Exploring Shock Waves in Protostar Formation: A Mathematical Perspective

[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.

 

[pmacias]

Thank you for sharing your thoughts on the role of shock waves in protostar formation. Your mathematical perspective is certainly intriguing and adds to our understanding of this complex process.

It is true that shock waves are often associated with the formation of stars through the explosion of nearby supernovae. However, as you have pointed out, your equation shows that shock waves can also be formed through the convergence of gas and dust particles under the influence of self-gravity.

This is an important consideration in the study of protostar formation and could potentially explain the presence of shock waves in regions where no supernovae have occurred. Further research and analysis will be needed to fully understand the role of shock waves in protostar formation, but your perspective is a valuable contribution to this field of study.

Overall, your thoughts highlight the importance of incorporating mathematical models and equations into our understanding of complex astronomical phenomena. Thank you for sharing your insights.