Solid angle to the unit sphere with center P, which can be calculated as the surface integral: where formula_5 is the unit vector corresponding to formula_6, the position vector of an infinitesimal area of surface with respect to point P, and where formula_7 represents the unit normal vector to . Even if the projection on the unit sphere to the surface S is not isomorphic, the multiple folds are correctly considered according to the surface orientation described by the sign of the scalar product formula_8. Thus one can approximate the solid angle subtended by a small facet having flat surface area ,
Surface-area-to-volume ratio icy or rocky material in outer space may, if it can build and retain sufficient heat, develop a differentiated interior and alter its surface through volcanic or tectonic activity. The length of time through which a planetary body can maintain surface-altering activity depends on how well it retains heat, and this is governed by its surface area-to-volume ratio. For Vesta (r=263 km), the ratio is so high that astronomers were surprised to find that it "did" differentiate and have brief volcanic activity. The moon, Mercury and Mars have radii in the low thousands of kilometers; all three retained heat well