A real-time applicative software consists of both aperiodic and periodic tasks. The periodic tasks have regular arrival times and strict deadlines. The aperiodic tasks have irregular arrival times and no deadline. The objective of an optimal aperiodic task server is to guarantee minimal response times for the aperiodic tasks and no violation of hard deadlines for periodic tasks. We consider a real-time energy harvesting system composed of energy harvester, energy storage unit, uniprocessing unit and the real-time tasks. We introduce a novel periodic task scheduler, namely SSP which is an extension of the slack stealing server so as to cope with fluctuations in energy availability. Experimental results show that the proposed algorithm achieves better performance in terms of aperiodic responsiveness. Simulations have been conducted for various settings of workloads and harvested energy profiles.