NavList:

A number of NavList contributions under at least 2 different post titles/headings have recently been published about the effect of Refraction onto the apparent angular distances between 2 points close in the sky. In particular such contraction of apparent distances has an impact onto the horizontal and oblique refracted vs. unrefracted Diameters and Semi-Diameters of the Sun and Moon.

These contributions can be summarized as follows :

(1) - For Horizontal Contraction :

(1.1) - Formula (2)

ΔD" = D° * π / 3

(1.2) - Earlier formula published here

dSD/SD = - 29 * E-5       which is identical in essence to the formula here-above.

(2) - For Oblique Contraction (including horizontal contraction)

(2.1) - Great Circle Method ("the" benchmark)

It implies comparing refracted and unrefracted distances computed through the classical GC formulae.

One very interesting variant is the Method advocated by Paul Hirose applicable to the limb of the Sun or Moon. Starting from the Body Center position and any given position angle, one first computes the corresponding unrefracted point on the limb through the classical GC formulae and then "refracts" it so as to further compare refracted and unrefracted oblique distances between center and limb.

(2.2) - Plane Geometry

For close in objects, i.e. under say 1° of separation, the difference between the Plane Geometry method and the GC method should remain well under 0.001" as is observed here. However Plane Geometry formulae are often considered easier than the GC formulae.

(2.3) - Other Methods

(2.3.1) - "Formula 1" method.

Showing up as D’ - D = ΔD = Ref(B) * cos (α + 2δ)  -  Ref(A) * cos α , this methods gives the angular distance contraction as a difference between its 2 correcting terms, one applicable at each end of the angular segment considered.

Although rather simple, this Formula actually does not prove of immediate practical interest simply because the required values of α and δ are not immediately or easily obtained. Nonetheless, when such quantities are accurately known, this method yields results quite close from the Plane Geometry method, as can also be verified here.

(2.3.2) - The "Another Approximation" method.

Described in §4 of this post, this method gives remedies to the shortcomings of the previous method.

Like Paul Hirose's method mentioned earlier, it simply starts with 3 initial variables : a departure point with its unrefracted height, and a position angle.

Through a series a simple algorithms numbered from (4.1) to (4.5) on page 2/2 of this enclosure, it enables a quick computation of the refraction induced contraction.

This approximation has the built-in limits applicable to the validity of its very simple if not archaic refraction model : it should not be used under 10° of height, not to even mention 15°. Even if slightly "degraded" compared to the previous methods, it nonetheless seems reliable to +/- 0.002 ".

A "magic" short and comprehensive formula is yet to be discovered in this field.

Assuming that it has not been published any earlier this  "Another Approximation" method is probably one which brings some innovation here since it involves only a series of simple chain calculations very easily carried out on a calculator.

Still this subject remains of limited interest for Classical Celestial Navigation.

Kermit