Oscillator Design - where are ESR and gmcrit formulars from?

[DavidKuo]

I am reading datasheet AN2867(Oscillator design guide). There are two formulars : ESR and gmcrit. It's common formular and easy to find it from internet. My question is where are these formulas from? (How to derive these formulas?)

1. AN2867 Rev 18, sheet 13, gmcrit = 4 x ESR x (2 x pi x F)^2 x (C0 + CL)^2
2. AN2867 Rev 18, sheet 15, ESR = Rm x (1 + C0/CL)^2

[STOne-32]

Dear @DavidKuo ,

Indeed, these formulas are key to select the right crystal matching our external oscillators such as LSE or HSE , then ensure the right margin to have startup oscillation and good stability .  As a general rule , smaller Crystal ESR and ESR max the better is the margin. If you are interested by the Theory of oscillators, it comes from Prof Eric A, VITTOZ here https://picture.iczhiku.com/resource/eetop/shigwIRlqyQHIxbb.pdf

have a good Sunday !