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    Re: Lunar trouble, need help
    From: Kent Nordstr�m
    Date: 2008 Jul 5, 13:19 +0200

    Georeg wrote [5635]: The next point of issue is the Moon parallax. Kent 
    writes "It is unclear how
    George measured his local parallax"
    
    It's simple enough. See navlist [5530]. We agree precisely on the figure for
    Moon HP, quoted by Kent to be 56m 36.5s, and by me, from Skymap, to be
    .9435�, which works out as 56.61', or 56m 36.6. Can't expect much better
    than that!
    
    Yes, I have taken the moon's HP from Umland. But it might be a little 
    confusion here so let me try to explain how I calculate the moon's parallax.
    Firstly, George is right about the value. The error in calculation is on me 
    (very embarassing) but the good news is that the model seems to calculate 
    correct.
    I used 61d 41m 4,19s as (wrong) input data which gave 26m 50.39s. The input 
    should have been 60d 41m 04,19s.  When using this the parallax will be 26m 
    42,3s (George 26m 44s). So it seems that we don't have a disagrement here.
    
    What my model does is as per below. As can be seen the calculation includes 
    two corrections for earth flatness (maybe a better English term is 
    oblateness?):
    - find the azimuth to the moon
    - find the difference between the geographic and geocentric latitude
    - multiply  this difference with cosine for the azimuth
    The azimuth is approx. 111d and the diff. between the latitudes is 5m 45s. 
    The product is +2m 6,46s, which gives a "local altitide" of  60d 38m 57,73s 
    + 2m 6,46s =  60d 41m 04,19s to be used for parallax calculation. Due to the 
    earth oblateness the value is added to the true local altitude if the 
    azimuth is greater than 90d (the moon is pointing away from the pole), 
    otherwise the value is negative.
    
    Next is a small correction to the moon's HP with the arguments latitude and 
    HP. This gives a "HP" of 56m 36s - correction 0,78s = 56m 35,22s.
    
    Now the parallax is calculated as arcsine (sine "HP" x cosine "local 
    altitide") = arcsine ( sine 56m 35,22s x cosine 60d 41m 04,19s ) = 27m 42,3s 
    (George 27m 44s).
    
    Geroge wrote:  Kent has ignored the augmentation factor, which allows for 
    the fact that an
    observer is significantly closer to the Moon, when it's high in the sky,
    than when its near the horizon. Indeed, he has taken "augmentation of Moon's
    semidiameter" into account, in making an exact calculation of Moon's
    altitude corrections (but when that has little or no influence on the final
    lunar distance) but neglected to do so in correcting the lunar distance for
    semidiameters (when it's crucial).
    
    I am not sure I understand this comment. So again let me try to explain what 
    my model does for the moon.
    
    The measured altitude is corrected for dip and the moon's semi-diameter: 
    61d 05m 16,4s - dip 10m 10,9s - SD 15m 25s = 60d 39m 40,5 s.
    To this two corrections are done:
    - augmentation of -13,3 s. The minus sign depends on the measurement of the 
    UL.
    - refraction correction of + 0,34s (diff .in refraction form UL to 
    geocentre).
    
    For finding the apparent distance (what I believe George defines as d) my 
    model does the following:
    Obs. distance + corr. for index error +/- SD for the moon +/-SD for the sun 
    (if used) +/- corr. for augmentation of the moon +/- refraction correction 
    for the moon +/- refraction correction for the sun (if used). The correction 
    for augmentation is based on the arguments moon's semi-diameter and apparent 
    altitude. Is this what George means... "but neglected to do so in correcting 
    the lunar distance for semidiameters (when it's crucial)."??? Is the 
    "augmentation factor "someting else?
    
    Georeg wrote: Kent doesn't hasn't yet told us what actual values he puts in 
    for his Sun
    and Moon refraction, or his Sun parallax, and we need to know these for a
    comparison (or at least, know how they are somehow to be included into the
    lunar distance calculation.).
    
    My input data were:
    - moon's refraction -29,82s
    - sun's refraction -1m 21s
    - sun parallax 7,3s
    
    Kent N
    
    
    
    ----- Original Message ----- 
    From: "George Huxtable" 
    To: 
    Sent: Wednesday, July 02, 2008 10:49 PM
    Subject: [NavList 5635] Re: Lunar trouble, need help
    
    
    
    I can see why the Kent's correction to the lunar distance for semidiameters
    differs, by a fraction of an arc-minute, from my own.
    
    We agree about Sun semidiameter, but differ, a bit, about the Moon's.
    
    Kent used-
    
    Sun SD: 15m 45s
    Moon SD: 15m 25s
    
    whereas mine were (translated into the same units)-
    
    Sun SD. 15m45s
    Moon SD (as seen from Earth's centre) 15m 25s
    Moon SD (augmented, to allow for altitude of 61 and-a-bit degrees) 15m 39s.
    
    Kent has ignored the augmentation factor, which allows for the fact that an
    observer is significantly closer to the Moon, when it's high in the sky,
    than when its near the horizon. Indeed, he has taken "augmentation of Moon's
    semidiameter" into account, in making an exact calculation of Moon's
    altitude corrections (but when that has little or no influence on the final
    lunar distance) but neglected to do so in correcting the lunar distance for
    semidiameters (when it's crucial).
    
    Kent has simply taken the semidiameters from Henning Umland's website, but
    as Umland doesn't know the altitude at which the Moon was being measured
    (because it depends on where the measurement is being made from), that
    correction has be applied next, by him.
    
    If Kent recalculates, taking augmentation into account, I suggest that he
    will find perfect agreement between us, at least as far as the lunar
    distance corrected for semidiameters.
    
    Perhaps we can draw a line under that one, and go on to the next discrepancy
    between us.
    
    ===============================
    
    The next point of issue is the Moon parallax. Kent writes "It is unclear how
    George measured his local parallax"
    
    It's simple enough. See navlist [5530]. We agree precisely on the figure for
    Moon HP, quoted by Kent to be 56m 36.5s, and by me, from Skymap, to be
    .9435�, which works out as 56.61', or 56m 36.6. Can't expect much better
    than that!
    
    Now, if we're bothered to, we can make the correction for the reduction in
    the Moon's HP on account of the spheroidal shape of the Earth, described by
    Kent as "Earth flattening". At such a low latitude of 15�, this amounts to
    only .01' (taken from a table in a modern Norie's), so we end up with a
    corrected HP of 56.60'
    
    Then multiply that by cos alt, to get the actual parallax at that altitude.
    True altitude of the Moon's centre, after allowing for dip and semidiameter,
    I get to be 60.6523�, or 60� 39' 08' in Kent's notation. This gives me Moon
    parallax of 27.74', or 27' 44". That has to be compared with the figure Kent
    quotes of 26m 50.39s. There's a significant difference here, of getting on
    for a whole minute, so my next question is : exactly how did Kent arrive at
    a Moon parallax of 26m 50.39, starting from an HP of 56m 36.5s?
    
    Kent doesn't hasn't yet told us what actual values he puts in for his Sun
    and Moon refraction, or his Sun parallax, and we need to know these for a
    comparison (or at least, know how they are somehow to be included into the
    lunar distance calculation.). Did Kent's final figure, for cleared lunar
    distance, come from some tables, or from a log-trig calculation, or a
    computer / calculator program, or what? Details would be of interest.
    
    George.
    
    contact George Huxtable at george@huxtable.u-net.com
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    
    
    
    
    
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