How bright can we make a sun jar? A sun jar is an object that stores solar energy in a battery and then releases it during dark hours through a led.
Assume:


*

*a $65cm^2$ solar panel 

*a 12h/12h light/dark cycle

*insolation of $2.61kWh/m^2/day$

*perfectly efficient components (i.e. without violating Entropy laws or other theoretical limits)

*light is emitted by an ideal "white" source



How bright can we make the jar so that it shines for 12 hours?

Useful links:


*

*Insolation

*Peukert's law

*Solar cell efficiency

*Luminous efficacy
 A: Using that ideal white source on the wiki link, that states 251 lm/W.
The insolation level is $2.61kWh/m^2d = \frac{2610}{24}W/m^2$
which across $65cm^2 (= 0.0065m^2)$ gives 0.7W. If everything were 100% efficient, then you'd have $0.7 \times 251 lm = 178 lm$
Now we derate on some maximum theoretical efficiencies. The biggest theoretical derating will be on the PV, and that will completely dwarf any loss on the maximum theoretical efficiency of a round trip into storage and back.
For a single-junction n-p PV cell in unconcentrated sunlight, the maximum efficiency (Shockley–Queisser, DOI:10.1063/1.1736034) is 30%, giving you about 53 lm. By layering multiple junctions, you could theoretically get 42% (2 junctions, 74 lm), 49% (3 junctions, 87 lm), tending to 68% ($n\to\inf$, 121 lm) (doi: 10.1088/0022-3727/13/5/018)
Now, if you are allowed to put a concentrating lens onto the $65 cm^2$ cell, so that the sun jar harvests light from a much larger area, then we can really go to town. From the second link, the maximum efficiencies for concentrating PV, as the number of junctions tends to infinity, is 86.8%. So then you've got to find out what the maximum concentration could be without the whole lot bursting into flames ... that 86.8% is based on a concentration factor of 45 900 , so it's a PV cell made of pure unobtainium, and I'll stop right there.
A: Well, basic answer is that it can be about twice "brighter" than sun (i.e. it will emit twice the amount of light absorbed), if LED emits only visible light, while battery absorbs everything from IR to UV.
Reality is of course much much worse.
A: 2.61kWh/m^2/yr * 65cm^2 = 0.002 watts; since the time to charge/uncharge is the same, this is both the power in and the power out.
Conversion to light at 300 lumens/watt: 0.6 lumens http://en.wikipedia.org/wiki/Light-emitting_diode
If it shines in all directions, about 0.05 candela - so about as powerful as one-twentieth of a candle.
So. Not very brightly, then.
