## Estimating Times of Second and Third Contacts

The times of second and third contact for any location not listed in this
publication can be estimated using the detailed maps found in the final
section. Alternatively, the contact times can be estimated from maps on which
the umbral path has been plotted.
Table 7 lists the path coordinates
conveniently arranged in 1deg. increments of longitude to assist plotting
by hand. The path coordinates in
Table 3 define a line of maximum
eclipse at
five minute increments in time. These lines of maximum eclipse each represent
the projection diameter of the umbral shadow at the given time. Thus, any
point on one of these lines will witness maximum eclipse (i.e.: mid-totality)
at the same instant. The coordinates in
Table 3 should be added to the map in
order to construct lines of maximum eclipse.
The estimation of contact times for any one point begins with an interpolation
for the time of maximum eclipse at that location. The time of maximum eclipse
is proportional to a point's distance between two adjacent lines of maximum
eclipse, measured along a line parallel to the center line. This relationship
is valid along most of the path with the exception of the extreme ends, where
the shadow experiences its largest acceleration. The center line duration of
totality **D** and the path width **W** are similarly interpolated from
the values of the adjacent lines of maximum eclipse as listed in
Table 3.
Since the location of interest probably does not lie on the center line, it is
useful to have an expression for calculating the duration of totality **d**
as a function of its perpendicular distance **a** from the center line:

d = D (1 - (2a/W)^2)^(1/2) s [1]
where: d = duration of totality at desired location (s)
D = duration of totality on the center line (s)
a = perpendicular distance from the center line (km)
W = width of the path (km)

If **tm** is the interpolated time of maximum eclipse for the location, then
the approximate times of second and third contacts (**t2** and **t3**,
respectively) are:

Second Contact: t2 = tm - d/2 [2]
Third Contact: t3 = tm + d/2 [3]

The position angles of second and third contact (either **P** or **V**)
for any location off the center line are also useful in some applications.
First, linearly interpolate the center line position angles of second and third
contacts from the values of the adjacent lines of maximum eclipse as listed in
Table 5. If **X2** and **X3** are
the interpolated center line position
angles of second and third contacts, then the position angles **x2** and
**x3** of those contacts for an observer located **a** kilometers from
the center line are:

Second Contact: x2 = X2 - ArcSin (2a/W) [4]
Third Contact: x3 = X3 + ArcSin (2a/W) [5]
where: xn = interpolated position angle (either P or V)
of contact n at location
Xn = interpolated position angle (either P or V) of
contact n on center line
a = perpendicular distance from the center line (km)
(use negative values for locations south of the
center line)
W = width of the path (km)

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