Can Shock Waves Initiate Protostar Formation?

[hunt_mat]

I have a thought whilst in the shower this morning, it is usually said that shock waves only take part in star formation when other stars have formed and have exploded as supernova sending out a shock wave. Now I have been thinking, the usual equation for self gravitation is given by:
$$\frac{d^{2}r}{dt^{2}}=-\frac{Gm}{r^{2}}-\frac{1}{\rho}\frac{\partial p}{\partial r}$$
Upon writing $v=dr/dt$, it is possible to write the acceleration as:
$$\frac{d^{2}r}{dt^{2}}=\frac{dv}{dt}=\frac{\partial v}{\partial t}+v\frac{\partial v}{\partial r}$$
So the equation becomes:
$$\frac{\partial v}{\partial t}+v\frac{\partial v}{\partial r}=-\frac{Gm}{r^{2}}-\frac{1}{\rho}\frac{\partial p}{\partial r}$$
The above equation is a first order hyperbolic equation which allows the formation of shocks, in this case allows the possibility of converging shock waves from the mathematical standpoint at least.

Thoughts?

 

[Lavabug]

You've basically written the 1-D Navier-Stokes equation where pressure and gravity are the only forces present. You can get shock waves applying different conditions like stationary flow. Not sure what you mean by converging shock wave though.

 

[Chronos]

The probability of shock waves from more than one supernova converging at any particular point in space is ... low.

 

[hunt_mat]

A converging shockwave is one that converges to a particular point, it doesn't spread out like a detonation wave. So a converging shock would compress the gas to a very small region therefore making it very hot indeed. This I think would be enough to start fusion more easily.

One of the things I keep hearing about is that population III stars didn't have shock waves to get then started which I think is wrong because the equation modelling them does support shock wave and the more I think about it the more I think that the gas coming together under gravity would naturally shock up.