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Dear Paul,

This recent post of yours prompted me into further studying semi-diameters contractions, whether horizontal or oblique ones.

While we have independently developed [deadly accurate  -:) ] vector solutions which are probably similar, I decided this time to tackle this problem differently through attempting to derive some "more or less" simple formula to compute such refraction induced contraction.

Not claiming here that the following is totally new, since it might be, once more, just an independent re-discovery. So many things have been published on this topic ...

Again, too many non-significant digits - that is once more fully acknowledged -  but I have kept them all to compare the pure mathematical performance and behavior of the various approaches / algorithms described here-after.

In the enclosed 2 page attachment, bottom of page 1/2 I have derived the following rather simple formula :

With Ref(x) = Refraction at point x, the refraction induced contraction ΔD is given as follows by

Formula (1) : D’ - D = ΔD = Ref(B) * cos (α + 2δ)  -  Ref(A) * cos α

Formula (1) is quite easy to implement as long as ... you know α  and δ ... the [lenghty] computation of which parameters is given on this page 1/2 .

For our most recent Ladd Observatory example (end of the post), and from my own derived values, on page 2/2 of the attachment, I have solved the Moon Semi-Diameter oblique contraction towards Regulus through 4 different methods :

(1) - Great Circle : for which I am finding  ΔD = -0.299 124" (benchmark value)

(2) - Plane Geometry : from which I am finding  ΔD = -0.299 154"  - Difference 0.000 030"

(3) -  Formula (1) here-above : from which I am deriving ΔD = -0.298 974"  - Difference 0.000 159"

(4) - Another Approximation (as explained on page 2/2 of the attachment).

This Approximation only requires 3 variables : Distance, Position angle, and unrefracted height.

From these data, I am "reconstructing" the other required quantities, and altogether with a rather simple approximation for Refraction -  Ref(h) in arc minutes = 1 / tan h - I am deriving the following value : ΔD = -0.297203" - Difference 0.001 921" : not that bad for such an approximation.

This Another Approximation performs quite well in the computation of horizontal contractions of the Moon or Sun SD's.

As indicated by André Danjon, such contraction is almost independent of the Sun or Moon altitudes as can be, once again, verified here.

Best Regards,

Kermit