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    Eclipse computations with Lunar4
    From: Paul Hirose
    Date: 2016 Oct 10, 21:41 -0700

    Eclipse duration has been mentioned here, so I'll point out that my
    Lunar4 lunar distance program can calculate the times of the four
    contacts of a total solar eclipse.
    
    Set Lunar4 to a time before first contact. Select near limb for both
    bodies. Don't enter altitudes. Enter zero lunar distance. Check the
    "solve for time" box.
    
    Lunar4 will automatically solve for time of first contact, i.e., the
    instant when a partial eclipse begins. Set a time after fourth contact
    and run Lunar4 again to get the actual fourth contact time.
    
    The reason for setting two different times is ensure you get the correct
    contact. The condition you entered (zero distance, near limb to near
    limb) occurs twice, but Lunar4 automatically finds the one that's
    closest to the time you entered.
    
    For the second and third contacts (beginning and end of the total
    eclipse) we must understand how Lunar4 identifies the near limb: it's
    the 180° segment of limb that's nearest to the center of the other body.
    That's true even when the Sun and Moon overlap.
    
    As a total eclipse proceeds toward second or third contact, the Moon
    near limb approaches the Sun far limb. So to find the contact times,
    select those limbs, and take care that the desired contact is the one
    closest to the starting time you set in Lunar4.
    
    To find the second and third contacts of an *annular* eclipse, select
    Moon far limb and Sun near limb. If you choose the wrong limbs in either
    type of eclipse, it's obvious. The solution won't converge, since the
    distance between the selected limbs is never zero.
    
    Speaking of convergence, if zero lunar distance is not attained (plus or
    minus your accuracy setting) in 20 iterations, Lunar4 aborts with an
    error message. That can happen if 1) your time input is too inaccurate,
    or 2) the relationship of lunar distance and time is too nonlinear. In
    either case you can use trial and error, or temporarily reduce accuracy
    so the program can converge on a good approximation of the correct time.
    Then input that time and increase the accuracy setting.
    
    It may be necessary to proceed entirely by trial and error. For example,
    you would do that to find the time of maximum eclipse if the eclipse is
    partial.
    
    I tested the program with the total eclipse of 2016 March 9 at N07 E144:
    
    00:09:39 UTC first contact
    01:37:55 second contact
    01:41:39 third contact
    03:14:18 last contact
    
    An online eclipse calculator agrees within 4 seconds. I used either 0100
    or 0200, as appropriate, for the time inputs, UT1-UTC = -.0359203 s per
    IERS Bulletin B, and the default .01′ accuracy. Lunar4 never needed more
    than 3 iterations to converge to that accuracy. It may be capable of
    more, but I didn't try. Note that I didn't follow my own instructions
    regarding the time settings, but they were still close enough that
    convergence was rapid.
    
    For utmost accuracy, check the "Moon center of figure correction" box.
    It applies the standard correction of +.5″ longitude and -.25″ latitude
    to the center of gravity positions that you get from an ephemeris. The
    Astronomical Almanac eclipse data includes this correction, but I don't
    think it's important and in fact didn't use it in the above computations.
    

       
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