söndag 5 juli 2020

DigiMat: Unphysical Gravitational Collapse

DigiMat offers a rich playground to explore the World. In DigiMat Model Workshop (70-74) you find a cosmological model in the form of fluid/thermodynamics with self gravitation. As an example consider the following simulation of the phenomenon of gravitational collapse:

What do we see? We see unphysical gravitational collapse with 
  1. gravitational potential peaking in the middle (light-blue)
  2. high density middle (blue)
  3. negative temperature middle (green)
with the unphysical nature signified by negative temperature. How do we know that the simulation is unphysical? This is because the negative temperature comes with negative Total Internal Energy E(t) (= -0.483 at a certain time t) with 
  • $E(t) = \int \rho (x,t)T(t,x)\,dx$,
  • where $\rho$ is density and $T$ temperature.
Now, total internal energy E cannot be negative, neither density nor temperature can be negative. This connects to the basic energy balance of a self gravitating fluid/thermodynamic system which takes the form 
  • $TE(t) = E(t) + K(t)$ with 
  • $TE(t) =$ Total Energy at time $t$
  • $K(t) =\frac{1}{2}\int\rho (x,t) |v(x,t)|^2dx$ Kinetic Energy, with $v(x,t)$ fluid velocity.   
Notable is that Total Energy has no contribution from gravitation. 

What we see in the simulation is accelerated fluid motion from a gravitational potential with central peak (light-blue) accompanied by an increase of kinetic energy balanced by a decrease of internal energy/decrease of temperature (green), while density is increasing (blue). But there is a limit to this process, because temperature/internal energy cannot become negative. 

When the simulation passes this limit and temperture/internal energy becomes negative, the simulation turns unphysical and thus produces numerical artefacts, not physics. 

Before collapse of the numerics, density and kinetic energy are increasing while temperature decreases. This means that the concentration of mass towards the center does not come with an increase of temperature, but instead with a decrease of temperature.

This shows that a common idea of "stars being ignited by gravitational collapse" (see Wikipedia quote below) is incorrect in the sense that the ignition does not come from increasing temperature. The ignition of stars thus can only be the result of a concentration of mass (triggering nuclear fusion balancing further concentration), and not any fire ball sucking energy from gravitation.

And the other way around, if a black hole is what results if star ignition/fusion does not take place, it must be a very cold place. 

To see physical uncollapse run the code with larger gas constant $\gamma$ (e.g. 1 instead of 0.1)!

Summary: With Total Energy = Internal Energy + Kinetic Energy without contribution from gravitation, there is no physics of unlimited sucking of energy from gravitational collapse.  There is no physics of unlimited sucking of energy from internal energy (allowing negative internal energy).

More precisely, the energy balance takes the form 
  • $\frac{dE}{dt} = -W+P+D,
  • $\frac{dK}{dt} = W-P-D,  
where $p=\gamma\rho T$ and $\phi$ gravitational potential (with minus sign),  and
  • $W(t) = \int p(x,t)\nabla\cdot v(x,t)\,dx$,
  • $P(t)=\int\phi\nabla\cdot (\rho (x,t)v(x,t))\,dx$, where $-\Delta\phi =\rho$ (and thus $\phi >0$), 
  • $D\ge 0$ turbulent dissipation, 
which as a 1st law of energy balance gives $E+K$ constant with $D\ge 0$ as a 2nd law. See Computational Thermodynamics.

Now you are ready to experiment yourself with DigiMat, and seek deeper understanding of how stars are formed...by thermodynamics, gravitation and fusion. 

Notice that fusion opens a new source of positive contribution to internal energy, which can be transformed into high temperature and high velocity. Without fusion this option is closed and the amount of internal energy is limited by given total energy.

We observe that energy without gravitation can be transferred from $E$ to $K$ (only) in expansion with $\int p\nabla\cdot v\, dx>0$ and vice versa from $K$ to $E$ (only) in contraction with  $\int p\nabla\cdot v\, dx<0$. Gravitation can transfer energy from $E$ to $K$ (only) in contraction with $\int\phi\nabla\cdot (\rho v)\,dx <0$ and the other way around. 

Is it true that 
  • DigiMat unlocks the grand challenges in science, education and industry?

Compare with Wikipedia presenting the standard view:
  • A star is born through the gradual gravitational collapse of a cloud of interstellar matter. The compression caused by the collapse raises the temperature until thermonuclear fusion occurs at the center of the star.


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