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From: Frank Reed (no email)
Date: Tue Nov 02 2004 - 01:35:05 EST
I wrote earlier:
>I general, good (slightly approximate) expressions for the required accuracy
>of the altitudes are:
>AccuracyBody = 6' * sin(Distance) / cos(BodyAltitude)
>AccuracyMoon = 6' * tan(Distance) / cos(MoonAltitude).
>
>I have never seen these expressions in print anywhere. Has anyone else
>encountered them?
And George H replied:
"No, and they don't 'feel' correct to me."
You need to do some real examples before you worry about how it feels...
George H wrote:
"... the geometry in which the lunar distance is MOST sensitive to changes in
altitude, of the Moon or other-body, is when the two bodies lie somewhere in
an arc that passes through his zenith, so they have azimuths that are
identical or else 180 deg apart. In that case the lunar distance is either the
difference or else (180-sum) of the two altitudes, depending on whether they are on
the same side of the zenith or opposite sides. If there's any change, or error,
in either altitude, then a corresponding change in lunar distance, of ±100%
of that amount, results. This is just as true for large lunar distances as for
small ones, as I see it."
Nope. You're assuming that the difference in azimuth is fixed. It isn't. We
don't measure the difference in azimuth. We measure two altitudes and a
distance. What you REALLY need to do before you start dismissing this issue is work
some examples. Let's take one that's close to 90 degrees (only "close" to 90 to
avoid another issue for the moment). Suppose the Moon is 29deg 00' high.
Suppose the Sun is 62deg 00' high. These are the pre-cleared altitudes; dip and
semi-diameter have been taken out. Suppose the measured lunar distance, again
cleared of semi-diameters is 87deg 00.0' exactly. Clear this lunar for parallax
and refraction using any method you like. Write down what you get... Now
suppose that your Moon altitude observer has bad handwriting and you realize that
the altitude might actually have been 24deg 00' instead of 29 degrees. So you
decide to work the observation again. Go ahead and do it. How does your result
compare with the earlier cleared distance?
I would also suggest that you start thinking in terms of the series expansion
for clearing a lunar distance in order to understand "intuitively" why these
things work as they do. That is, the relationship between a cleared distance
d' and the measured distance d is:
d' = d + dh1*A + dh2*B + higher order terms
where A and B happen to be equal to the corner cosines (the cosine of the
angle between the measured lunar arc and the local vertical) and dh1 is the
Moon's altitude correction while dh2 is the other body's altitude correction. The
altitude corrections depend on the measured altitudes so an error in the
measured altitude yields a corresponding error in the altitude correction. But the
corner cosines A and B ALSO depend on the measured altitudes. An error in
either altitude when the objects are close together in the sky necessarily yields a
bigger error in A or B or both. If you draw yourself a diagram for the Moon
and Sun at roughly equal altitudes and 20 degrees apart, and then draw the same
diagram for 10 degrees apart, it's very easy to see that a 1' error in the
altitude of either object has a bigger impact on the corner angle at 10 degrees
distance than it does at 20 degrees distance.
I wrote earlier:
>Finally, I should note that these
>expressions are "slightly approximate" because they assume that refraction
is
>insignificant. As as long both objects are above 10 degrees or so in
altitude,
>that's a reasonable assumption.
And George H commented:
"I question that. By my reckoning, changes in refraction with altitude are
actually GREATER than changes in parallax with altitude, at altitudes up to
about 14 degrees, and don't become "insignificant" until the altitude increases
significantly beyond 14 deg."
No, what I was getting at is that the exact expressions that I gave should be
(and can be) modified at lower altitudes because of the effect of refraction.
For example, when the other body is at a low altitude, that "90 degree
miracle" I refered to shifts over to become a "95 degree miracle". But for the great
majority of cases, this is irrelevant. The more complicated expression that
covers lower altitudes, too, obscures the simple facts that apply to the
majorityof cases.
And added:
"It's often recommended that altitudes above 10 degrees are avoided. But that
doesn't imply that refraction is insignificant above 10 deg."
That's not what I said, and it's not what I was getting at in any way.
Frank R
[ ] Mystic, Connecticut
[X] Chicago, Illinois
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