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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: Averaging
From: Fred Hebard
Date: 2004 Oct 21, 17:53 -0400

```Herbert,

I'm glad somebody familiar with the parameter space stepped in.  I
started to formulate a table from your written response, and then got
stupid and simulated the whole thing from meridian altitudes of almost
90 down to 15 degrees, in 15 degree increments.

To do this, I computed altitudes for various times after meridian
passage.  I determined meridian passage empirically as the time of
maximum altitude, which is off a bit, but appropriate for this
discussion.  The object was the sun, with meridian passage at 12:03:18
UT1 on 1/1/04 at 0d 0.0' W.  The latitude was varied to vary meridian
altitude.  I know this can be done much more elegantly than this
brute-force simulation, but it might help show dummies like me what's
going on.  If you or Alexandre want to step in and present an
analytical solution, I would be delighted to see it.

In the table, "Sc dd interpolate" is the average of the proceeding and
following altitudes (Sc dd), which are separated by 5 minutes.  The
error is then the difference between the computed and the interpolated
altitude, which is expressed in arcminutes.

As you rightfully pointed out, at 90 degrees the error of 1.0
arcminutes is serious at meridian passage, but negligible 10 minutes
after.  The magnitude of the error declines slightly at 75 degrees to
0.7 arcminutes but its duration above 0.5 arcminutes extends to a
half-hour after passage.  By 60 degrees the error is under 0.5
arcminutes, although its duration above 0.2 arcminutes persists for an
hour.  I expect at sea that errors under 0.5 arcminutes are
insignificant, especially by people who don't shoot sights all year and
from small craft.  Thus the U.S. Power Squadron's recommendation to
avoid hight altitudes.

Fred

> Fred Hebard wrote:
>
>> as best as I can tell, the only time you
>> will see significant (>0.1' of arc) systematic shift will be within 10
>> minutes of meridian passage.
>
> This is only true for extremely high altitudes, namely over 89.5 deg!!
>
> Dependent on the meridian altitude of a celestial body, the diurnal
> arc that
> it traces in the sky as seen from an observer on earth can be anywhere
> between a smooth curve not changing curvature much, or a straight line
> up in
> the east and straight line down in the west, with a sharp bend
> in-between.
>
> As a general guideline, therefore, the higher the meridian altitude,
> the
> more pronounced the non-linearity effect will be in the vicinity of the
> meridian and the faster it will drop to comparatively lower levels (but
> still considerable in absolute terms) away from the meridian. The
> lower the
> meridian altitude, the smaller the maximum effect, but also the longer
> the
> period during which the error persists.
>
> For a body with meridian altitude = 75 deg, this error for two sights 5
> minutes apart is 0.8' at meridian transit. It drops to 0.3' within the
> next
> hour and to 0.1' within the next 40 minutes after that.
>
> For a body with a meridian altitude of 45 deg, the same error is only
> 0.2'
> at meridian transit, but an error of 0.1' persists within 3 hours on
> each
> side of meridian passage.
>
> Herbert Prinz
>

```

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