So connecting a discharged capacitor will short-out your solar panel, until the capacitor voltage rises as it charges. With a supercapacitor, it will take a very long time to charge - so the voltage will remain low for a long time. Until the capacitor has charged to at least the forward voltage of the LED, the LED is not going to light
Yes, you can use capacitors with solar panels. But, only the supercapacitors are eligible to perform with solar panels. The supercapacitors can discharge the high-voltage current from the solar cells, which is much higher than the loading current. It will help the system when there is an intermittent load.
The resistor is useless. Your solar panel already has a voltage decreasing when current increases (that is, it is not an ideal voltage source,) and the maximum current your small panel produces should be no issue at all for the capacitor. There is no reason to dissipate power as heat The 1N4148 diode you use is not adapted for your application.
When putting the solar panel very close to a source of light this 0.4 value slowly rises up. I think you are right, i have a second solar pannel i might try to use both to charge it, I saw some people talking about a diode to not let the current flow back to the solar panel is this right ? A discharged capacitor is, essentially, a short circuit.
For exact calculation of the charging-discharging of the capacitor, we would need: The link to the datasheet of your solar panel. Information on the load attached to it (link if possible, minimum and maximum voltage.) You'll have to get more than 3V out of your panels and more than 3V on the cap/battery to get some seconds of 3V 500mA out of it.
The integration of capacitors into solar power systems stands as a potent strategy for enhancing their efficiency and operational longevity. Capacitors, essentially energy storage components, function by storing and swiftly releasing electrical energy.