#### CbIncus

##### Member

It seems to me that I've discovered some important errors in Carnicom's article "Halo Measurements": http://www.carnicominstitute.org/articles/halo1.htm. It states:

1. Carnicom calculates the angular dimensions of a 22-deg. halo with A = 360 deg/6 (hexagonal prism) and n = 1.31.

The equations are correct, but the result is valid

2. There should be a mistake in the calculations of the absolute error in

Partial derivatives (D in degrees):

a) for d: f(d) = d/21*2*arctg(5.25/16.7) = 1.662061807*d. The partial derivative is 1.662061807 and the part of absolute error 1.662061807*0.08 = 0.1329649446 (degrees).

b) for f: f(f) = 12.8/21*2*arctg(5.25/f) = 1.219047619*arctg(5.25/f). The partial derivative in radians is -20479954636/(3199992912*f^2+88199804637) (computed with http://www.numberempire.com/). Substituting f = 16.7 gives us -0.0208842 (the absolute value is 0.0208842) and the absolute error = 0.0208842*0.3 = 6.26526e-3. In degrees this is 0.3589729556.

Total error is 0.1329649446+0.3589729556 = 0.4919379 (degrees) = 0 deg. 29 min. 30.98 sec.

The real value of D computed by Carnicom is 21 deg. 16 min. +- 0 deg. 29 min. 30.98 sec. The precomputed value of 21 deg. 37 min. 37.77 sec is within the error margin.

I'm interested if some errors exist in my computations.

If measurements indicate a deviation from that result, it

informs us that the materials forming the

aircraft-generated halos, cirrus and cirro-stratus cloud

decks are no longer composed solely of ice as is often

claimed. The measurements do indicate such a deviation. Initial

halo measurements suggest that the hexagonal prisms of

uniform size and associated cirrus and cirro-stratus cloud

decks are not composed solely of ice as is usually

claimed.

1. Carnicom calculates the angular dimensions of a 22-deg. halo with A = 360 deg/6 (hexagonal prism) and n = 1.31.

D = 2 [arcsin(nsin1/2A)]-A or D = 21deg. 50min. 30”sec.

The equations are correct, but the result is valid

__only for the middle of the visible spectrum.__Later he measures distance at the photos between the center of the sun and inner part of halo ring. This means that n refraction index should be changed to 1.3072 (red part of the visible spectrum). The D angle now becomes 21 deg. 37 min. 37.77 sec.2. There should be a mistake in the calculations of the absolute error in

**D = (d / 21cm) * 2arctan(5.25/f) with errors of deltad = 0.08 cm and deltaf = 0.3 cm.**We'll use the Lagrange formula for obtaining this result:Partial derivatives (D in degrees):

a) for d: f(d) = d/21*2*arctg(5.25/16.7) = 1.662061807*d. The partial derivative is 1.662061807 and the part of absolute error 1.662061807*0.08 = 0.1329649446 (degrees).

b) for f: f(f) = 12.8/21*2*arctg(5.25/f) = 1.219047619*arctg(5.25/f). The partial derivative in radians is -20479954636/(3199992912*f^2+88199804637) (computed with http://www.numberempire.com/). Substituting f = 16.7 gives us -0.0208842 (the absolute value is 0.0208842) and the absolute error = 0.0208842*0.3 = 6.26526e-3. In degrees this is 0.3589729556.

Total error is 0.1329649446+0.3589729556 = 0.4919379 (degrees) = 0 deg. 29 min. 30.98 sec.

The real value of D computed by Carnicom is 21 deg. 16 min. +- 0 deg. 29 min. 30.98 sec. The precomputed value of 21 deg. 37 min. 37.77 sec is within the error margin.

I'm interested if some errors exist in my computations.

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