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    Lunar occultations - a graphical method
    From: Geoffrey Kolbe
    Date: 2015 Nov 1, 13:10 +0000

    Well, the lunar occultation of Aldebaran on the 29th of this month
    came and went. I was watching out for it, but the weather here in the
    Borders of Scotland was overcast and while I could see the moon
    through the cloud, stars of any magnitude were obscured and so I could
    not see Aldebaran. My only consolation is that the the "Cartes du
    Ciel" planetarium program seem to agree with the results for the times
    of immersion and emersion that I obtained using the graphical method I
    outlined in a previous post reasonably well.
    
    On the basis of my experience in crunching through the numbers and
    plotting the two graphs required for the graphical method for this
    particular example, I would estimate that accuracy of the time (UT1)
    obtained from this method should be of the order of +/- 10 seconds,
    possibly a little better, though there are certain physiological
    considerations (see below) which mean it would be best to obtain the
    times of both the observed immersion and emersion, split the
    difference and then compare this time to the similar time derived from
    calculation (whether this be the graphical method mentioned above or
    any other method).
    
    On the subject of the graphical method which I outlined in a previous
    post, there was a wrinkle I omitted to include and so for the sake of
    completeness - and for anybody who wishes to use the method to obtain
    time - I will restate the method here.
    
    1) Determine the calculated altitude and azimuth for the moon and the
    star at various times before, during and after the occultation. I
    calculated for every 10 minutes. The progression or change in both
    azimuth and altitude for the moon and Aldebaran was quite linear, as
    one might expect, so crunching the data at ten minute intervals may
    seem excessive. But as the data were derived from the Nautical
    Almanac, whose accuracy is limited, there was some scatter on the data
    and it was worth having lots of data points through which a best fit
    could be plotted.
    
    2) Apply the corrections (for parallax) for the Moon's calculated
    altitudes. The Nautical Almanac lists the corrections as a function of
    observed altitude in order to produce something to compare to the
    calculated geocentric altitude. However, we want to know what the
    observed altitude should be as this is what we will actually observe,
    so the corrections need to be applied in opposite sign to the
    calculated altitude to produce an "observed" altitude. (Strictly, the
    correction applied needs to be such that when it is applied in the
    normal way to the "observed" altitude, the calculated altitude is
    reproduced. However, the increase in accuracy this iterative process
    will produce is probably not worth the effort. Similarly, it is not
    worth correcting for refraction as this will be virtually the same for
    both Moon and star and it will make no difference to the result.)
    
    2) For these times, note the difference in azimuth and the difference
    in altitude between the star and the Moon.
    
    3) The azimuth differences need to be converted to Horizontal Angular
    Differences - that is the angle between the star and the azimuth
    meridian through the centre of the Moon. This was the step missed out
    on my previous posting. To do this, multiply the azimuth difference by
    Cos(star altitude). The reason for this step is set out below*.
    
    4) Plot Horizontal Angular Difference (horizontal axis) against
    Altitude Difference (vertical axis) Draw a circle at the origin of
    radius equal to the semi-diameter (SD) of the Moon for the time of
    occultation. Draw a best fit between the plotted points and note the
    Horizontal Angular Differences at which the line intersects the Moon.
    
    5) Plot Horizontal Angular differences against time and draw a best
    fit line through the points. Note the times for the intersection
    angular differences determined in step 4. These are the times for
    observed immersion and emersion.
    
    There is a demonstrated physiological effect which is that the
    threshold brightness of the dimmest star we can see is a function of
    the ambient light in the sky at that time. So it is that when the Moon
    is up, most of the dim stars and the Milky Way disappear because of
    the high background light. Because of this effect, we should expect
    that a star will be observed to be occulted before it actually is, and
    its observed emersion will be somewhat after the actual event.
    Splitting the difference between the observed times of immersion and
    emersion, as recommended above, will (to first order) cancel out these
    errors.
    
    *The horizontal angular width of the moon, 30.2', is only the same as
    the azimuth width (the difference in azimuth from one side of the Moon
    to the other) when the Moon is on the horizon. When the Moon has some
    non-zero altitude, the azimuth width of the Moon will be greater than
    its angular width. See the attached drawing which attempts to
    demonstrate this. The increase in azimuth width of the Moon goes as
    (horizontal angular width) x Sec(observed altitude). Thus the azimuth
    width will be the same as the angular width at zero altitude, when the
    Moon is on the horizon, and the azimuth width will (would) be
    essentially infinite, or 360 degrees, if the Moon was at the zenith.
    
    The Moon's azimuth width will vary with altitude, then, and this adds
    a complication to the whole problem unless we 'normalise' the Moon's
    azimuth width. We do this by converting to horizontal angular widths.
    Now, we can use the fixed angular width of the Moon on the first
    graph, and plot the Horizontal Angular Differences between the star
    and a vertical line going through the centre of the Moon, which is the
    Moon's azimuth meridian.To convert azimuth difference into horizontal
    angular difference:
    
    Horizontal Angular Difference = (Azimuth Difference) x Cos(observed altitude)
    
    
    --
    Dr Geoffrey Kolbe, Riccarton Farm, Newcastleton, Scotland, TD9 0SN
    Tel: 013873 76715
    

       
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