How would a solar powered outboard work?

I guess you could, but it wouldn't be very fast.

Think about those little electric trolling motors they use for fishing?

So you get one of those, a small deep cycle battery, a charge controller, and a solar panel on the roof or deck to recharge the battery as you go. I guess you could drop the battery and charger, but if it clouded over you would stop. With a battery you can motor for another hour or 2 and get home.

But it's not going to be very fast as you have maybe a 1 hp engine? But I guess it beats rowing...

This company makes commercial solar electric boats, but probably a bit pricey for the high performance ones. They sell smaller wooded dinghies with electric outboards which is likely what you are thinking of? Some ideas to get you thinking anyway.

http://www.aquawatt.at/GB/startseite_GB.html

You can get 12v outboard motors which run from a battery. If you put a solar panel on the vessel to recharge the battery, you have a solar powered boat.

In reality you won't get a big enough solar panel to power the outboard without discharging the battery, but the battery would recharge while not in use.

Would you really feel safe going to sea in a solar powered boat though with no backup?

If it gets cloudy you are at the mercy of the tide and wind.

I would not exactly include the term "hydro" electric because that would imply that it was powered by water (like a dam or water wheel). Where does the complex solar robot come into play, or is that to keep the solar panels perpendicular to the rays of the sun regardless of which way the boat is headed? You would need to calculate what solar panels you would need to power the electric motor and would also need a battery to get the motor started because a motor uses the most current to get started and is almost like a direct short when not turning. For example I could not even turn a tiny low power 3 volt motor with a 5 watt solar panel that generated 15 volts I think at no load.

In a way sailing is indirect solar power, since heat of the sun creates rising thermals and air comes in under that to create wind.