Jud wrote: << ... When is [a planet whose orbit is inside the Earth’s] the brightest? What function describes the brightness in terms of the angle between the Earth and the planet viewed from the Sun? (Assume that the orbits are circular and co-planar.) There could be two versions of this problem: (1) assume that all sunlight portions of the planet are equally bright as seen from the Sun (2) take into account the differences in brightness of parts of the planet (but still assume that all parts of the planet inherently reflect the same amount of sunlight).
Some thoughts: Physically, maybe a fair way to answer this question is to say which position of the planet in its orbit reflects the maximum amount of the sun's energy to Earth. (Or more to the point, we could take a single point of earth at a given time on a given day of the year, and calculate the energy density of the sun's energy that reaches that point after reflecting from the planet.) But if we use the most basic model of reflectivity used in computer graphics, we assume a given surface's reflectivity characteristics is a convex combination of "diffuse" and "specular" reflection. Pure specular reflection is as though the surface were a perfect mirror (and is assumed to be independent of which colors comprise the incoming light ray). Pure diffuse reflection is assumed to reflect equally in (the hemisphere of) all directions leaving any point of the surface. (It's also assumed to have a reflectance spectrum given by assigning a number in [0,1] to each of R, G, and B -- since that's what CRT monitors are based on.) Light sources are usually modeled as point sources, which works fairly well. Another reasonable simplification that's used is the light source is assumed to be "at infinity" so that its rays are emitted all parallel to each other, making angle calculations easier. (The moon is often given as an example of a pure diffuse reflection -- as you look at the lunar disk of a full moon, the reflected light is virtually independent of whether you look, from the the center of the disk to right near its edge.) If we make the simplest assumption -- that the planet in question has pure diffuse reflectivity -- then a good approximation to how bright any point of it will look is proportional to 1/(d(S,P)^2 * d(P,E)^2), where d = distance, S = the sun, P = planet, E = Earth. But of course, only the side of the planet facing the sun will be seen from Earth, so the above brightness should be multiplied by the area of the lune of the planet's disk that is sunlit and can be seen from earth. For the other extreme -- pure specular reflectivity -- and continue assuming the sun is a point light source, then only the points of the planet whose line to the sun makes an equal & opposite angle with a line that meets the earth will appear to be illuminated at all. In this case only a typically small portion of the planet will reflect light that reaches the earth. (This will be approximately the result of taking the spherical disk of the planet's surface that is centrally projected to Earth, and moving each of its points halfway to the point Q on the planet that's closest to the sun.) It will look like an oval from the earth, possibly a bit kidney-shaped. Is there anyone out there who's willing to take on the task of filling in the details and programming this up to get a brightness function determined by the location of the center of a planet with respect to Earth and the sun? Then this function could be maximized when restricted to the planet's orbit. --Dan A.