Reelight SL-120 circuit theory and making it flash flasher

I noticed the other day that my front, white, Reelight SL-120 had a crack in the plastic. Since it was partially cracked and needed to be fixed I decided to take it apart to see how it operated.

Reelights work through magnetic inductance. Two magnets on the spokes of the wheel spin around and briefly pass by the light. The light has an iron core that meets the magnets as they pass by. The iron core is wrapped with enamel coated wire. As the magnets pass by the iron core, the collapsing magnetic field creates and electric current that travels through the wire. The electric current in the wire travels to a capacitor where the energy is stored. The energy in the capacitor is then used to power a 555 timer circuit to flash the two white LEDs.

Simple right?

But what if you wanted to make your Reelight SL-120 flash a little faster?

First we need to look at the LM555 datasheet to figure out all the elements of the circuit and how Reelights has implemented the circuit.

It looks like the SL-120 uses an Astable waveform. The frequency is 2.2 Hz (roughly – I counted the number of pulses over 60 seconds).
The astable equation looks like
2.2 Hz = 1 / T = t1 + t2.
t1 = 0.693(Ra + Rb)C
t2 = 0.693(Rb)C

Looking at the Reelight SL-120 I see that 560k is Ra and 56k is Rb but the cap is un-marked. Damn. Gotta do this the hard way…
2.2 Hz = 1/0.45
0.45 = 0.693(Ra + Rb)C + 0.693(Rb)C
0.45 = 0.693(616)C + 0.693(56)C
0.45 = 426.89C + 38.8C
0.45 = 465.69C
C = 0.000966 (roughly 1uF)

If we wanted to make the lights flash more often, we would need to change Ra.

For fun I threw a 100k resistor on top of the 560k resistor. What happens is that Ra is now 84.85 [1/(1/560 + 1/100)] so the frequency would be:
1/(0.693(84.85 + 56)0.001 + 0.693(56)0.001) = 7.33 Hz