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We present a computationally inexpensive analytical model for simulating celestial polarization patterns in variable conditions. We combine both the singularity theory of Berry et al (2004 New J. Phys.6 162) and the intensity model of Perez et al (1993 Sol. Energy50 235–245) such that our single model describes three key sets of data: (1) the overhead distribution of the degree of polarization as well as the existence of neutral points in the sky; (2) the change in sky polarization as a function of the turbidity of the atmosphere; and (3) sky polarization patterns as a function of wavelength, calculated in this work from the ultra-violet to the near infra-red. To verify the performance of our model we generate accurate reference data using a numerical radiative transfer model and statistical comparisons between these two methods demonstrate no significant difference in almost all situations. The development of our analytical model provides a novel method for efficiently calculating the overhead skylight polarization pattern. This provides a new tool of particular relevance for our understanding of animals that use the celestial polarization pattern as a source of visual information. [ABSTRACT FROM AUTHOR]