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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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