Why are vapour cones from aircraft cones?

[lowea001]

Does the vapour cone from a high-speed aircraft correspond to the Mach angle of its sonic boom? What is the general physics responsible for a vapour cone, and why does it resemble the shape of a cone? Basically I'm wondering if this :

corresponds to this :

 

[Simon Bridge]

Have you googled for "vapor cone mach angle" yet?
https://en.wikipedia.org/wiki/Mach_wave

 

[cjl]

The vapor cones occur where there are expansion fans around the object, since expansion tends to cool down the gas, inducing condensation if the water vapor concentration is high enough. You can see a decent amount of the math regarding this expansion here: https://en.wikipedia.org/wiki/Prandtl–Meyer_expansion_fan