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Pondering the Fedderath particle formation requirements

Particle creation requirements

violates the truelove criterion
is on the highest level of refinement,
surrounding flow is converging in each direction,
has a central gravitational potential minimum,
is Jeans-unstable,
is bound, and
is not within racc of an existing sink particle

and 2. are the same as Krumholz

seems reasonable to avoid particle formation in shearing layers

seems like it would favor particle creation in the center of the grid…

Consider a 1D system with two sink 'planes'. The Greens function is just \frac{|x|}{2} and let's consider three points at x=-1,0, & 1. The potential will then be \frac{|x+1|+|x|+|x-1|}{2}

As you can see only the central point would form a sink particle even though there are 3 equal masses. The paper is not entirely clear about the local gravitational potential and could possibly be referring to a gas potential calculated using only the local mass as opposed to the local gas potential of the entire domain.

Indeed, for check # 5 it calculates the self-energy of the control volume and does calculate a potential using only the local contributions… So they probably are referring to this same local-mass potential with regards to check 4. However, any cell surrounded by a uniform region that violates the true love criterion will have a minimum for the potential at the center of the region because the control volume will be centered on that cell. One could use the periodic approach where the mean density of the region is first subtracted off - in which case the potential would be zero everywhere… and would require the density to be the largest at the center… However, a better check - and one that wouldn't be dependent on the centering of the control volume might just be to make sure that the density is the highest in that cell compared to the surrounding cells in the control volume. This will keep particles from forming in very close proximity. We should update the code with one of these two approaches.

This requires that the surrounding control volume be as dense as the central cell - otherwise it will violate the true-love criterion without forming a particle… we have…

E_{grav}=-\frac{3GM^2}{5R_c}=\frac{16}{15}\pi^2G\rho^2R_c^5

E_{th}=\frac{3}{2}M\sigma^2=\frac{6\pi}{5}\rho R_c^3 c_s^2 where \rho \sigma^2=P

Jeans density criterion is that \rho > \frac{\sqrt{P \pi \gamma/G}}{R_c} which is equivalent to \rho > \frac{\sqrt{\rho c_s^2 \pi/G}}{R_c} \rightarrow R_c > \sqrt{\frac{\pi c_s^2}{\rho G}} = \lambda_J

Jeans energy criterion is that

2E_{th} < |E_{grav}| \rightarrow \frac{12\pi}{5}\rho R_c^3 c_s^2 < \frac{16}{15}\pi^2G\rho^2R_c^5 \rightarrow \sqrt{\frac{9}{4\pi^2}\frac{\pi c_s^2}{\rho G}} < R_c or that \rightarrow R_c > .4775 \lambda_J or in other words, if a region violates the density criterion, we would expect \frac{2E_th}{E_{grav}} = .4775^2

I've checked in the code that when the density is uniform and just above the jeans criterion that the ratio is .48^2 which is consistent with the various assumptions and discretizations… at least when gamma=5/3. When gamma = 1 this changes slightly to closer to .7^2.

So if there are only a few cells inside of the control volume near the jeans density, it may not satisfy the energy constraints initially… But it seems unlikely that it would artificially fragment without having enough mass present to satisfy this criterion.

After criterion 4, criterion 6 tends to be the most restrictive - as a collapsing region will not form a particle until enough kinetic energy has been converted into thermal energy at the center, and then cooled.

astrobear

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Pondering the Fedderath particle formation requirements

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