Metal dependences of two convection theories for cool stellar envelopes

Most theories of turbulent convection in stellar envelopes assume incompressible how, and so require the assignment of a characteristic length scale from external evidence. In mixing-length theory, this length I is usually assigned to be a constant, alpha, times the local pressure scale height, H-P, or, alternatively, times the distance from the top of the convection zone, z. The new full-spectrum-of-turbulence theory of Canuto & Mazzitelli uses l = z, and therefore is formally parameter-free, Chieffi, Straniero, & Salaris have recently suggested that alpha in mixing-length theory depends on metallicity, Z, but they considered only low-mass stars. We do a similar analysis for stars of higher mass. Specifically, we compare predicted and observed effective temperatures of red giants and red supergiants of widely differing metallicities, but identical luminosities, within the mass range 5-10 M(circle dot). The stars utilized belong to several open clusters in the Galaxy with Z approximate to 0.02 and to the clusters NGC 330 and NGC 458 in the Small Magellanic Cloud with Z = 0.002-0.004. It appears that either alpha in mixing-length theory is independent of metallicity or, since the empirical effective temperatures of the SMC stars may have been underestimated, alpha increases slightly with decreasing metallicity. On the other hand, Canuto & Mazzitelli's theory with l = z is found to perform quite well in all cases, within the possible errors of the observations and of the low temperature opacities.