A diode irradiation with 10-MeV protons was performed to measure the silicon current related damage rate at a temperature of 50 C. This measurement was fundamental to predict the performance of the detectors which will be used in the X-ray spectrometers of the Simbol-X and BepiColombo space missions. These detectors consist of arrays of large area silicon drift chambers with integrated depleted p-channel field effect transistors. The leakage current increase due to radiation damage and its consequent energy resolution degradation can be critical for these missions, specially for BepiColombo. These effects cannot be predicted because, during the whole missions, the sensors will be kept at temperatures below 40 C, and the existing models are based on measurements on structures which underwent annealing at higher temperatures.An irradiation experiment was performed to measure the current related damage rate at 50 C, and the obtained value was (11 1 0 2) 10 17 A/cm. This result implies that it will be possible to achieve the Simbol-X energy resolution, whereas some annealing strategies will be needed for the BepiColombo mission. The annealing behaviour at 60 C was studied as well and the results are in agreement with the already available measurements.