Some considerations on the image of the Sun produced by a mirror -with some delay : -)
The shape, the dimension and the characteristics of the image of the Sun reflected by a mirror on an opaque screen or on a wall, depend on many factors: - the dimension and the shape of the mirror - the distance between mirror and screen - the angle between the rays of the Sun and the mirror - the angle between the reflected rays and the screen - the indirect illumination on the screen We have the simplest case when the mirror is circular and the reflected rays are perpendicular to the screen (as in a planetarium, with an horizontal mirror in the center of the half-sphere) Also in this case the image is not circular, but has the shape that an observer sees looking from the direction of the rays of the Sun: if the mirror is a circle of diameter Dm, this shape is an ellipse with the axes a=Dm (great axis) and b (small axis). The different quantities that interest are: Dm the diameter of the mirror Lm the distance between mirror and screen Do the diameter of the image produced by a mirror of infinitesimal dimensions (a point) Do = Lm *0.009308 = Lm / 107.4 The value 1/107.4 is the mean angular dimension of the Sun in radiant (= 32') Taking the diameter of the Sun = 30', we obtain the value 114.59 given by Bill Gottesman Jo the illumination on the mirror Ji the illumination of the surface in shade, due to the indirect light I call Lo = 107.4*Dm the "fundamental" distance of the system We can have 2 different cases: 1) Lm <Lo The mirror is near to the screen (less then 100 times its dimension). In this case the image is formed from a central zone with maximum illumination, surrounded by a band (or strip) of constant width in which the light is dimmed (zone of penumbra or of penlux ) >From a point of the central zone an observer can see, in the mirror, the whole disk of the Sun and therefore in this zone the illumination Jo is that produced from the Sun This zone (of elliptic shape) has the greatest axis = (Dm - Do), while the external strip has the width = Do The total image has the dimension = (Dm + Do) In this case the shape of the image is similar to that of the mirror (as seen by the Sun): for instance if the mirror is square also the image will have 4 vertexes and it will have the shape of a parallelogram or that of a rectangle We can notice that in the width Do of the strip of dimmed light, the illumination passes from the value Jo to the value Ji. For this reason, if the indirect light Ji is great (for example in open air) we can see only a part of this strip, while if Ji is small (in rooms or churches with small windows) we can see a widest zone of it. The image seems therefore smaller in open air, that in a closed place ( with few light). 2) Lm> Lo The screen is far from the mirror (the phenomenon are more evident if the distance Lm is at least 200 - 300 times Dm) Also now the image is formed from two zones: a zone of almost circular shape, surrounded by a strip of non constant width, in which the light is dimmed. >From a point of the central zone an observer can see, in the mirror, only a part of the solar disk (he sees the mirror under an angle <32 '). The illumination is smaller J = Jo*(Lo/Lm)^2 < Jo The strip of dimmed light has a varying width from a maximum = Dm to a minimum = b The dimension of the central zone in a given direction = (Do - Dm) and the dimension of the whole image = (Do + Dm ) In the perpendicular direction the dimensions are (Do - b) and (Do + b ) If the distance between screen and mirror is great (Do >> Dm), the image appears almost circular whatever is the shape of the mirror . If the indirect illumination is great and we increases the distance Lm, the illumination of the central zone decreases and the image is no more visible. In conclusion: - if the mirror-screen distance is enough small (as in many reflection sundials on a wall) the image has the shape of the mirror and the central zone it is well illuminated - if the mirror-screen distance is great (as in many reflection sundials in internal rooms), the image is almost circular and the central zone has a little illumination. - A mirror with a diameter of around 2 cm (0.8") is suited for almost all the applications. Two numerical examples: Lm = 500 cm (200") Dm = 2 cm (0.8") Lo = 214.8 cm (86") Do = 4.7 cm (1.86") Central zone = 2.7 cm (1.06") Penumbra = 2 cm ((0.8") J= 0.18 * Jo Lm = 500 cm (200") Dm = 10 cm (4") Lo = 1074 cm (430") Do = 4.7 cm (1.86") Central zone = 5.3 cm (2.14") Penumbra = 4.7 cm ((1.86") J= Jo If we put in the center of a circular mirror a non reflecting circle (for instance with Dm=2cm, Dnr=1 cm) we can see very well the dark spot in the center of the bright image, that marks the center of the image (if the screen is near) On the contrary if the screen is far, the mean illumination of the image decreases, but in the center there is a maximum of brightness that can help, also in this case, to find the center of the image. Note - In all cases, obviously, we would have to keep track of the angle between the reflected rays and the screen, that seldom is = 90 degrees. - The image produced by a mirror that reflects the rays of the Sun has the same properties and characteristics of that produced by a hole in an opaque screen - For the principle of Babinet (Optics) the same properties can also be extended to the shadow produced by a sphere on a screen. Best wishes Gianni Ferrari
