Abstract
Cities exhibit both beneficial and detrimental characteristics, many of which stem from agglomeration effects and are, to a first approximation, influenced by population size. However, urban density also plays a critical role. For example, cities with similar population sizes but higher densities tend to emit less carbon, while simultaneously exhibiting a more pronounced urban heat island (UHI) effect. This trade-off highlights the need for a balanced approach that simultaneously minimizes both carbon emissions and the urban heat island (UHI) effect. To address this challenge, we examine how both carbon emissions and UHI intensity are influenced by the population size and spatial extent of the cities. As objective function we define the some of both quantities where city population and area are variables. Considering the scaling relation between area and population as constraint, we derive a theoretical expression leading to an optimal city size. To validate our approach, we analyze carbon emissions data from cities in Germany and consider UHI parameters from the literature. We find that, in the specific case of German cities, achieving an optimal city size that simultaneously minimizes both carbon emissions and UHI intensity is not physically feasible. From a methodological perspective, only the UHI intensity parameters, together with the exponent of the relationship between population and area, determine whether an optimum exists or not. We argue that instead, the scaling relation between population and area itself should be understood as an optimum.