SPU 26 Ref 11: Equation of time

Transcription

1 SPU 26 Ref 11: Equation of time The sun moves across the fixed background of stars at a rate of approximately one degree per day. Our system of time is based on what is called mean solar time. This is constructed so that, on average, the sun will be directly over the Prime Meridian at noon, Greenwich Mean Time (GMT). The modern terminology for time is called UTC for Universal Time, Coordinated. To be precise, with atomic clocks in usage, GMT and UTC differ by some number of seconds, but we ll take them as identical for our purposes. Our modern system of time is constructed to place the sun at approximately the local meridian at noon for all time zones. The reason for this is pragmatic. People wake at sunrise; work throughout the day, and go to bed when the sun sets. Since local noon depends on one s longitude, time zones are arranged so that the local noon occurs approximately at the meridian passage for a person in that time zone. Since the sun moves to the west at 15 o per hour, the time zones are shifted by one hour every 15 o from the Greenwich Prime Meridian, with a width of +/- 7.5 o on average. Time zones boundaries will dart around to avoid cutting major cities in half, but this works on average. Time zones are referenced to UTC (Greenwich mean time). If you re in Greenwich, your time zone is UTC. If you re in central Europe, your time zone is UTC+1, if you re on the east coast of the United States, your time zone is UTC 5. The number designates whether your clock is early or late compared to Greenwich. Bear in mind that UTC does not shift for daylight savings time. If you re in eastern daylight savings time, it s UTC-4, and when we shift to standard, it s UTC 5. For example, if it is 16:00 in UTC, it is 12:00 in EDT (Eastern Daylight Savings Time). The shift to daylight savings time was largely political, based on maximizing the number of hours worked in shifts in factories. There s no astronomical basis for it. While our watches work on mean solar time. The sun itself doesn t precisely obey this, and will speed up or slow down with respect to mean solar time. Navigators have to account for this, particularly in the determination of longitude, which relies on mean solar time. There are two factors that shift the actual timing of the sun from mean solar time. The axial tilt of the earth s rotation will cause the sun to move in a path across the face of the earth that s slightly angled except at the solstices. The earth s orbit is an ellipse, where the sun is not at the center, but offset. The earth will speed up and slow down in it s orbit. The result of these effects will cause the sun s position in the sky to differ from mean solar time by as much as 16 minutes. If you have a watch or means of determining time, you can use the timing of the sun s meridian passage to find longitude. Since the largest excursion from mean solar time is about 16 minutes, this is equivalent to four degrees of longitude, and translates into 240 nautical miles at the equator, so it s an important correction! It is important to note that both the axial tilt and eccentricity of the earth s orbit come into play here.

2 Figure 1 Mean sun in the sky represents the mean solar time- the basis for our clocks. The true sun can cross the meridian passage early or late, depending on the time of year. The first effect is the obliquity of the ecliptic. That s a fancy way of saying that the earth s axis is tilted with respect to its orbit. At the solstices, the sun s path across the earth is parallel to the equator. That s because at that moment, the sun has halted its northward or southward march. At the solstices, the change in the position of the sun in the sky is directed completely as a change in right ascension (SHA or, as I ve called it stellar longitude ). At the equinoxes, the sun is moving north or south at a maximum rate and the change in the right ascension is slower than at the solstices. The net effect of this is to create a shift from mean solar time over the course of a year. The sun slows down by 20 seconds per day relative to mean solar time during the solstices and speeds up by 20 seconds per day at the equinoxes. The largest shift due to this effect is about 10 minutes and occurs at the halfway points between the solstices and equinoxes. Figure 2 The difference between the path of the mean sun and true sun at the vernal equinox, creating a shift from mean solar time.

3 Figure 3 Contribution of the obliquity of the ecliptic to the equation of time. The days on the horizontal axis are the number of days past January 2nd. The second effect that comes into play is the eccentricity of the earth s orbit. Mean solar time effectively averages the earth s orbit into a circle, but the eccentricity causes a speeding up and slowing down of the actual orbit. The earth s closest approach to the sun is in early January typically January 2 nd or 3 rd. At that time, the earth is moving fastest in its orbit about the sun and arrives sooner at a position in its orbit than mean solar time would predict.. Half a year later, it s at the most distant point in its orbit and is moving more slowly. The effect is accumulated over the course of the orbit, and the net effect is to place the shift from mean solar time to be zero at the perihelion (point of closest approach) and at the aphelion (most distant point).

4 Figure 4 Contribution of the eccentricity of the earth's orbit to the equation of time. The sum of these two effects give use the equation of time, which is a squirrely looking curve that has both components included. Figure 5 The equation of time is the sum of the contributions of the eccentricity and obliquity of the ecliptic.

5 Figure 6 Equation of time and trapezoidal approximation on top of it (dasged). The sun can be as much as 14 minutes slow and 16 minutes fast compared to the mean solar time. The above figure shows the equation of time. As with the deviation from the approximation to declination, the curve can be memorized as a set of lines, again with a mnemonic. In David Burch s book, Emergency Navigation, he has a jingle to memorize this, but I found it difficult to remember, particularly when something is positive and negative. My own scheme for memorizing the equation of time is summarized in one word: chocolate. Then remember the numbers 14 and 16. There are two times during the year when chocolate is important: Valentines Day and Halloween. Being a guy, I tend to hoard chocolate. I think of the sun like myself. During Valentine s Day, I m slow to part with chocolate, and am late. Since Valentine s Day is the 14 th of February, I m late by 14 minutes. At Halloween, I m fast to get my chocolate, so I m early, and early by 16 minutes. These define two of four inflection points (where the curve turns over). The other two inflection points occur 3 months after Valentine s day, and is 4 minutes early (think -14 to 4), and 3 months before Halloween and is 6 minutes late (think +16 to -6). If you draw a horizontal line at +/- 2 weeks around each inflection point, and then connect the lines with diagonals, you can get a good approximation to the equation-oftime. In step-by-step instructions:

6 1.) Draw a graph with days of the year on the horizontal axis, time in minutes on the vertical axis. 2.) Find Feb. 14 th,(valentine s day) and draw a horizontal line at time = - 14 minutes (slow), + and 2 weeks on either side of this. 3.) Find Oct. 31 st (Halloween) and draw a horizontal line at time = + 16 minutes (fast), + and -2 weeks on either side of this 4.) Find 3 months before Oct. 31 st, take -6 minutes and draw a horizontal line + and 2 weeks on either side of this. 5.) Find 3 months after Feb. 14 th, take + 4 minutes and draw a horizontal line + and -2 weeks on either side of this 6.) Connect the lines with diagonals. The resulting trapezoidal approximation ends up being good to about 1 minute (15 arc minutes in longitude) for the equation of time. The oldest written description of what is effectively the equation of time comes from Ptolemy and other Greek astronomers. One can ask how the Greeks came upon such information with inaccurate clocks. The answer is two-fold first they understood from theory the change in time from noon to noon due to the axial tilt. Secondly, the moon moves across the sky fast enough to act effectively like a kind of clock. Ptolemy used Babylonian records of three lunar eclipses to demonstrate the equation of time. This is because one can use the time difference between sunset and an eclipse as a kind of clock. If the position of the sun in the sky is off by some amount in the timing will differ from the prediction. The Babylonian predictions were off by a measurable amount. Ptolemy noted that the above effects producing the equation of time could account for the Babylonian discrepancies. Although clocks that keep good time over the course of a year didn t exist during this era, water clocks, called clepsydra by the Greeks, were sufficient to measure short time intervals with reasonable precision. These were employed by the Babylonians to establish the timing between sunset and a lunar eclipse Water clocks were employed by ancient Egyptians, Babylonians, ancient Indians, Chinese, Greeks and Romans, and were useful in astronomy, as with other tasks that required timekeeping. More precise mechanical clocks began to come into use in the 1500 s. Before that sundials were what most people used to keep time. In the late 1700 s, pocket watches became affordable, but were not entirely reliable, as the jiggling motion of walking could alter the rate at which they kept time. Often times, a traveler would have to reset his pocket watch by a more accurate stationary town clock, or with a sundial, which was much more common. In resetting the watch to a sundial, the traveler had to be aware of the equation of time, however, as the sundial only showed the actual sun s time, as opposed to mean solar time. An almanac from 1793 discusses the method of adjusting a watch by a sundial: It is well known that common dials give what the English call apparent, and the foreign astronomers true time; which, on account of the unequal motion of the sun, is unequal: the natural day being sometimes longer, and sometimes shorter, than the mean day, shewn by clocks and watches

7 which go equably. In order, therefore to find whether a clock or watch goes right, by a common sun-dial, or to set a clock or watch to mean time, when wrong, it is necessary to add to, or subtract from, the time shewn by the dial, a certain number of minutes and seconds, usually called the Equation of time; in order to find the time which the clock or watch ought to shew or to which it should be set i. Analemma The analemma is a curious figure 8 shape that results from the simultaneous plot of the declination of the sun and the equation of time. It is effectively what you would see if you photographed the sun in the sky every day of the year at precisely the same time. It s important to note that over time, the earth s orbital parameters will change, due to effects like the precession of its axial tilt, so this has to be recalculated annually, although the above approximations will be reasonable for primitive navigators over many, many years. It is possible to design a sun-dial that takes into account the equation of time and the declination of the sun. Such sundials are called analemmatic sundials. In order for these to work properly, one has to reposition the gnomon roughly every month. The positioning of the gnomon requires the latitude where the observer is located, but will work reasonably well for any date. For details on how to construct an analemmatic sundial, there are a number of websites available. See, for example:

8 Figure 7 Analemmatic sundial. The gnomon has to be repositioned each month. The design compensates for latitude, declination and equation of time. i Francis Wollaston, Directions for making a universal Meridian Dial, The Monthly Review or Literary Journal V7, London, R. Girffiths September December 1793 P.336