Saturday, January 11, 2020

Finding UTC of LAN using an Analemma

Finding latitude at noon—when the sun is at its peak height in the sky as it crosses our meridian bearing either due south or due north—is the most basic technique in celestial navigation, but it is a relatively long process. We can minimized this time by estimating when it will occur based on our DR longitude. Then we can be on deck and ready to go maybe just 10 minutes earlier, and not waste a lot of time, often in extremely hot temperatures under the high sun.

In another post we show how to figure the time of LAN from our compact perpetual sun almanac by just looking up the time that the GHA of the sun is equal to our DR-Lon. In this note we look at another alternative to the traditional procedures that requires a Nautical Almanac.

The traditional procedure is to look up the UTC of meridian passage (mer pass) at Greenwich from the daily pages of the Nautical Almanac and then correct that for our longitude, and then convert that UTC to watch time WT.

The Nautical Almanac gives this time accurate to the second, but we have no need for that precision, not to mention that our DR-Lon could be off enough to shift this time a few minutes—in the tropics, a 15 mile DR error would be a 1 minute time error.

A quick way to get the UTC of mer pass at Greenwich without an almanac is to have at hand the unique figure shown below called an analemma.

These figures are usually drawn slightly differently to match the track of the sun in the sky throughout the year, but we made this one years ago specifically to be used to find real values of the sun's declination and the Equation of Time (EqT) based on the day of the month. The link above tells more about the origin of this famous figure—well known to those who know it well.

To use this drawing, estimate the date of interest along the curve that marks the first of each month, then the sun's declination is on the left scale and the EqT is on the bottom. Each dot is 1 minute of time or 1 degree of angle.

The Equation of Time is not the relativistic secret to the universe that it might sound like, but rather the more humble difference between 1200 UTC and the actual UTC that the sun crosses the Greenwich meridian. In other words:

UTC of mer pass at Greenwich = 1200 UTC ± EqT


Thus if the Nautical Almanac tells us that the UTC of mer pass at Greenwich is 1207 on some specific date, it means the EqT is +7 min.  If the Almanac says UTC mer pass is 1144, it means EqT is -16 min.

We want to use this the other way around. We know a date we care about, then we use the diagram to find the EqT and apply it to 1200 to find the UTC of mer pass at Greenwich on that date.

Once we know the time the sun goes by Greenwich, we can figure when it will get to us. The earth turns 360º in 24 hr beneath the sun, which means the geographical position (GP) of the sun  moves west at the rate of 360º/24h = 15º/1h = 15'/1m. These can be further rewritten at 1º = 4 min and 1' = 4 sec.

An example: The date is July 19 and my DR-Lon is 138º 25', what time do I expect the sun to be at its peak height in the sky, bearing due south?  My watch is set to PDT, zone description (ZD) +7.  The analemma tells me the UTC at Greenwich is 1207, so we are just left with converting 138º 25' to time and adding that to 1207.

The usual solution here is to refer to the Arc to Time Table from the Nautical Almanac, where we find that 138º = 9h 12m, and 25' = 1m 40s, so our DR-Lon is equivalent to about 9h 14m. We add this to 1207 to get 21h 26m UTC, and for watch time we undo the ZD  to get 14h 26m.

Without such a table, we can figure it manually. One shortcut is just divide by 15 to get the hours and then figure the minutes. In this case 138/15 = 9.2 hr = 9h 12m and then add on the arc minutes part of the Lon: 25' x 4s/1m = 100s = 1m 40s or about 2 min, so the answer is 9h and 14m, which is what we get from the tables. (See section of an Arc to Time table at the end here.)

In eastern longitudes, we subtract our DR-Lon time from the Mer Pass time at Greenwich, because the sun moving west goes by us first before reaching Greenwich.

Note the above example did not have a date. These times do vary slightly over the leap year cycle, then repeat every 4 years, but this variation is just a minute or so, which is not crucial to our planning needs.

That is the end of the procedure discussion. Below is a bit more on the motion that causes this.

___________

There are two reasons the sun's GP does not circle the earth at an exactly constant rate throughout the year, which leads to the varying times of LAN. One is the earth's orbit is not a circle, but rather it is slightly elliptical, and orbital speed changes slightly at various parts of the ellipse. The other reason is the tilt of the earth's axis relative to the plane of its orbit, about 23.4º. This adds a N-S component to its actual path across the earth, leading to a varying westerly speed. This also leads to the tropics band on earth (23.4N to 23.4S) that covers all latitudes where the sun might be directly overhead.

Both of these effects are regular cyclic patterns, but they are not in phase, which leads to the unusual shape of the EqT shown below as well as to the odd shape of the analemma.


_________________






No comments: