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    Re: Moon - Antares
    From: Frank Reed
    Date: 2009 Jan 26, 19:07 -0800

    You wrote:
    "Mr. Reed, I believe that civil discourse requires a bit of formality, 
    particularly due to some of the invective found in the archives.  Topics that 
    involve intellectual discourse can become testy, so for now, I will stick to 
    the nee-plus-ultra formality.  "
    Sure, Brad. Whatever floats your boat is fine with me! :-)
    And you added:
    "I have been considering the statement by Mr. Reed that the changes in 
    elevation at the moon's limb may result in "seconds" of variation with 
    respect to the time of an event"
    I do have one request: please do not refer to me in the third person. I 
    PERSONALLY consider that impolite. Ok? :-)
    You calculated the Moon's distance as:
    "356,043 km = 14959871 km * 0.00238"
    That's a bit low. The Moon is about 240,000 miles or 385,000 km away. Since 
    the orbital eccentricity is around 0.055, the apogee and perigee distances 
    are about 5.5% greater or less than this (somewhat variable because the 
    Moon's motion is very far from a simple Keplerian ellipse).
    You wrote:
    "16.775 arc-minutes = DEGREES(arctangent (1737.4 km / 356043 km))* 60"
    Just FYI, you don't need to dig out trig functions when you're calculating 
    small angles unless you need an extraordinarily exact value. For small angles 
    just take the distance across and divide by the distance out. Then multiply 
    by 3438. That gives you the result in minutes of arc. It's convenient and 
    quick, and you don't need a calculator. It's also very useful in marine 
    And you wrote:
    "The angular rate of the moon is given in the Nautical Almanac,  in the 
    increments section, as 14 degrees 19 minutes per hour (examine the increment 
    for 59 minutes and 60 seconds)."
    Now here you have made a mistake. That number you've taken from the almanac is 
    the base rate at which the Moon changes its GHA (the total rate, on average, 
    is about 10 minutes of arc per hour faster since you have to add "v"). But we 
    don't want the rate of change of GHA. We need to know the rate of change of 
    the Moon's position relative to the stars or in other words its rate of 
    change of SHA. That is a MUCH small number. Subtract the Moon's average rate 
    of 14.5 deg per hour from 15 degrees per hour... The rate of change of the 
    Moon's position relative to Antares or any other star is around 0.5 degrees 
    per hour, or dividing both by 3600, around 0.5 seconds of arc per second of 
    time (the actual rate varies somewhat both because of the Moon's varying 
    orbital speed and also because of changing parallax during an observation).
    You wrote:
    "Next we should find the highest mountains on the moon."
    Actually what you need are the differences in altitude between the highland 
    features near the Moon's limb and the lowland plains. The lowland plains are 
    places like Mare Smythii and Mare Orientale. The highlands aren't usually 
    named but there are some very high areas both north and south of Mare 
    Orientale. The difference in altitude between the plains and highlands is 
    about 10km. So the mountain heights you found should be doubled to get a real 
    sense of the extremes of topography near the limb. Also note that the "limb" 
    is a bit of a "moving target" because of lunar librations.
    And you wrote:
    "Time, seconds = Angle Subtended * Moons Angular rate
            Mons Huygens        0.222 seconds
            Mons Hadley         0.186 seconds
            Mons Bradley        0.169 seconds"
    Again, because of the mistake I noted above, you need to multiply these 
    numbers by about 30. You can easily get a difference in occultation timings 
    of three seconds due to limb features.
    "In short, the tallest mountains on the moon, even if on the limb, fall far 
    short of "seconds" of time."
    As I hope you can see now, several seconds would not be an unusual timing 
    difference due to limb features. For occultations, there's also another 
    factor. Imagine a case where Antares is just barely occulted. That would 
    happen when the Moon's center passes, for example, 15 minutes of arc just 
    north or south of the star. The star will be seen to approach the limb at a 
    shallow angle. In that case, a small difference in angular height of limb 
    features can have an even larger impact on the timing of occultations. If the 
    occultation is "grazing", a star can disappear and then briefly re-appear as 
    it lines up with a deep valley near one of the Moon's poles.
    And you concluded:
    "From a different standpoint, lunar calculation for longitude could not have 
    been successful had the limb provided such variation in precision as seconds 
    of time.  Observers, of necessity, bring the object to the moon at different 
    limb locations as a function of the close proximity of the moon to the earth. 
    If seconds of variation existed, then the longitude could not have been 
    determined with the degree of accuracy necessary."
    Not true. At best, observers in the late 18th/early 19th centuries expected to 
    determine GMT using lunar distances within 15 or 20 seconds, and even 30 or 
    60 seconds of time was considered acceptable error. The few seconds, plus or 
    minus, caused by the variations along the lunar limb did not matter to them. 
    Whether it matters to modern lunar distance enthusiasts is an open 
    Navigation List archive: www.fer3.com/arc
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