# NavList:

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

**Re: Precomputed lunar distances**

**From:**Frank Reed

**Date:**2005 Apr 18, 22:20 EDT

Bill you wrote:

"Yes, that is my image. A two-dimensional representation of
three

dimensions. What a camera would see."

dimensions. What a camera would see."

OK. And this can be useful so long as you remember that the sides are
actually straight as an arrow.

And you wrote:

"What deeply confuses me is as follows. Using two hypothetical stars
with

equal declinations and an LHA between them, I calculate true separation as

34d 27.7'. I raise the equal Hc's of the two stars from a staring point of

1d 36.8' in increments of 11d 02.9' (11d 02.8 for last step) and calculate

refraction separation correction. The results are as follows:

Hc Refraction Correction

1d 36.8' -18.2' -0.31796

12d 39.7' -4.1' -0.57133

23d 42.6' -2.2' -0.59930

34d 45.5' -1.4' -0.60260

45d 48.4' -0.9' -0.57422

56d 52.3' -0.7' -0.64021

67d 54.2' -0.4' -0.64310

78d 57.1' -0.2' -0.63534

89d 59.9' 0 0

They do not seem to reflect refraction moving along a straight line to me,

where I might expect the corrections to be similar to a curve derived from

refraction values at those altitudes."

equal declinations and an LHA between them, I calculate true separation as

34d 27.7'. I raise the equal Hc's of the two stars from a staring point of

1d 36.8' in increments of 11d 02.9' (11d 02.8 for last step) and calculate

refraction separation correction. The results are as follows:

Hc Refraction Correction

1d 36.8' -18.2' -0.31796

12d 39.7' -4.1' -0.57133

23d 42.6' -2.2' -0.59930

34d 45.5' -1.4' -0.60260

45d 48.4' -0.9' -0.57422

56d 52.3' -0.7' -0.64021

67d 54.2' -0.4' -0.64310

78d 57.1' -0.2' -0.63534

89d 59.9' 0 0

They do not seem to reflect refraction moving along a straight line to me,

where I might expect the corrections to be similar to a curve derived from

refraction values at those altitudes."

I take it that you're suspicious of these results because it seems as if
the correction is just about 0.6 minutes of arc across a wide range of
altitudes. Strange, huh? Strange but true... in this case where both stars have
the same altitude (and as long as the altitudes are above about 12 degrees). The
correction will be 0.6 minutes at every altitude. This happens in this special
case because the portion of refraction acting along the arc between the objects
is increasing at a rate that exactly counter-balances the decrease in refraction
at higher altitudes. At low altitudes, the arc between the stars is nearly
perpendicular to the vertical arcs so only a small portion of the rather large
refraction down there acts along the arc you're measuring. But consider the case
where the stars are 40 degrees apart and both 70 degrees high. Now they are
necessarily on opposite sides of the zenith and the great circle arc between
them passes from one star through the zenith to the other. That means that all
of the refraction acts 100% along the arc. It's simply an accident of Nature
that the rate of change of refraction with altitude is exactly equal to the rate
of change of the geometric factors with altitude. And this accident doesn't work
for altitudes lower than 12 degrees (or so).

In some old lunar methods and in some approaches to clearing star-star
distances, it's been useful to pull out a specific equation for refraction
instead of using a table or similar method. Above 12 to 15 degrees, refraction
in minutes of arc is very nearly equal to 0.95/tan(alt). Try it and compare the
results with a standard refraction table. When we use this specific form for
refraction, the equation for correcting the distance between two objects for
refraction simplies quite a bit. Letcher in his 1978 book "Self-Contained
Celestial Navigation with H.O. 208" showed what happens under these
conditions. The result for the correction to the distance is

r = 1.90*(x - cos(D))/sin(D)

where D is the measured distance and x is calculated from the
altitudes:

x = (sin(h1)/sin(h2) + sin(h2)/sin(h1))/2.

And that's it. This is all you need to correct star-star distances given
the conditions above. In the example you gave in your earlier message, h1 and h2
are equal in every case so x=1. And if you plug in 37 degrees for D, you will
get 0.6 minutes for the refraction correction. Letcher also includes a neat
little graphic trick for reading off x with a pair of dividers, but that's
superfluous in these days of cheap computation. By the way, this is not a
"different" way of calculating star-star distances from the one you've already
seen. It's derived directly from it under the assumption of that specific law of
refraction above (a law which happens to be accurate above 12 degrees or so but
not below).

-FER

http://www.HistoricalAtlas.com/lunars

http://www.HistoricalAtlas.com/lunars