Hi Gianni,
It is always good to hear from you on this list.
Most of what I know about nodus design is a result of reading
your papers so your message is of very great interest.
You made this comment...
> I think that it is advisable that the plane of the hole is a
> polar plane with the hole axis on the intersection among the
> equator and the meridian planes.
Ah. You want the axis of the hole to point at the position
of the equinoctial sun at 12 noon. At that instant, the other
end of the axis meets the dial where the noon line intersects
the equinoctial line.
This idea has good features and less good features...
GOOD FEATURES
The design is very symmetrical. The axis aligns with the sun at
a position halfway between the extremes of declination and halfway
between sunrise and sunset.
Having the hole in a polar plane is particularly good if the dial
is also in a polar plane. Of course, the hole is then parallel
to the dial.
LESS GOOD FEATURES
Unfortunately, having the hole in a polar plane does not work
well in general. An extreme example is a direct east-facing wall.
The hole would then be in a plane perpendicular to the wall. You
get no spot of light falling on the dial when the sun is due east.
I think you were considering horizontal dials. Your suggestion
certainly lets more light fall on the dial at the winter solstice.
Let's look at an example...
A CAMERA OBSCURA MERIDIANA
To make the calculations easy, consider the situation at 12 noon.
As an example, I shall consider a camera obscura meridiana at:
Latitude 45 degrees
Hole diameter 20 mm
Height of hole above the pavement 20,000 mm
Although millimetres were before their time, Cassini and Bianchini
would have been familiar with these dimensions!
The brightness of the spot of light falling on the pavement at
12 noon is proportional to the solid angle of the hole as seen
from points in the spot.
In a really simple case:
The hole would be horizontal
The altitude of the sun would be 90 degrees
In this impossible case, the spot of light would be at the point
perpendicularly below the hole. The solid angle is:
RSA = (pi x 10 x 10) / (20000 x 20000)
I call this RSA for Reference Solid Angle. This is the area of
the hole divided by the square of the hole-to-pavement distance.
In reality, the altitude of the sun varies from 45-23.5 degrees
(at the winter solstice) to 45+23.5 degrees (at the summer solstice).
In general at latitude 45 degrees and at 12 noon:
altitude = 45 + dec (where dec = declination)
When the altitude is changed from 90 degrees, the hole-to-pavement
distance increases by a factor 1/sin(altitude) AND the hole will
appear elliptical when you look at it from the spot of light.
The semi-minor axis of the hole will be 10 x sin(altitude) if the
hole is still horizontal.
You suggest that the hole should be 45 degrees to the horizontal.
Let's make this angle arbitrary:
Let ang = angle of hole to the horizontal
The semi-minor axis of the hole is now 10 x sin(altitude+ang)
The solid angle is now:
SA = RSA x sin(45+dec) x sin(45+dec) x sin(45+dec+ang)
In the impossible case where the sun is in the zenith, dec = 45
and, if ang=0, we are back to SA = RSA the reference solid angle.
The factor RSA is a constant. It is much more interesting to
look at the other terms. Consider the function:
sa(dec,ang) = sin(45+dec) x sin(45+dec) x sin(45+dec+ang)
If the hole is horizontal ang = 0 and two special cases are:
sa(+23.5,0) = 0.805 and sa(-23.5,0) = 0.049
The spot of light is 16 times brighter at the summer solstice
than it is at the winter solstice.
Now let's try the Ferrari angle. When ang = 45 we have:
sa(+23.5,45) = 0.794 and sa(-23.5,45) = 0.123
This give a MUCH better balance. The spot of light is only 6.5
times as bright at the summer solstice as it is at the winter
solstice and there is only a slight reduction in brightness
at the summer solstice. Your suggestion looks VERY good...
We can do even better. Why not align the axis of the hole with
the winter solstice point on the noon line? Here ang = 68.5
and:
sa(+23.5,68.5) = 0.590 and sa(-23.5,68.5) = 0.134
Now the spot of light is only 4.5 times as bright at the
summer solstice as it is at the winter solstice. From the
winter solstice point the hole now appears circular. This
point is, of course, much further from the hole so it still
receives less light.
If you really want to achieve balance you can try the King
angle. This is over-the-top in at least two senses but it
really works. You set ang = 104.267 when:
sa(+23.5,104.267) = 0.109 = sa(-23.5,104.267) = 0.109
The hole faces due south and is actually leaning backwards
away from the noon line on the pavement. Also, the punto
perpendiculare will probably be outside the building but,
with modern surveying technology, this should not be a
problem!
We are severely restricting the light at the summer solstice
point but you can see that the brightness at the winter
solstice point is the same as at the summer solstice point.
Equality at last!
You will say that this is a bad idea! There is too little
light at both the winter solstice AND the summer solstice.
True, but we can easily adjust this...
Use an elliptical hole! The minor axis of the hole would
still be horizontal and 20mm across. The major axis (which
is 14.267 degrees off the vertical) could be, say, 100mm long.
You would then get 5 times the brightness AND still have an
image of the sun on the pavement at noon every day of the year.
My arrangement would not be suitable for an ordinary
sundial because you would get no spot at all when the
sun was north of due east or west. It should be OK
for a noon line though.
SUMMARY
There is no magic best angle for a disc nodus. You have
to ask `What do I want to optimise?' In some cases the
polar plane is the best answer. In other cases this is
not the best answer.
FOOTNOTE
> In Cassini sundial in S. Petronio (Bologna) the ratio
> R = ( Distance hole-spot / Diameter of the hole) changes
> from 1070 to 2670 during the year
Yes, this is true but, because the hole appears as an
ellipse, you get a better comparison if you use the
apparent minor axes at the two positions. You then
compare:
1070 / sin(summer solstice altitude at noon)
and
2670 / sin(winter solstice altitude at noon)
This compares the angles subtended by the apparent minor
axis of the (circular) hole (which appears to be an ellipse).
The angle subtended by the minor axis at the winter solstice
is VERY small.
The figures for Bianchini's meridiana at S. Maria degli
Angeli (Rome) are similar.
Molti cordiali saluti.
Frank
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