Yes, your solar powered plan is possible.
Quickly reviewing the maths, a set of 2300mAhr batteries which last 40 hours are supplying around 50mA. So any charger which is required to keep pace with the power demand would need to provide some 50mA at somewhat over 3V.
This would require a solar panel with an area equivalent to, or larger than the headband, and would add considerably to the mass and the cost - not to mention the styling, and the reflection off the panel, which in turn would reflect off the shiny bonce of your bald captain only to blind you!
As a certain manufacturer of ANR headsets (Based On Sound Economy, or BOSE for short) advertise the lightness of their headset, and their price is top Dollar already, I doubt this is attractive to them.
A set of alkaline AAs can be bought for around 50p, which works out to be about 1p per hour. Rechargeable NiMH batteries cost under 1p to charge for a full 50 hours use. So where is the economy in adding a solar panel, a charging circuit and rechargeable batteries, which might add some £100 or more to the cost, as well as adding 50% to the weight?
I'd enjoy the lightweight headset as it is, and bask in the sunshine with a smile on your face, comfortable in the knowledge that by recharging your AA batteries on domestic mains each hour's use is costing you less than 1/50th of a penny!
Good luck
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