1. This question asks you to solve the steady-state heat equation -V. (kVT) = a (1) in a sphere, under the assumption that T depends only on distance r from the centre of the sphere. The question assumes first that k is also (given) a function of r, and later, as an alternative, that k depends on T. Assume that the rate of heat production a = ao is constant in the sphere, that the sphere has radius R and has a given surface temperature Ts. (a) Write down the steady state heat equation (1) as an ordinary differential equation for T(r) (that is, in terms of derivatives with respect to r). You do not need to derive this equation. Do not assume that k is constant, but do assume that it is either explicitly a function of r, or a function of T(r) (and therefore depends on position (x, y, z) only through r). (b) Suppose you don't know the value of ao but know the value of the heat flux at the surface of the sphere. Call that value qgeo. Show that 39geo R regardless of whether thermal conductivity k is constant or not. (c) Let k be of the form k = k exp [(R r)/Ro], where ko and Ro are constants. For given values of ko, ao, Tsurf and R, find an equation that must be satisfied by Ro to ensure that hte temperature at the centre of the sphere is T. This equation cannot be solved analytically, but for given parameter values can be solved numerically or graphically. If we take values appropriate for Earth as ko = 2 W m K, T = 280 K and T = 3000 K, qgeo = 0.04 W m2 and R = 6380 km, what is Ro to two significant figures? Sketch the corresponding temperature distribution in the Earth. Recall that the integral fr exp(x) dr can be done through integration by parts. (d) Let k be of the form k = ko exp [(T Ts)/To] - where To is a constant. For given values of ko, ao, Tsurf and R, again find an equation that must be satisfied by To in order to produce a a tempera- ture at the centre of the Sphere of Tc. For the numerical parameter values given above, find To to two significant figures. Sketch the corresponding temperature distribution in the Earth.