The problem of penumbra

Originally, the eyepiece, which served as the gnomon, was drilled in a copper plate placed horizontally, on the right of the interior façade of the building. This plate has long since disappeared. The plate is positioned at a right angle so as to cast a large shadow on the ground. The horizontal position of the eyepiece has numerous advantages: besides the fact that the height of its center can be very precisely determined, it is the same for the base of the gnomon on the ground in that it marks the starting point of the meridian line, corresponding to an angular altitude of exactly 90°.

Facsimile of a typical observation made using the meridian line. What we see here are the two readings of the tangent of the zenith distance at the top and bottom edges of the image of the Sun. These two readings are then corrected by a constant amount equal to 50, which is basically half the diameter of the gnomon. (credit: Library of the Paris Observatory. Journal of observations made at Paris Observatory, 1683-1798, D4 :5-6)

However, the main advantage of this configuration is that it allows for a correction of the slight penumbra encircling the bright image of the Sun. Indeed, the Sun has a substantial apparent size in the sky (about half a degree, which is the angle covered by a two-euro coin placed at a distance of only 2.8 m) in contrast to other stars which appear to us merely as points of light because of their distance from us.. Thus there is no clear separation between the bright area, which forms the image of the Sun, and the shaded part on the ground; there is always a narrow transition region from light to shade, which is called the penumbra. The problem then is to know precisely where to make the separation between the penumbra and the light since the accuracy of the measurement depends on it.

Cassini I resolved this issue in 1655 when he built his great meridian in the basilica of San Petronio in Bologna. He hit upon the idea of placing the eyepiece horizontally since this means that the area of the penumbra on the ground matches the exact size of the diameter of the eyepiece. This is not the case if the eyepiece is placed in a different position, whether vertical or tilted.

The diagram shows the general geometry of the meridian line and the elliptical image of the Sun surrounded by a penumbra. The center C of this image does not match the center M of the Sun on the line but it deviates very little from it, at most 7 mm at the winter solstice, which represents just less than 4 arcseconds (The arcsecond is 1/3,600 of a degree). What needs to be measured is the apparent zenith angle zm of the center of the Sun. This method consists of measuring the positions x0 and x1 of the upper and lower edges of the solar image to which we add or remove the radius of the eyepiece, d/2, in order to obtain the values ??of the zenith distances z0 and z1 of the upper and lower edges of the the Sun. This allows us to obtain the value of zm directly as well as that of the apparent diameter of the solar disc, as can be seen here below: