So let us begin with this calculation of John Baez, which gets the entropy wrong - it would decrease during star formation, i.e. the gravitational collapse of the matter which makes up the star. What the formula leaves out is the entropy of the outgoing radiation, but I would like to stay in a simple Newtonian model with classical point particles only.
In this case the "missing entropy" must come from the particles with velocities above the escape velocity of the star, which leave the collapsing cluster of particles. (The positions and velocities of the particles are actually not bounded, violating an assumption of this calculation, as he noted at the end of his page.) In other words, the formula John uses can only be an approximation, there is actually no decreasing volume V which encloses all particles and if one defines V considering a sphere which encloses all particles which cannot escape, the number N he uses would not be constant. So how does one really calculate the entropy?
A simpler question would be: If the initial number of particles was N, contained in a volume V, what fraction will escape within a small time interval dt? The Maxwell distribution would tell us the number of particles with velocities above the escape velocity and approximately 1/2 of them would escape, if they are within a distance dt*v from the surface ...
But all this seems a bit unsatisfactory; does anybody have the reference to a full calculation of this problem or do I have to run a computer simulation?
added later: A simple simulation of N=1000 particles, initially contained within a sphere of radius 1 and with zero initial velocity, suggests that after long enough time almost all particles escape to a location outside the initial sphere, due to the simulated gravitational interaction. Of course, my program (quickly cobbled together) could be wrong or inaccurate. The chart below shows the fraction of escaped particles on the y axis after so many time steps on the x axis (I have no explanation for the kink after 500 time steps).
The distribution of particles (projected onto a 2d plane) after hundred time steps ...
... one can see a "halo" of escaping particles surrounding the majority of particles in the collapsing star.
9 comments:
Below is a link to an article entitled "Trends of stellar entropy along stellar evolution." The authors try to calculate how stellar entropy evolves in various classes of stars, but I don't think it clearly answers the question you're asking.
https://arxiv.org/pdf/1509.05262.pdf
Thank you for this link!
I think CIP's question may have started something, because now I really want to understand how star formation (and/or galaxy formation) actually happens.
Btw this is actually an important area of ongoing research ...
I suspect that your calculation leaves out some stuff, because globular star clusters are pretty stable. Also I'm interested in your simulation. In particular is it symplectic? However it should be true that it eventually evaporates a lot of stars.
If other forces than gravity are involved, it's an important point that there is no star formation without cooling, which practically means radiation.
I'm not sure, but the kink may have something to do with virialization. You might try starting your particles with random velocities and kinetic energies just slightly less than the average potential energy. Also, how does the time step compare with mean cluster crossing time?
Lee - Thanks for your link. It's just as I feared - the formation of the black hole represents an abrupt and catastrophic increase in entropy that doesn't seem to be correlated with any interesting thermodynamics outside the black hole. That seems strange in the light of the fact that we know we can extract free energy from the BH by lowering stuff into it.
I wonder if the author's calculations take into account the fact that baryons in a neutron star are about 20% lighter than their free counterparts?
CIP,
good questions.
>> globular clusters
Obviously, my simple simulation is not realistic (zero initial velocity), but it is one of the things I am curious about - why are galaxies actually quite stable?
>> symplectic
It was simple 1st order and I need to improve that ...
>> star formation
In a realistic simulation, at some point nuclear reactions set in, so obviously none of that
>> time step .. cluster crossing
I suspect, as you seem to do, that this is one explanation, thermalization/virialization or a bug is the other option.
I will keep you guys posted ...
We did a slightly smaller (500 particle) simulation in my stellar dynamics class. I got evaporation but not on the scale that you are seeing it. I think I did 100 crossing times and maybe 10000 time steps.
I think the mass of my particles is too high ( = the initial size of my cluster too small) to be a realistic simulation of e.g. a galaxy cluster.
My "star" is small and very heavy and quickly heats up and therefore creates a lot of "radiation" (i.e. escaping particles) very fast.
While this is unrealistic (compared to real stars or galaxies) it demonstrates the point I was trying to make very well 8-)
CIP,
>> as I feared - the formation of the black hole represents an abrupt and catastrophic increase in entropy that doesn't seem to be correlated with any interesting thermodynamics
That's the way it looks to me too. When you made that comment on your blog I almost posted the link above, but decided it best not to interfere in your conversation with Wolfgang.
>> That seems strange in the light of the fact that we know we can extract free energy from the BH by lowering stuff into it.
I'm not quite sure, but I think you must be thinking of the Penrose process. I guess the thing that kind of bothers me about your reasoning is that that same process could be used around any spinning massive object to gain energy from the object's angular momentum. The thing that is different about the process in black holes is that there are geodesics confined within the ergoregion with the property that particles following them have potential energies that are so negative that they outweigh in magnitude their rest mass and kinetic energies combined. But that seems to me to be a special property of spinning black holes which again indicates a step change.
Post a Comment