Trigger Monostable 555 When Capacitor Raises above 0.5 Volts
The threshold triggering could be done using a rail-to-rail comparator, such as the Linear Technologies LT1711, using a 0.5 Volt reference voltage.

Depending on the precision required, the voltage reference can be obtained by using a sub-bandgap voltage reference IC such as the Analog Devices AD130 which has a 0.5 Volt output, or through a resistor voltage divider between the supply voltage and ground.

For a hybrid compromise, a Zener diode shunt reference of say 1.8 Volts can be used with a resistor voltage divider connected across it:







simulate this circuit &ndash; Schematic created using CircuitLab

The triggering slope (rising or falling) can be switched by interchanging the inputs to the comparator.