136
If the optical axis of the pyrheliometer is not directed to the solar centre, than the angle measured from
the solar centre (z1) differs from the angle measured from the optical axis (z). The transformation:
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 measured in the plane of the receiver, it is zero if
the radiance comes from the solar centre.
D 1.2.1 Radiance along the solar disk
Photospheric models of the Sun produce one-dimensional radiance distribution across the solar disk,
that is the so called limb 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 some observations too (Zirin 1988) and using z1 as variable instead of the aforementioned
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 atmosphere affects the absolute value of the radiance coming 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 pyrheliometer has to be 0.0001 degree to obtain
0.1 W m accuracy.
-2
D 1.2.2 Radiance along the circumsolar sky
For several atmospheric aerosol content and solar elevation angle the radiances coming from the
circumsolar sky have been calculated by Putsay (1995). To make our calculation more practical, second
order polynoms have been fitted to the logarithm of the two selected circumsolar 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 circumsolar) sky functions are shown for the two selected
atmospheric models.
D 1.2.3 The penumbra functions
To make the computations faster, the penumbra functions have been approximated by third order polynoms
in the interval between the slope and limit angles. Again, the fit is not perfect, but this has small 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 circumsolar sky could be separated. On
Figure D 1.2 and D 1.3 the actually direct irradiance of the pyrheliometric sensor can be seen. If the
pointing error is smaller than the slope angle, the irradiance is not affected. If the solar disk is in the
penumbra region of the pyrheliometer, the irrdiance decreases rapidly with the increasing pointing
error.
On Figure D 1.4 the irradiance coming from the circumsolar sky is seen for all pyrheliometers and for
both atmospheric conditions. The decrease is continuous but the effect is not significant compared
to that of the solar disk.
D 1.4 Conclusions
(1) If the pointing error of a pyrheliometer is smaller than its slope angle, the effect is negligible.
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