## Project I: A Model for the Length of Daylight

This project concerns using a mathematical formula called DAYLIGHT to estimate the number of hours of daylight on the x-th day of the year at a given latitude LAT.

[1] Since the author of this report lives in Omaha, Omaha will be substituted for all questions involving Lincoln. In particular, the Omaha World-Herald states for October 1, 1995 that sunrise occurs at 6:20 am, and sunset occurs at 6:06 pm. Thus there are 11 hours and 46 minutes (or 11.77 hours) of daylight on October 1.

To use the DAYLIGHT function to compute the number of hours of daylight, we first determine that October 1 is day x=274. Second, using a map we determine the latitude LAT of Omaha to be 41.5 degrees North latitude. With these inputs, DAYLIGHT gives the length of daylight for October 1 to be 11.52 hours, for a relative error, compared to the World-Herald data, of .25/11.77=2.12%.

[2] Since the newspaper only gives times of sunrise and sunset up to the nearest minute, the length of daylight as derived from newspaper data can be expected to have an error of as much as 1 minute. Since the change in the length of daylight from one day to the next is on the order of a few minutes, to get a reasonably accurate daily rate of change of the length of daylight we must compare the lengths of two days some days apart. From [1] we know that October 1 had 11.77 hours of daylight. The World-Herald likewise gives the length of daylight for October 8 as 11 hours and 27 minutes. Thus the length of daylight decreased 19 minutes over 7 days, for a daily rate of change of -2.71 min/day.

Since the output of DAYLIGHT is very precise, we can compare the length, 11.52 hours, on October 1, with the length, 11.47 hours, on October 2, to find a daily rate of change of -.05hr/day, or -3 min/day. The relative error here is .29/2.71=10.7%.

[3] Here is a graph of DAYLIGHT as a function of the day x, using LAT=41.5:

From this graph we can see that the longest day of the year occurs between day 170 and day 180. By checking the values of DAYLIGHT for x between 170 and 180, we find that the longest day occurs on day x=173, or June 22, and that that day is 15 hours and 32 seconds long. Similarly, the shortest day occurs for x=356, or December 22, and that day is 8 hours, 59 minutes and 27.7 seconds long.

To see what happens in Nome, AK or Miami, FL, we first check a map to get their latitudes (obtaining 65 degrees N for Nome and 25 degrees N for Miami). Using DAYLIGHT we find that the dates of the longest and shortest days are the same for Omaha, Miami and Nome, but the lengths of those days differ, with Nome's longest day being 21.29 hours long and Miami's being 13.55 hours, and Nome's shortest day being 2.71 hours long and Miami's being 10.45 hours. This shows that the length of the longest day (i.e., of the summer solstice) increases as you move away from the equator, while the length of the shortest day (i.e., of the winter solstice) decreases as you move away from the equator, but the dates of these days is independent of latitude. (Actually, this is not quite true, since the date of the summer solstice in the Northern hemisphere is the date of the winter solstice in the southern hemisphere, and vice versa, but for all latitudes in the same hemisphere, the dates are the same.)

[4] We now present a table of data showing the rate of change of DAYLIGHT as a function of the day x, using LAT=41.5. This data was obtained by subtracting the value of DAYLIGHT at x from the value at x+0.1, and dividing by 0.1 to obtain the rate of change. To make the data easier to relate to, the result was multiplied by 60 to convert to units of min/day.

______day x___________rate of change of DAYLIGHT (min/day)
________0_______________0.52
_______30_______________1.99
_______60_______________2.84
_______90_______________3.00
______120_______________2.47
______150_______________1.26
______180______________-0.38
______210______________-1.88
______240______________-2.80
______270______________-3.02
______300______________-2.54
______330______________-1.38
______360_______________0.24

We also present a graph of the rate of change of DAYLIGHT:

An examination of the data shows that the rate is greatest on March 22, the spring equinox (the reader is reminded that the equinoxes occur on the dates with equal parts daylight and night), at 3.03 min/day, and the rate is least on September 22, the autumnal equinox, at -3.03 min/day. The dates on which the rate is 0, June 22 and December 22, are also the dates of the greatest and least hours of daylight (the solstices), found in [3]. This makes sense, because at a maximum or minimum, a function is neither increasing nor decreasing, so its derivative (i.e., its rate of change) should be 0.

[5] Since there are always at least 12 hours of light in Spring and Summer, and at most 12 hours in Fall and Winter, days with 10 hours of light in Omaha must occur in Fall or Winter. Our graph in item [3] verifies this, and shows that a 10 hour day occurs twice, on about day x=40 and day x=305, or February 9 and November 1. If the rate of change in the length of daylight is positive, say 2 to 3 min/day, then the season must be Winter or Spring, since only in these seasons does the amount of daylight increase each day. Thus a 10 hour day with a rate of change of 2 or 3 min/day can only occur in Winter, and thus must be around February 9.

[6] Even if all you knew was that there were 16 hours of daylight today, you would at least know you cannot be near the equator, since there are always 12 hours of daylight at the equator. In fact, by evaluating DAYLIGHT for various latitudes with x=173 (the date on which the number of hours is greatest for locations in the Northern Hemisphere), you see in fact that you must be at least 49 degrees away from the equator, but you cannot determine the latitude any more precisely, since 16 hour days occur at all latitudes farther from the equator than 49.

If, however, you also knew that the rate of change in the number of hours of daylight was about 5 min/day, then you would know that you cannot be exactly 49 degrees away from the equator, since if you were at 49 degrees N latitude, a 16 hour day only happens on the summer solstice (x=173), when the rate is 0 min/day. As you go farther north, the date of the 16 hour day falls increasingly earlier in Spring, and at the same time the rate of change grows larger and larger. For example, at a latitude of 55 degrees N, the spring day that is 16 hours long occurs on day x=136, but the rate (3.33 min/day), although larger, is still too small. To achieve a day that is 16 hours long and increasing at the rate of 5 min/day, one must go even farther north. At 60 degrees N, the spring day that is 16 hours long happens for x=124, and the rate is close to 5 min/day. Farther south, the rate would be too small; farther north the rate would be too large. Thus to have a 16 hour day with a rate of change of 5 min/day, you must be at about 60 degrees N latitude, with the only other possibility being below the equator at 60 degrees S latitude.

Knowing the length of a day and its rate of change would still be enough to pin down your latitude, even if you're at the equator. Since every spot on earth has 12 hour days at some point during the year, if you measure the day to be 12 hours long (as you must if you are in fact at the equator), your latitude could be anything. But it is only at the equator that you can have a 12 hour day and have the rate of change in the length of the day be 0. Anywhere else the 12 hour day occurs at an equinox, and the rate of change will be nonzero.

Knowing the length of a day and the rate of change of the length need not be enough to pin down your latitude if you're above the Arctic Circle (recall the Arctic Circle is defined as the most southerly North latitude which can have 24 straight hours of sunlight), since at certain times of year there can be successive 24 hour periods of light or darkness. Thus you could measure the day to be 0 or 24 hours long with a rate of 0 min/day, based on which all you could be sure of is that you are above the Arctic Circle (or below the Antarctic Circle), but you couldn't be sure how far above (or below).