A reference or "no-feedback" radiative response to warming is fundamental to understanding how much global warming will occur for a given change in greenhouse gases or solar radiation incident on the Earth. The simplest estimate of this radiative response is given by the Stefan-Boltzmann law as -4 sigma(T) over bar (3)(e) approximate to -3.8 W m(-2) K-1 for Earth's present climate, where (T) over bar (e) is a global effective emission temperature. The comparable radiative response in climate models, widely called the "Planck feedback," averages -3.3 W m(-2) K-1. This difference of 0.5 W m-2 K-1 is large compared to the uncertainty in the net climate feedback, yet it has not been studied carefully. We use radiative transfer models to analyze these two radiative feedbacks to warming, and find that the difference arises primarily from the lack of stratospheric warming assumed in calculations of the Planck feedback (traditionally justified by differing constraints on and time scales of stratospheric adjustment relative to surface and tropospheric warming). The Planck feedback is thus masked for wavelengths with non-negligible stratospheric opacity, and this effect implicitly acts to amplify warming in current feedback analysis of climate change. Other differences between Planck and Stefan-Boltzmann feedbacks arise from temperature-dependent gas opacities, and several artifacts of nonlinear averaging across wavelengths, heights, and different locations; these effects partly cancel but as a whole slightly destabilize the Planck feedback. Our results point to an important role played by stratospheric opacity in Earth's climate sensitivity, and clarify a long-overlooked but notable gap in our understanding of Earth's reference radiative response to warming. Plain Language Summary Earth's climate is stable because a warmer planet loses more energy to space, at infrared wavelengths invisible to the naked eye. The rate of change of this energy loss as the planet warms provides an estimate how Earth's energy balance responds to warming, which is simple enough to write on a small piece of paper. When scientists investigate the warming predicted by climate models, they often start from a similar but not identical calculation of how Earth's energy balance responds to warming. This calculation, based on model output, is about 15% less stabilizing than the simple pencil-and-paper estimate. In this paper, we explore the causes of this 15% difference between the pencil-and-paper estimate and the calculations using climate models. We show that the difference is primarily caused by the lack of assumed warming in climate models high in Earth's atmosphere, where temperatures are not closely linked to surface warming. This lack of warming acts as a hidden destabilizing feedback in current analysis of climate models.
