can the arduino start up when solar panels get light.

I was wondering if a battery charged solar system was left on over night, and if it had a low voltage/current sensor so to not dissipate the battery. would it totally switch off? And would it turn back on again once the sun came out the following morning??? Oh lifes little wonders. Thanks, Mick

Yes. The feature is called Low Voltage Disconnect (LVD). Not all solar chargers have it.

Here is a PWM model:

Here is a MPPT model: