in the topography determine the geometric position where the Sun disappears. Given the heights of the eye and the physical horizon, we can calculate the dip of the apparent horizon — but we must take refraction into account.

Refraction and dip

If the lower atmosphere had a constant lapse rate, as I assumed for simplicity on the dip and distance to the horizon pages here, refraction might be tractable; for the effect of refraction from air below eye level on the dip of the horizon would be just one fourth of its effect on the Sun at the horizon. First of all, because rays are symmetrical about their lowest points, the Sun's horizon-grazing ray is bent as much beyond the horizon as it is between the eye and the horizon; so the bending between eye and horizon is half the total bending of the solar ray below eye level. But ray bending near the horizon adds very little to the dip, because of a perspective effect (see the discussion of parallactic refraction on the terrestrial refraction page); if the eye is low enough that we can assume the ray is an arc of a circle, this decreases the contribution of refraction to the sea horizon's dip by another factor of 2.

Thus, refraction near the horizon itself — i.e., near the sea surface — hardly affects the dip; but it affects the distance to the horizon, and the Sun's refraction, with full force. As a result, the dip turns out to be affected by the mean curvature of the ray; while the distance to the horizon, and the solar refraction, are affected by the (reciprocal of) the mean reciprocal of the curvature. So, in the real world, where the lapse rate varies with height, the dip depends on just the (arithmetic) mean lapse rate, while the Sun's refraction depends on the reciprocal of the mean reciprocal — i.e., the harmonic mean — of the lapse rate. (See my 1998 paper with George Kattawar on the dip diagram for technical details.)

Therefore, while we can find the dip from just the temperatures at eye level and at the sea horizon, without knowing the detailed temperature profile in between, the refraction of the setting Sun depends on all the details of the temperature profile. In general, the factor of four discussed above is a lower limit; if the ray curvature is similar to the Earth's curvature near the sea surface, the solar refraction can become very large, even though the dip of the sea horizon is hardly affected. (This actually happens in Fisher's Type B sunsets.)

Variations in dip and refraction

The variations in dip of the horizon are several minutes of arc, and are easily visible to the naked eye. For example, Riccò (1889, 1890) noticed the changes in apparent height of the sea horizon, compared to nearby architectural features, while looking out his office window.

As the variations in astronomical refraction at the horizon are larger than the dip variations by a factor of at least four, they clearly amount to a few tenths of a degree ordinarily, and sometimes reach a few degrees. These angles correspond to a minute of time or so at low latitudes, and several minutes at higher ones. In the polar regions, the observed times of sunrise and sunset can differ from predictions by several days; see the section of the bibliography devoted to the Novaya Zemlya effect.

So it is hardly surprising that the standard tables of sunrise and sunset times are given only to the nearest minute. That's about as accurate as we can expect a prediction to be. As G. M. Clemence, the Director of the Nautical Almanac Office, wrote in 1951,

… the indeterminacy is a geometrical one, and the refraction at any instant may differ by several minutes of arc from the most accurate value that can be calculated.

Evidently, the variability of refraction is the big problem. But can't we just use the atmospheric profiles measured by radiosondes twice a day to calculate the refraction accurately?

No. It turns out that standard meteorological instrumentation isn't accurate enough to allow the calculation of refraction very precisely, even at the astronomical horizon. And below it, at the apparent horizon, the refraction is still more sensitive to the details of atmospheric structure that are missed by the radiosondes, as Bruton and Kattawar explained in their papers of 1997 and 1998. In any case, the radiosondes aren't launched from sea level, but from (usually) airports, at locations many meters above the sea; so even if the instruments had perfect accuracy and unlimited resolution, they couldn't provide data for the most important part of the profile: where the sunset ray is horizontal — i.e., the part below eye level. (See my 2004 paper on low-altitude refraction for more details.)

For an excellent and thorough discussion of refraction near the horizon, see Fletcher's great article.

Waves and dip

An additional complication in computing the dip of the sea horizon is that the apparent horizon corresponds to the highest wave crests, where the air temperature is different from the sea-surface temperature. According to Freiesleben (1949), the typical height of the waves is 40 cm, and the apparent horizon occurs at a height of about 50 cm. (An English summary of this work appears in his excellent 1950 review paper, and another brief summary of his results appeared in English in 1954.)

This subject is discussed further in Lutz Hasse's thesis (in German; but summarized in English in 1964.)

Mirages and dip

Hasse (see above) was the first after Bravais (1853) to point out explicitly that anomalous refraction can even shift the apparent horizon above the apparent one — though “negative dip” had in fact been observed by several people, notably Koss & Graf Thun-Hohenstein (1901) and von Schrötter (1908). (Searching the bibliography for the phrase “negative dip” will produce several other examples.)

Hasse's unique contribution was to point out that a strong thermal inversion can produce a Kimmfläche or “horizon surface” above the horizontal. This “false horizon” can best be thought of as the upper edge of Wegener's blank strip associated with a superior mirage. If that mirage contains only an inverted image of the distant sea surface, it may not be noticed. However, as Wegener (1918) pointed out, this strip blocks the observer's view of the Sun (and other celestial objects); so its upper edge will be where the setting Sun disappears, if the strip's lower edge is hidden by the physical horizon. Because of the very great refraction at this false horizon, the sunset will appear to be greatly delayed.

A similar elevation of the apparent horizon can sometimes be produced by a distant layer of coastal stratus. If its upper surface is flat enough, it can be mistaken for the sea horizon. This may account for some reports of anomalously small refraction at sunset (i.e., early sunsets).

Also, if there is an inferior mirage at the horizon, the “last glimpse” of the Sun occurs a few minutes of arc above the apparent horizon. (See, for example, an animated simulation of an inferior-mirage sunset — or the same sunset at high resolution.) As the diurnal motion carries the Sun through 1 minute of arc in 4 seconds of time, that alone will make the sunset appear several seconds before the expected time.

Nominal Predictions

The standard predictions of sunset times are certainly useful to observers and green-flash hunters; just don't take them too literally, or expect more accuracy than they can deliver. They assume the true depression of the Sun's center is 50 minutes of arc; that's the sum of an average radius of 16 minutes, and a nominal horizontal refraction of 34 minutes.

The place to start is the Naval Observatory's website, which provides sunrise and sunset predictions for each day, and can generate tables for every day in the year. Dip of the apparent horizon is completely neglected; the calculations are for the astronomical horizon, not the apparent horizon. (See the section called “Technical Definitions and Computational Details”, about 2/3 of the way down the page of “Rise, Set, and Twilight Definitions”.) Data are provided for U.S. cities only; but rising and setting times for the Sun, Moon, planets, and bright stars are provided for any place in the world at http://www.usno.navy.mil/USNO/astronomical-applications/data-services/mrst-world.

For arbitrary places on Earth, you can also get similar results from either the FAA calculator or NOAA's calculator.

Sarah Wahlberg has pointed out another page with many links to solar positional and other information. It leads to nominal predictions for moonrise and moonset, as well as twilight. Just be aware that these values are as arbitrary (and as inaccurate) as predictions of sunrise and sunset times.

© 2010, 2012 Andrew T. Young


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