In order to obtain high accuracy solar spectral irradiance in a limited sun observation window, it is necessary to ensure that the instrument has completed warm-up when the sun enters the observation window. The start time of warm-up can be predicted in real time to ensure the consistency of the warm-up. Therefore, a short-time high-precision forecast is made for the time when the sun begins to enter the instrument observation window. A method for predicting the solar angle from the current broadcast time of the satellite platform and the instantaneous root of the orbit is described in detail. Using this method, we calculate the solar angle of a sun-synchronous orbit. Te calculated results are compared with the STK simulation, indicating that the maximum angle error of the proposed forecasting method is 0.5° during the warm-up period. The maximum deviation of the warm-up time caused by the angle error is 20 s, which meets the requirement of 1 min. The main error sources of the proposed method are analyzed, which provides reference for on-orbit short-term solar angle forecast of other satellite loads.