DigiMat: Physical Gravitational Collapse

We continue our DigiMat study of star formation by gravitational collapse into a high density hot spot. 

 

In a fluid at rest without self gravitation a hot spot cannot develop without external forcing by transfer of energy from a cold surrounding. This is because without forcing, heat energy can only transfer from hot to cold, by the 2nd law of thermodynamics.  

We now ask if in a fluid with self gravitation a hot spot can develop by surrounding cooling. We use DigiMat to give an answer: We thus simulate the motion of a fluid with self gravitation starting with the fluid at rest with constant  density, temperature and internal energy, assuming periodic boundary conditions for fluid variables. We set the gravitational potential to zero at the boundary and thus create a gravitational force directed to the center of the domain. Here is the result at a certain time:

What do we see? We see (crosscut of) central accumulation of mass/density (blue) combined with central heating and far-away cooling given by temperature (green), with black line the initial distribution. We thus effectively see a transfer of heat energy from cold to hot. This transfer is accomplished by gravitation, which from an initial state with zero fluid velocity attracts the fluid towards the center thereby increasing kinetic energy there, which then is transfered into heat energy as pressure finally retards the flow (and reverses the flow into expansion). 

We thus find that the presence of a gravitational field can be viewed as a form of external forcing making possible transfer of heat energy from cold to warm. Gravitational collapse thus appears possible thermodynamically.   

Gravitational collapse with transfer of heat energy from a cold exterior to a hot interior fits with the observation of interstellar dust at low temperture, about 20 K. The price of shining stars is chilly space between stars, by conservation of total energy. 

     

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 How are stars formed?

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.

True?