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Lunars: Altitude Accuracy (revisited)
From: Ed Popko
Date: 2018 Apr 9, 09:37 -0700

In an ancient (by blog standards) 2004 post[1], Frank Reed makes the following case:

"When shooting lunars, accuracy in the altitudes doesn't matter much. We've all heard that. But how much does it matter? I propose that you can get by with a 6 arcminute error for lunar distances around 90 degrees, assuming you want no more than an 0.1 minute resulting error in the cleared distance, but that generally the required accuracy (for constant altitudes) is proportional to the sine of the distance which means that for short distance lunars, the altitudes have to be significantly more accurate. This is yet another reason why short distance lunars would not have been popular historically."

He offers two formulas for estimating required accuracy:

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 amassed about 70 lunars test cases drawn from historic references and from my own lunars and there is a wide variety of altitudes and LDs to test against.  So I tried some to see if these accuracy estimates actually work out. And I was more than a little surprised just how insensitive the final cleared lunars were to changes in height within these accuracy estimates.

Here are a some cases I experimented with. For each case, I list the original observation, its pre-cleared and cleared result
using Letcher's clearing method. I then find the height accuracy for both moon and other body using Frank's formulas.
Three variations of the original observation heights are run varying first the moon, then the other body and then both within their respective accuracy limit.
The resulting cleared distances are found for each variation and compared to the original. Most variations change the cleared lunar no more than 0.1', the worst was 0.2'

Conclusion - it's ture, when shooting lunars, accuracy in the altitudes doesn't matter much.

Perhaps Frank will cover this important point in his forthcoming Lunars Workshop at Mystic Seaport.

Ed Popko

[1] Blog post: fer3.com/arc/m2.aspx/Lunars-altitude-accuracy-FrankReed-oct-2004-w19454

Using Letcher's Clearing Method (easiest to use when analyzing with calculator)

CASE 1

Original Observation, pre-Clear and Cleared
Moon Ho       Moon HP      Body Ho       LDpc        LDc
22°50.7'            58.4'          59°08.0'     59°48.5'    59°05.5'

Height Accuracy Bracket Reed Formula
Moon Ho                    Body Ho
11.2'                            10.1'

Height Variation Tests
Moon Ho       Moon HP      Body Ho       LDpc        LDc        Effect of Height Change on LDc
22°40.0'(-)      58.4'           59°08.0'       59°48.5'    59°05.4'   -0.1 difference
22°50.7'          58.4'           59°00.0'(-)   59°48.5'    59°05.5'    None, LDc same as original observation
22°40.0'(-)      58.4'           59°00.0'(-)   59°48.5'    59°05.5'    None, LDc sane as original observation

CASE 2

Original Observation, pre-Clear and Cleared
Moon Ho       Moon HP      Body Ho       LDpc        LDc
53°19.5'           59.6'           45°00.7'     69°59.4'    69°33.3'

Height Accuracy Bracket Reed Formula
Moon Ho                    Body Ho
29.6'                               8.0'

Height Variation Tests
Moon Ho       Moon HP      Body Ho         LDpc        LDc        Effect of Height Change on LDc
52°40.0'(-)       59.6'           45°00.7'        69°59.4'    69°33.1'    -0.2'
53°19.5'           59.6'           44°53.0'(-)    69°59.4'    69°33.4'    +0.1'
52°40.0'(-)       59.6'           44°53.0'(-)    69°59.4'     69°33.2'    -0.1'

CASE 3

Original Observation, pre-Clear and Cleared
Moon Ho       Moon HP      Body Ho       LDpc        LDc
50°29.9'           59.4'           21°08.1'    107°22.9'   106°49.3'

Height Accuracy Bracket Reed Formula
Moon Ho                    Body Ho
30.1'                              6.1'

Height Variation Tests
Moon Ho       Moon HP      Body Ho       LDpc             LDc         Effect of Height Change on LDc
50°00.0'(-)      59.4'            21°08.1'       107°22.9'   106°49.4'      +0.1'
50°29.9'         59.4'            21°02.0'(-)    107°22.9'   106°49.4'      +0.1'
50°00.0'(-)      59.4'           21°02.0'(-)    107°22.9'   106°49.5'      +0.2'

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