To complete the analysis we need one more fact or approximation. We have assumed that the density of the melt differs little from that of the solid. The Birch-Murnaghan equations (Birch, 1938; 1952; Anderson, 1989) show that the bulk moduli of solid systems change in a predictable way as a function of the changing density. A similar result for simple liquids known as Rao's rule (Rao, 1941) also shows that the bulk modulus of many pure (i.e., single constituent) liquids is also a simple function of the density. Based on these results, if the density of the melt is the same as that of the surrounding solid, then we expect the bulk modulus of the melt to differ very little (on the order of a few per cent) even though the shear modulus has dropped from a finite value to zero. We have shown in the preceding paragraph that, if the fluid inclusions have the same bulk modulus as the solid, then Ksat = Kg and c1 = 0. We expect this approximation to have the same level of validity as the approximation that the density is constant (which is to say that we think of it as a reasonable first approximation). With this approximation substituted into (dlnvp2), we find that
d v_sd v_p 3. This is correct as long as . But the shear modulus will necessarily decrease as the melt fraction increases, since, for fixed overall shear distortion, less shear energy can be stored in the system, and this implies that c2 > 0. Computing the actual value of c2 requires a model of the microstructure, but the result (veldecrement) shows that the precise value c2 is not required to obtain the desired result in (veldecrement), since it cancels out of the final formula.