This question was asked before, but I am struggling with the same question and I can't figure out "how to deal with it in the real world" as someone put it at the time.
Olin's approximation is the first idea I came up with: assume the LED is a voltage source, assume the capacitor initially charges up to 5V - Vled_drop, and assume a simple RC discharge. However simulating the circuit seems to show that the approximation is nowhere close to being valid. With this circuit:
If I assume a simple RC discharge model the simulator gives
RC = - 3s / Ln(V(3s) / (5V - Vs)) = 4.5
after 3s, when the calculated value would be
RC = 1.8mF * 1k = 1.8
(the simulator uses Vs = 1.64V for a red LED).
In other terms the capacitor discharges much slower than the RC=1.8 would indicate. Is there any better way to predict what will roughly happen without a simulator, or without solving the actual differential equations?