Sawtooth wave generator using 555 timer working at low frequencies
A common alternate starting place is to plug the circuit into simulator and see what you get. I did this with the following similar circuit:



The simulated saw tooth waveform came out as:



Note that in my circuit I replaced the zener diode with two forward biased signal diodes. One diode drop approximates the Vbe drop of the PNP transistor. This means that the R1 resistor has an ~0.65V drop across it which defines the current through the resistor. This is an easy way to make a current source that works with a small drop across the resistor.