Kipp&Zonen BSRN Scientific Solar Monitoring System - Manual (Page 148)

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If the optical axis of the pyrheliom eter is not directed to the solar centre, than the angle m easured from
the solar centre (z1) differs from the angle m easured from the optical axis (z). The transform ation:
cos(z1) = cos(d) cos(z) + sin(d) sin(z) cos(n),
where d = the deviation between the solar centre and the optical axis, that is the
pointing error,
n = an azim uth angle m easured in the plane of the receiver, it is zero if
the radiance com es from the solar centre.
D 1.2.1 Radiance along the solar disk
Photospheric m odels of the Sun produce one-dim ensional radiance distribution across the solar disk,
that is the so called lim b darkening function. According to theoretical calculations (Allen 1985, Zirin
1988) the radiance depends near linearly on the cosine of the zenith angle at the solar "surface". Taking
into account som e observations too (Zirin 1988) and using z1 as variable instead of the aforem entioned
zenith angle, the following radiance distribution along the solar disk has been used:
L(z1) = Lo (0.3 + 0.7 SQR(1-(z1/0.26) )) 2
where Lo = the radiance at the solar centre,
0.26 = the radius of the solar disk in degrees.
This way the atm osphere affects the absolute value of the radiance com ing from the solar disk, but
not the relative distribution along it. If the direct radiation is 1000 W m , then Lo = 2.01565*10 W (m *sr) ,
-2 7 2 -1
while at 461 W m it is 9.29216*10 . Since the gradient at the solar edge is very large, the integration -2 6
step for the calculation of irradiance received by a pyrheliom eter has to be 0.0001 degree to obtain
0.1 W m accuracy.
-2
D 1.2.2 Radiance along the circum solar sky
For several atm ospheric aerosol content and solar elevation angle the radiances com ing from the
circum solar sky have been calculated by Putsay (1995). To m ake our calculation m ore practical, second
order polynom s have been fitted to the logarithm of the two selected circum solar sky function. The
fit is not quite perfect, but it is not significant since we want to obtain the effect of the shift caused by
the uncertain pointing.
On Figure D 1.1 the whole (solar and circum solar) sky functions are shown for the two selected
atm ospheric m odels.
D 1.2.3 The penum bra functions
To make the com putations faster, the penumbra functions have been approxim ated by third order polynom s
in the interval between the slope and lim it angles. Again, the fit is not perfect, but this has sm all effect
on the deviations of the values calculated for different pointing uncertainty.
D 1.3 Results
In the calculations the effect of the solar disk and that of the circum solar sky could be separated. On
Figure D 1.2 and D 1.3 the actually direct irradiance of the pyrheliom etric sensor can be seen. If the
pointing error is sm aller than the slope angle, the irradiance is not affected. If the solar disk is in the
penum bra region of the pyrheliom eter, the irrdiance decreases rapidly with the increasing pointing
error.
On Figure D 1.4 the irradiance com ing from the circum solar sky is seen for all pyrheliom eters and for
both atm ospheric conditions. The decrease is continuous but the effect is not significant com pared
to that of the solar disk.
D 1.4 Conclusions
(1) If the pointing error of a pyrheliom eter is sm aller than its slope angle, the effect is negligible.