Friday, April 19, 2019

Finding UTC of LAN with the Starpath Custom Sun Almanac

We published a short book last year that is intended to be the bare minimum celestial navigation training that will serve as a backup to a loss of GPS. In keeping with the backup concept, the book is presented in such a way that it is totally self-contained, which means the book also covers how to use the simple Davis Mark 3 sextant. This is a small device, for under $50, that could be used to safely circumnavigate the world.

Our promise is that this book could be opened up and read for the first time when it was actually needed, and it would be adequate to teach how to take the sights, and then find your position from them using only data in this small book—a large part of which is a custom Sun Almanac designed to make the position solutions especially easy.

The techniques taught in this book are finding Lat and Lon from "noon sights" (local apparent noon,  LAN) as well as Lat by Polaris in the Northern Hemisphere. What we forgot to include in this first printing was a way to determine an efficient time to start taking the sights near midday. LAN occurs when the sun crosses our meridian, bearing due north or due south at its peak height in the sky.

When the sun is less than halfway up the sky at noon, we can approximate its motion along the horizon as a bearing change of 15º per hour.  So if we want to start about 30 min before the sun reaches the horizon, we would start taking sights when the sun was bearing about 173 T.  After that first round of sights, we will know the time of LAN and the next day we can fine tune the starting time.

When the noon sun is much higher in the sky it is more difficult to predict its bearing change rate as it approaches noon, and as we head off toward the tropics the sun will indeed be much higher at noon, so it is valuable to have a systematic way to predict the time of LAN to plan around. The custom Sun Almanac offers an easy way to do this. In fact, it is easier than the standard methods we use when teaching the "full cel nav" course.  The full theory is in the picture below from our textbook Celestial Navigation: A Complete Home Study Course.

As the earth rotates toward the east, the position on earth directly below the sun (its geographical position, GP) moves west at the rate we are rotating, namely 360º of Lon in 24h = 15º of Lon per hour.

The custom Sun Almanac tells us the longitude of the GP (called its Greenwich hour angle, GHA) every hour of every day. So we can go into Sun Almanac on the right day to see which whole hour of UTC has the sun's GHA just east of our location.  Then we subtract that GHA from our DR-Lon to see how far it has to go (as an angle) to get to us, and then we just covert this angle to time at the rate we are turning, which can be derived from 15º = 1 hr, 1º = 60 min/15 = 4 min, etc.

                      1º = 4 min

                      1' = 4 sec.

Here are two practice problems from our textbook (page 33.)

Example 1.  July 25 from DR-Lon = 122º 18'W.

DR  = 122º 18'   =  121º 78.0'
20h = 118º 21.7' = -118º 21.7'
            diff =    3º 56.3'

3º = 3º x 4m/1º                 = 12m
56.3' = 56.3' x 4s/1' = 225.2 s =  3m 45s
                            SUM = 15m 45s

Final time is 20h 15m 45s, which we would round to 2016 UTC, as we never need to know these times more precise than that, not to mention that the DR position that it is based on has some uncertainty.

This is the UTC of LAN observed from 122º 18' W. If we wanted the watch time (WT) of the event for a watch set to zone description (ZD) + 7, then we would have to back out that ZD.

Recall the definition of ZD, which comes from this equation UTC = WT + ZD, where WT is the watch time being used for navigation. Thus if the ZD = +7 and the WT = 1200, then the UTC = 1900. Likewise if we know the UTC is 1900 on a watch with ZD = +7, then the WT = 1200. 

In the above example, LAN = 2016 UTC would correspond to WT = 1316 for a watch set to ZD = +7.

Example 2. Find  UTC of LAN on October 27 viewed from 136º 10.5' E.

This is an eastern longitude. In east longitudes the GP of the sun is still moving west with increasing time, so the longitude it crosses gets smaller as it passes across the eastern half of the globe. But that does not matter to us, because, unlike longitude, GHA does not decrease on the eastern half of the globe. GHA is defined as 0º to 360º, measured west from  Greenwich.  Our job is to determine what GHA does 136º 10.5' E correspond to. Starting at Greenwich we head west to 180º W (the dateline), and then proceed from 180º E to 136º 10.5' E, or we cover 180º - 136º 10.5'  =  179 60 - 136 10.5 = 43º 49.5', which is what we must add to 180º to get the GHA equivalent of this eastern longitude. The answer is 223º 49.5'.

Now we are back to solving the problem just as we did for western longitudes.

 "DR" = 223º 49.5'E 
  02h = 214º  1.7'  
 diff =   9º 47.8'

9º = 9º x 4m/1º                  =  36m 00s
47.8' = 47.8' x 4s/1' =    191.2s =  3m 11s
                              SUM = 39m 11s

We add this to 02h to get 02h 39m = 0239 UTC,

If we need to convert back to WT, we refer to the definition: UTC = WT + ZD, or with ZD = -9, we WT = UTC +9h = 1139 WT. 

For those who might like extra practice, you can use the USNO sun data computer (Form B) to randomly select locations and dates to compute UTC of LAN (they call it "sun transit"), and compare that with what you get from the custom Sun Almanac.

In principle the answer depends on the year, but our custom Sun Almanac averages out yearly differences, but even with this, the time of LAN found from the these tables will always be right to well within one minute.

• the USNO site is down until April 30, 2020
• the new printing of this backup book now includes an arc to time table as well as an analemma as an alternative means of finding time of LAN.

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