I was after how you derived resistor values and why.Briefly.
When the solar cell is below .5 volts Q1 transistor is OFF which allows Q2 to conduct which then turns ON Q3.
When the solar cell is above appx .52 volts Q1 turns ON which then turns OFF Q2 and Q3.
The circuit is known as a schmitt trigger. It will rapidly switch ON or OFF when activated by those voltages listed.
R6 is added to limit the current to 100ma on the LED, adjusted for the small voltage loss across Q3 (appx 200mv).
Circuit verified on the breadboard with the components listed.
