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That is around an amp and a quarter, for 4 sun hours per day (assuming it's pointed at the sun!) you are going to get 60 watt hours from a 15 watt panel, when I put a meter on my car to see how much power everything drew, just the cars ECU drew around 80 watts, head lights are 65 watts each, it seemed like while the engine was running I could expect around 200 watts draw on the battery so 60 watt hours going in to the battery is not going to charge it much more then about 10 to 15 minutes of driving after you factor in the starter as well.
As for your original question of if it will cook the battery or not? no, it should not cook the battery, even a small 40 amp hour starting battery is going to be able to handle a 1.25 amp charge and as the battery reaches full charge that charge rate is going to drop off to less then an amp.
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