How to understand this filter circuit

I made this circuit, with the intention of it lighting and then fading out the led after very briefly closing the switch:

This is intended for a CR2032 coin cell battery power source and the led is a normal 20ma through hole red LED, probably Vf =~ 2v.

Ignoring the one resistor in series with the power supply, what function describes how the transistor and the values of capacitor and resistor determine the led brightness and how long it takes to fade out? Like if I wanted to make it fade out 50% more quickly or more slowly, how would I change the resistor and/or capacitor?

Does the characteristics of the transistor matter?

• You don't mention the value of the voltage source, nor the rated (or maximum continuous) current through the LED. We need those values before being able to say anything about the transistor and its base resistor. – HarryH Jun 12 '18 at 3:07
• updated: it's a 20ma red led, voltage = 3V coin cell – J Halcres Jun 12 '18 at 4:13

$R*C = \tau$