The battery pack in the laptop is not just a simple two-terminal battery; it has about 10 connections, with internal temperature sensors, and who knows what else. I've no idea how to charge it externally, or what all those contacts do. The only way I know how to charge it is via the two-terminal coaxial plug on the rear of the laptop.
The external power supply for my laptop is just a switching power supply which puts out 19.5V at 3.3A. Any DC power supply which puts out 19.5V could be substituted and the laptop would not know it...
In my case, my laptop's external power supply will operate from 12-15Vdc from a vehicle battery (it steps up the voltage to 19.5V), so if I wanted to solar power it, I could get a panel, a 12V solar charge controller, and a 5 to 10AH 12V SLA battery. Since the external power supply is about 80% efficient, I would loose about 20% of the power between the 12V battery and the laptop...