Problem 2.16: If the intensity of radiation incident on a surface is uniform from all directions and denoted by the constant I , verify that the total flux is π I , as stated by (2.60). Note that this approximately describes the illumination of a horizontal surface under a heavily overcast sky. It also describes the relationship between the flux and intensity of radiation leaving a surface, if that surface is emitting ra- diation of uniform intensity in all directions. Problem 2.17: As noted above, when the radiant intensity incident on a flat surface is isotropic with intensity I , the resulting flux is π I . The power intercepted by a circular flat plate of radius r is therefore P = π 2 r 2 I , assuming that it is illuminated from only one side. How much power is intercepted by a sphere of radius r exposed to the same source? Hint: There are various ways to reach the same conclusion, some of which are more cumbersome than necessary. Try to find a simple but compelling geometric argument. Problem 2.18: Compute the flux from an overhead spherical sun, as seen from a planet in an orbit of radius D , given that the sun has radius R s and a uniform intensity I s . Make no assumptions about the size of D relative to R s . Consider the radius of the planet itself to be negligible. Use two different methods for your calculation: (a) Method 1: Integrate the intensity over the solid angle sub- tended by the sun, with the usual cosine-weighting relative to the local vertical. You will need to derive an exact expression for the solid angle subtended by the Sun’s disk for arbitrary D > R s (see Fig. 2.5). (b) Method 2: Compute the flux density emerging from the sur- face of the sun, translate that into a total power emitted by the sun, and then distribute that power over the surface of a sphere of radius D . Do your two solutions agree?
Applications 49 Solar Flux S 0 Intercepted Flux Φ= S 0 π R E 2 R E Fig. 2.7: The total flux of solar radiation intercepted by the earth is equal to the product of the incident flux density S 0 and the area of the earth’s shadow. 2.8 Applications to Meteorology, Climatology, and Remote Sensing Of fundamental importance to the global climate is the input of en- ergy from the sun and its spatial and temporal distribution. This input is a function of two variables: 1) the flux of solar radiation incident on the top of the atmosphere, and 2) the fraction of that flux that is absorbed by either the surface or the atmosphere at each point in the earth-atmosphere system. The second of these depends in a complex way on distributions of clouds and absorbing gases in the atmosphere, as well as on the absorbing properties of the sur- face. These are all issues that will be taken up in the remainder of this book. The first variable, however, can already be understood in terms of the material presented in this chapter.