Previous estimates of the microwave background anisotropies produced by freely falling spherical clusters are discussed. These estimates are based on the Swiss-Cheese and Tolman-Bondi models. It is proved that these models give only upper limits to the anisotropies produced by the observed galaxy clusters. By using spherically symmetric codes including pressureless matter and a hot baryonic gas, new upper limits are obtained. The contributions of the hot gas and the pressureless component to the total anisotropy are compared. The effects produced by the pressure are proved to be negligible; hence, estimations of the cluster anisotropies based on N-body simulations are hereafter justified. After the phenomenon of violent relaxation, any realistic rich cluster can only produce small anisotropies with amplitudes of order 10−7. During the rapid process of violent relaxation, the anisotropies produced by nonlinear clusters are expected to range in the interval (10−6, 10−5). The angular scales of these anisotropies are discussed
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