Estimating the 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
(Figures 5-11,
and 14-17). Alternatively, the contact times can be
estimated from maps on which the umbral path has been plotted. Tables
7 and
8 list
the path coordinates
conveniently arranged in 30' 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 plotted on 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 - ( a/
W)2)1/2
seconds [3]
where:
| d | = | duration of totality at desired location
(seconds)
|
| D | = | duration of totality on the center line
(seconds)
|
| a | = | perpendicular distance from the center
line (kilometers), and
|
| W | = | width of the path (kilometers)
|
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
| [4]
|
| Third Contact: | t3
| = | tm + d/2
| [5]
|
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 xx2 and x3 of those contacts for an observer
located a kilometer from the center line are:
| Second Contact: | x2
| = | X2 -
arcsin (2a/W)
| [6]
|
| Third Contact: | x3
| = | X3 +
arcsin (2a/W)
| [7]
| |
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 (kilometers)
(use negative values for locations south of the center line), and
|
| W | = | width of the path (kilometers)
|
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