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Horizontal Contraction induced by Refraction (further development from previous post)

In the case of horizontal Diameter contraction,

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

can be further simplified as follows :

Here the position angle α is equal to 90° , then Ref (B) = Ref (A) and cos α = 0 with cos (α + 2δ)  = -sin 2δ

Hence Formula (1) boils down to : ΔD = - Ref(A) sin 2δ

With D being the Diameter, the horizontal Azimuth ΔZ covered by the Diameter seen at height "h" is : ΔZ = D / cos h.

Since 2δ =  ΔZ sin h , then 2δ = D tan h . With D in degrees, in order to express 2δ in radians :

2δ (in radians) = D° tan h * π / 180 and with 2δ being a small angle - still in radians - (sin 2δ) / 2δ = 1

Then : sin 2δ = D° tan h * π / 180

On the other hand Ref(A) in arc minutes is equal to : Ref(A) ' = (1 / tan h) and with ΔD = - Ref(A) sin 2δ (see above)

then in arc minutes : ΔD' = D° * π / 180 .

To get ΔD in Arc seconds, we finally derive the following :

Formula (2) for Horizontal contraction :  ΔD" = D° * π / 3 =  1.047 * D° , a definitely quite simple result.

Numerical example compared to Bottom of Page 2/2 of the previous post Attachment :

for SD = 32' , get ΔD" = 32/60 * π / 3 = 0.5585" , to be compared with identical value at 0.5585" derived by the Approximate formula on page 2/2.

Accordingly - and by comparison to Approximate formula on page 2/2 -

Horizontal contraction Formula (2) : ΔD" = D° * π / 3  is expected to be accurate at +/- 0.002" .   

Kermit