A colleague has sent me the following forwarded message. I'm afraid that
teaching and doing some astronomy doesn't qualify me to answer this
question about solar coordinates, but I thought that someone on the list
might very likely be able to help him more than me!
If you can help, please reply to me off list & I will forward it on.
Cheers
Margaret Mazzolini
Margaret,
In a Botany project, my daughter has been looking at the way a leaf
can orient itself to adjust its interception of radiation from the sun.
The cosine of the angle of incidence is of course a measure of the
fraction of the direct solar beam that is intercepted, and she has found
equations which should allow her to calculate this in terms of various
directional coordinates for the leaf and the sun at any time on any day
of the year at any terrestrial location. She wants to use them to plot
graphs showing how the radiation intercepted will vary over the course of
the day for leaves pointing in different directions at different times of
the year at several different terrestrial locations.
The equation for the cosine of the angle of incidence is -
cosi = cosasinA + sinacosAcos(B - b)
where a = angle of leaf surface (above horizontal)
A = altitude of sun (above horizontal)
b = azimuthal angle of normal to leaf surface
B = azimuthal angle of sun
(azimuths measured east from north as zero degrees I assume)
For substitution into this equation she can calculate A from -
sinA = sinLsinD + cosLcosDcosh
where L = latitude of observer
D = solar declination
h = solar hour angle = (t-12) x pi/12 (in radians)
and t = mean solar time in hours.
However, to calculate the solar azimuth B to substitute into the
cosi equation, three references have given her three conflicting
equations -
(1) B = -arcsin(-cosDsinh/cosA)
(2) B = arcsin(cosDsinh/sinA)
(3) B = arcsin(cosDsinh/sinL)