Wednesday, January 8, 2020

A Modern Regiment of the North Star

The height of Polaris, the North Star, can be used to find latitude, whenever the star is visible, which is typically above latitudes of about 5º N.  The star is too faint to be seen very often at lower latitudes. The method dates back to the mid 1400s, as noted in our short review of the history of this method, which describes how early mariners read the necessary correction directly from the relative positions of other stars, in particular Kochab, the brighter of the two Guards stars in the Little Dipper.

In modern cel nav we do not use such methods, but rather look up the corrections in the Nautical Almanac, which change from year to year. The total correction is made up of three parts (a0, a1, and a2) which are applied to the observed altitude Ho of Polaris. This process is unique in that it is the only navigation technique that is fully described in the Nautical Almanac, whose instructions otherwise deal only with the astronomical data it presents.

In our recent book called GPS Backup with a Mark 3 Sextant we distill celestial navigation down to the easy-to-learn-and-apply basics needed to find your position should other methods (notably GPS) not be available for any reason. This leaves us with Lat and Lon from a noon sight, globally, plus Latitude from Polaris in the Northern Hemisphere.

As we are dealing with a backup solution, and indeed using a sextant with an accuracy of just a couple miles (the Davis Mark 3 for about $49), we do not need the precision of an almanac solution of the Polaris measurement. Instead, we can go right back to the fifteenth century basics that went on to be used for the discovery of most of the world as we know it.

This method is time and date independent, which is just what we want for a backup method. To this end we have created a modern regiment of the North Star based on the location of the star Alkaid (trailing star of the Big Dipper) or we can use Segin, the trailing star of Cassiopeia on the other side of the pole. Having options on opposite sides of the pole is required for this method because we use convenient reference stars that are well separated from Polaris, which means one might be below the horizon. The key star Kochab used in the traditional solution is 16º away from Polaris, which means that method works down to about 16º N; our method works anytime Polaris is visible.

The geometry of the new rules is shown below.
Our task is to determine the bearing of Alkaid or Segin as if they are laid onto a compass card.

The corrections are then read from this table, which is the one in the book cited above. We add one with finer steps below.

Here are several examples.

You can get a rough guess of the angle S or A by just looking at the stars.  Here is a sample.

First identify the stars, then estimate the angles as if you were looking at Polaris through a hole in the center of a compass rose, with the North-South line perpendicular to the horizon.

When I did this test by eye alone, I called S = 20º to the right of vertical (i.e., S = 020), and A = 30º to the left of vertical (A = 210). I know they should be 180 apart, meaning 20º to the left, but to my eyes on this picture, A looks bigger without any further aids.

 In our GPS Backup Kit we include a Douglas protractor and a roll up ruler that together can be used for alignment. To optimize this viewing at night, we found that a white tape along the bottom edge of the clear protractor made it easier to hold this bottom edge parallel to the horizon. In the kit we use a short piece of a plastic page binding clip that can be slid on and off so the protractor can still be used for other navigation.

We have learned that it is relatively easy to measure these angles fairly accurately using a tool of this type.

This is a schematic view. It will not look quite as nice as this, and indeed having a ruler to hold

 up to it will help. 

With such a tool we can read off better values, as noted below

Angles measured to determine the Polaris correction with the Backup regiment.

Now we could interpolate the Q-Table in the book, or make a new one with expanded scale, as shown below.
Our S = 024 would imply a Q= -31. It is plenty accurate to round these, or take the one in the middle if closer.  [Technical note to the careful observer: this formulation of the table did not reproduce exactly the values of all entrants of the original tables, though still well within expected uncertainties. As soon as possible, we will track down the best values and make the needed updates.]

If in this example we were using a sextant with an index correction IC = 2' OFF from an eye height HE = 10 ft, and we measured the sextant height of polaris to be Hs = 48º 56.9', we would make the corrections for IC (+2.0'), dip (-3.1'), and Refraction (-0.8') to get an  Ho = 48º 55.0', and then

Lat = Ho ± Q = 48º 55' -31' = 49º 24' N.

Here are two practice exercises with answers. Remembering in looking for the key stars, that the Big Dipper, home of Alkaid, is about the same distance from Polaris as Cassiopeia (home of Segin) is on the other side.  These two constellations are thus spread across a large span of the sky.  When the S-A line is roughly parallel to the horizon, the two stars span the northern quadrant of the sky, near 90º apart.

Example 1.

July 19, 2021, UTC = 05:56:28
DR: 44N, 141W
Hs Polaris = 43º 23.4’
Find Lat

Example 2.

July 14, 2020, UTC = 14:42:44
DR: 25N, 151W
Hs Polaris = 25º 29.4'
Find Lat


Pros and Cons
Pros: The regiment method does not need almanac, nor time, nor any dated tables. Even no tables work if you refer to the figure at the top and remember that Polaris is 15º forward of the Alkaid to Segin line, with the Polaris on the Cassiopeia side. Then draw it out and you can create the Q tables knowing the max offset is 40'.

Cons: we have to make the S or A angle estimate, which takes a bit of practice,  and in rare cases we might see Polaris for a good altitude, but not either of the two needed stars.

The traditional almanac procedure with known UTC has the advantage of being pure cookbook, no such star angle estimates needed, and indeed the results will be more accurate. With the Regiment we consider 3 or 4 miles accuracy as good. With timed sights and almanac reduction we expect 1 mile accuracy as good.

How to practice reading the angle
(1) Go outside at night and with low background lights study the north sky. Then by eye alone, estimate the angles S and A and write them down with the time.  Then use what tools you find best to estimate the angle of Segin or Alkaid, usually relative to horizon or the vertical is easiest. Write those values down and note the time.

This angle practice can be done anytime of night, even though we must eventually take the sights at twilight when we can see the horizon. For this practice you just need some way to discern a vertical line or horizontal one, even though you may not have a sharp enough horizon for the sight. Indeed, you can practice this when the horizon is blocked by buildings. You can often use their roofs for a reference level.

(2) Back at home or at work, on the computer, set up Stellarium (free super program, Mac or PC) with your time and date, Lat and Lon, and see the same sky you were looking at.  Then measure S and A from the screen in some manner, or print and use a protractor.  Mac users can use the excellent PixelStick app for this.

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