I'm glad you asked that, LB. That is what determines the overall expense. To run one 24/7 you would need large battery and high wattage of solar panels. Your and ML's new, small and shallow ponds are ideal for solar aeration without great expense. One thing is that the sun is your friend, in that summer and long days gives more time for charging, and more time for algae bloom to produce O2 in late spring and summer.

I couldn't speel airashun...aration...whatever until I came upon this great forum, so the experts in the field can elaborate and/or correct, but I feel that night aeration with an algae bloom is essential due to it and other plants which may(but hope not)develop taking in oxygen at night. They also give off O2 during the day, but with high temps., above 90, I would aerate for water circulation during the hottest time of day. I'm thinking midnight to 7AM and 2-6PM. That is 11 hrs per day. I aerate 24/7 becaues I have electricity and an old pond with silt and muck. 10-12 hrs. in late spring and summer and less at other times should be good.
There are many past posts on when and how long to aerate and determining factors. I suggest you look at Ted Lea or Cary Martins posts on the subject. Also Bill Cody, Bruce and Cecil, and many others with vast experience. Once you determine your objectives, I can tell you how much wattage and battery size you will need. Also if you will change out battery occasionally or want a self maintaining system.