Approximations of the Planck Function for Models and Measurements Into the Submillimeter Range

A brightness temperature is defined as a linear function of the Planck radiance, with the linear coefficients optimized to minimize the difference between the brightness temperature and the physical temperatures of atmospheric and terrestrial emitters. Radiative transfer (RT) calculations can be accelerated by formulating the integration in terms of this brightness temperature while producing output in terms of radiance or brightness temperature. Approximation errors are $? 0.012$ K for RT model applications up to 400 GHz, for any upward, downward, or limb-view geometry, which is about an order of magnitude smaller than for the common brightness temperature derived from a second-order expansion of the Planck function. When products of an RT model that uses this optimized Planck approximation are compared with measurements and the measured radiance is high (equivalent brightness temperature is $? 170$ K), it can be advantageous to apply a complementary approximation to the measurements to benefit from error compensation between the model and the measurements. Alternatively, error compensation can be obtained if the calibration and RT equations use consistent brightness temperature approximations.