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Re: Lunars: altitude accuracy
From: Frank Reed CT
Date: 2004 Nov 2, 01:35 EST
From: Frank Reed CT
Date: 2004 Nov 2, 01:35 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
>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
