For keeping the radio on for hours, or running a thermoelectric cooler, etc, the only way to go is a second battery. A solar panel of any reasonable (dash board) size is only good for keeping the battery topped off while siting idle for days/weeks.
Running appliances via solar requires a lot more investment. Look at it this way:
A coleman thermo-electric cooler draws 50 watts. That's 1200 watt-hours/day.
A 30-watt solar panel puts out 30-watts for only about 8 hours/day, even if it's in a perfect (no shade ever) position. That's only 240 watt-hours/day.
A standard optima deep-cycle has 120 reserve amp-hours (at 12v) of power, which is about 1440 watt-hours, just enough to power that cooler for a full day, and you'd totally drain the battery, which it doesn't like!
Constant loads put a surprising amount of drain on a system. Even just the radio, which need a fraction of what the cooler wants, will drain a battery after a few days of all-day use, and a 5-30 watt solar panel won't help much.