Again on the culmination - a mathematical point of view
As clearly Franco Martinelli has explained, there is a small difference
between the "meridian transit" and the "culmination" of a celestial body,
even if the Nautical Almanac describes them as coincident.
If we consider the well known formula that binds the Latitude L, the
Declination D, the hour angle W and the height H of a celestial body:
sin(H)=sin(L)*sin(D)+cos(L)*cos(D)*cos(W)
and make the derivative taken with respect to the hour angle W (that is
with respect to the time) we have:
cos(H)*dH/dW =
sin(L)*cos(D)*dD/dW -cos(L)*sin(D)*dD/dW - cos(L)*cos(D)*sin(W)
where all the derivative are partial ( with the angles in rad)
Since when H is maximum his derivative =0, to look for the maximum of H it
is enough to set =0 the second member of the equation.
We obtain :
cos(W)*tan(D)*dD/dW+sin(W)=tan(L)*dD/dW
In the case of the Sun the maximum change of D is near the Equinoxes and
its value is about 0.4d/day, that is 0.4/360 =0.0011111 rad / rad (positive
in Spring)
Supposing that D=0 exactly at noon in the Spring Equinox, the following
values can be calculated:
for L = 30d the maximum is at 8.8 sec after noon and the height of the Sun
= 0.07" ( = 1/48700 d) more than that on the meridian
for L = 40d the maximum is at 12.8 sec after noon and the height of the Sun
= 0.11" more than that on the meridian
for L = 50d the maximum is at 18.3 sec after noon and the height of the Sun
= 0.15" (=1/24000 d) more than that on the meridian
As Martinelli has already notice, the errors that we commit neglecting this
effect ( with the Sun) are small indeed.
Gianni Ferrari
