Owing to the energy scattered or absorbed by the constituents of earth's atmosphere and self-absorption in the outer layers of the sun, the spectrum of solar flux at earth's surface is different from that of a blackbody. Consequently, the second law of thermodynamics for heat engine cycles operating between thermal reservoirs needs to be revised to determine the maximum conversion efficiency. A thermodynamic model similar to those for multi-temperature plasmas and non-isothermal particle-exchange heat engines is proposed to estimate the maximum conversion efficiency of a mechanical or solid-state heat engine subject to a radiation flux not having a blackbody spectrum. An example is given to illustrate the calculation of the maximum power that can be converted from a solar flux with considerable gas absorption. Copyright (C) 2011 John Wiley & Sons, Ltd.