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Good idea!, I just did some research on solar water heating. look up an elsid pump. They are a pump that only has one moving part, the impeller, which is spun by a rotating magnetic field at VERY low power consumption. You can build a heat collector box around a car radiator painted black on the roof, and plumb the water down into another car radiator inside the coop (or some coils of copper tubing inside water supply. Use a smallish solar panel to run the elsid pump and to charge your battery, you may even have enough power to run some sort of a tiny heater on a thermostat after dark to keep the water from freezing.
I wonder if you just got a big heat sinked resistor and sunk it in your water dish and put a thermostat on it to apply current when the temp drops below 40ish if that would be enought to keep it from freezing? You would not have to worry about a power inverter, and it would work right off your 12v battery. According to Ohms law a 50 ohm resistor at 12v should create just under a 3 watt load, you could put a 5 or 6 watt resistor in there and it would generate a little heat. You would have to play with your load a little to see if it would work, but I think the resistor would work better for you than an inverter and 110 fish tank heater.
heat sinked resistor? what exactly is that? Is it something low wattage? If so, that might not be a bad idea. I'd ideally prefer to ditch the inverter if possible because it would use some of the power up converting.