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In this message I will explain a method to compute the refracted
semidiameter of the Sun or Moon (or any other body). Input data are the
unrefracted altitude of the center, unrefracted semidiameter, and
position angle on the limb. The position angle is measured from the
zenith, i.e., 0 is exactly at the upper limb and 180° at the lower limb.

First create a vector to the nadir: (0, 0, -1) in rectangular
coordinates or (0, -90°) spherical.

Create a second vector in spherical coordinates, with lambda (the
east longitude angle) equal to the desired position angle and phi (the
latitude angle) equal to -90° + unrefracted semidiameter. In other
words, this vector's separation angle from the nadir equals the
unrefracted semidiameter.

Rotate the coordinate reference system of both vectors about the y axis
by an angle equal to 90° + the unrefracted altitude of the center of the
body. (This rotation is counterclockwise, seen from the +y axis
looking toward the origin.) Now the vectors represent the unrefracted
relative directions of the center and limb, in the observer's horizontal
system.

To apply refraction, disassemble the vectors into spherical coordinates
lambda and phi. Apply refraction to phi. Re-assemble the vectors.

Finally, compute the separation angle between the vectors to get the
refracted semidiameter. (Separation angle can also be computed directly
from the spherical coordinates.) That is the refracted semidiameter.

For example, compute the refracted semidiameter of the Sun at position
angle 90° (the 9 o'clock position on its limb) when its unrefracted
center is at altitude 20°. Assume the unrefracted semidiameter is
exactly 16′.

The nadir vector is (0, 0, -1) in rectangular coocdinates. The limb
vector is (90°, -89.733333°) in spherical coordinates, or (0, .0046542,
-.9999892) rectangular.

Y-rotate the coordinate systems of both vectors 110° (i.e., 90° +
unrefracted altitude) via the rotation matrix

[cos 110°  0  -sin 110°]
[   0      1      0    ]
[sin 110°  0   cos 110°]

to obtain vectors to the Sun center (.9396926, 0, .3420201) and the
point on the limb (.9396824, .0046542, .3420164). In spherical (lambda,
phi) form, these are (0°, 20°) and (.283780°, 19.999774°).

Apply refraction to the altitudes. At 1013.25 mb and 15 C, refraction is
.044138 at center of Sun. Refracted center altitude is 20.044138°. Limb
refraction is .044139° and refracted altitude 20.043913°.

Re-assemble the vectors, with the refracted altitudes. The refracted
vectors are (.9394289, 0, .3427439) and (.9394187, .0046529, .3427402).

The refracted semidiameter is the separation angle between the vectors:
.266592°, which is .000075° (.27″) less than the unrefracted value 16′.
That agrees with Chauvenet: "for all zenith distances less than 85° the
contraction of the horizontal semidiameter is very nearly constant and
equal to one-fourth of a second."

I have ignored the effect of refraction on position angle, so the
calculation is not rigorous. Imagine a giant clock face in outer space,
with 12 oriented exactly to our zenith. Due to refraction, we see it
slighly distorted. The 11 (for example) mark is not exactly on a great
circle between the clock center and, for instance, the Moon at 30°
position angle (in refracted coordinates) with respect to the clock. But
my method assumes that it is.

Let's evaluate the error of that assumption at position angle 45°.
Unrefracted altitude is 20° and unrefracted semidiameter 16′ as before.
Then by the method I described, refracted semidiameter is .266320°. The
refracted vector to the Moon is (.9394289, 0, .3427439) and to the point
on the Limb (.9382934, .0032901, .3458247). The position angle between
those points is 45.058°.

That is .058° greater than the desired angle, so adjust the unrefracted
position angle to 45° - .058° = 44.942° and try again. Now refracted
semidiameter is .266320° as before, and refracted position angle is the
desired 45.000°. So the difference between refracted and unrefracted
position angles had insignificant effect on refracted semidiameter, at a
precision of a millionth of a degree (.004″).

Chauvenet gives an approximation for the contraction in semidiameter due
to refraction, when the position angle and the contraction in vertical
semidiameter are known. The latter is found (assuming 20° unrefracted
altitude and 16′ semidiameter as before) from the difference in
refraction at unrefracted altitudes 19°44′ and 20°16′. The refractions
are, respectively, .044770° and .043521°. The difference is the
contraction in diameter, so half that is the average contraction in
semidiameter, .000624°.

With that known, the contraction in semidiameter at given position angle
is the squared cosine of the position angle, times the vertical
contraction. Chauvenet says this approximation "is sufficiently exact
for all purposes to which we shall have occasion to apply it." In this
example, cos squared of 45° is exactly .5, the vertical contraction
.000624°, therefore contraction at position angle 45° = .5 * .000624° =
.000312°, compared to my .000347°, or 1.12″ vs. 1.24″.

Chauvenet's approximation gives no contraction at 90° and 270° position
angles; the "one-fourth of a second" horizontal contraction quoted above
is worked out separately in his book.

In the lunar distance chapter, Chauvenet explains how the Moon and Sun
altitudes and the lunar distance enable us to compute the position
angles, and then the contraction of the semidiameters. For example, Moon
apparent altitude h′ = 30°, Sun altitude H′ = 40°, lunar distance d′ =
100°. All angles are refracted, measured to center of body. (Because the
contraction is so small, no great accuracy is required of the position
angles. It is sufficient to use unrefracted semidiameters to reduce the
sextant measurements to the centers of the bodies.)

Let m = half the sum of the sides = 85°. Then according to Chauvenet,

sin^2 (q/2) = (cos m sin (m - H′)) / (sin d′ cos h′)
= .0723, therefore q = 31.2° = position angle of Sun with respect to
Moon, and

sin^2 (Q/2) = (cos m sin (m - h′)) / (sin d′ cos h′)
= .0837, therefore Q = 33.6° = position angle of Moon with respect to
Sun.

THAT IS WRONG. Angle Q should be about 36°. If you carefully compare the
equations in the Chauvenet book

http://books.google.com/books?id=6JlzJHd15vQC&pg=PA397

(top of the page) you will probably see something wrong even if you
don't understand how the equations work. Remember that q and Q are the
position angles of the Sun with respect to the Moon and vice versa.
Therefore it's reasonable that the right hand equation will put h′ in
the numerator and H′ in the denominator, opposite their positions in the
left equation. But only the numerator is different.

After changing h′ in the denominator to H′, the equation gives the
correct result. In my example, with the Sun 40° high, it probably makes
no significant difference. Even at low altitude, refraction has only a
little effect on semidiameter. It would be easy to overlook the mistake
in Chauvenet's equation.


NOTES

I used an HP 49G for the above computations. Refraction came from the
Bennett formula, with the additional term for more accuracy, as
described by Meeus in "Astronomical Algorithms."

All the needed vector and coordinate conversion algorithms are online at
the SOFA site in C and Fortran:

http://www.iausofa.org/index.html

I have no experience with the USNO NOVAS package, but it seems similar
to SOFA.

http://aa.usno.navy.mil/software/novas/novas_info.php

In practice I think the benefit of a semidiameter correction for
refraction is arguable. However, if coded into a program the correction
requires no extra effort by the user. The method I described works even
in degenerate cases, such as when the zenith is inside the limb. To me
it is more obvious and direct than Chauvenet's method. Admittedly, I
have enormously more computational power in my hands!

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