Solar panel selection
It's an energy equation.

38.5Watts, 3.4 hours is equivalent to 130.9Wh - watt hours - unit of energy.

To be net energy zero, you need at least 130.9Wh of energy per day. Now this depends a lot in your location but a good start is that you get 1000 effective hours of sun per year. 

With 130.9Wh per day, you have 47.8kWh per year. If you have 1000 effective hours this would mean a solar panel of at least 47.8W. 

Then it's more complicated because in winter you will have less, summer more, etc, etc.. but you get an initial approximation at least of what you would need.

Sizing the battery then also depends on location, weather, etc. That's another point altogether.

Hope this helps as a starting point. It doesn't consider efficiency in your converters, MPPT, charge/discharge efficiency, etc. All this could be grouped into a "safety coefficient"